professor stewart's casebook of mathematical mysteries stewart's casebook of... · the...
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ProfessorStewart’sCasebookofMathematicalMysteries
ProfessorIanStewartisknownthroughouttheworldformakingmathematicspopular.HereceivedtheRoyalSociety’sFaradayMedalforfurtheringthepublicunderstandingofsciencein1995,theIMAGoldMedalin2000,theAAASPublicUnderstandingofScienceandTechnologyAwardin2001andtheLMS/IMAZeemanMedalin2008.HewaselectedaFellowoftheRoyalSocietyin2001.HeisEmeritusProfessorofMathematicsattheUniversityofWarwick,wherehedivideshistimebetweenresearchintononlineardynamicsandfurtheringpublicawarenessofmathematics.Hismanypopularsciencebooksinclude(withTerryPratchettandJackCohen)TheScienceofDiscworldItoIV,TheMathematicsofLife,17EquationsthatChangedtheWorldandTheGreatMathematicalProblems.Hisapp,ProfessorStewart’sIncredibleNumbers,waspublishedjointlybyProfileandTouchPressinMarch2014.
BytheSameAuthorConceptsOfModernMathematicsGame,Set,AndMathDoesGodPlayDice?AnotherFineMathYou’veGotMeIntoFearfulSymmetryNature’sNumbersFromHereToInfinityTheMagicalMazeLife’sOtherSecretFlatterlandWhatShapeIsASnowflake?TheAnnotatedFlatlandMathHysteriaTheMayorOfUglyville’sDilemmaHowToCutACakeLettersToAYoungMathematicianTamingTheInfinite(AlternativeTitle:TheStoryOfMathematics)WhyBeautyIsTruthCowsInTheMazeMathematicsOfLifeProfessorStewart’sCabinetOfMathematicalCuriositiesProfessorStewart’sHoardOfMathematicalTreasuresSeventeenEquationsThatChangedTheWorld(AlternativeTitle:InPursuitOfTheUnknown)TheGreatMathematicalProblems(AlternativeTitle:VisionsOfInfinity)Symmetry:AVeryShortIntroductionJackOfAllTrades(ScienceFictioneBook)
withJackCohenTheCollapseOfChaosEvolvingTheAlien(AlternativeTitle:WhatDoesAMartianLookLike?)FigmentsOfRealityWheelers(ScienceFiction)Heaven(ScienceFiction)
TheScienceOfDiscworldSeries(WithTerryPratchett&JackCohen)TheScienceOfDiscworldTheScienceOfDiscworldII:TheGlobeTheScienceOfDiscworldIII:Darwin’sWatchTheScienceOfDiscworldIV:JudgementDay
iPadappIncredibleNumbers
ProfessorStewart’sCasebookofMathematicalMysteries
IanStewart
Firstpublishedin2014byPROFILEBOOKSLTD3AExmouthHousePineStreetLondonEC1R0JHwww.profilebooks.com
Copyright©JoatEnterprises2014
10987654321
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eISBN9781847654328
CONTENTS
Acknowledgements
IntroducingSoamesandWatsupNoteonUnitsTheScandaloftheStolenSovereignNumberCuriosityTrackPositionSoamesMeetsWatsupGeomagicSquaresWhatShapeisanOrangePeel?HowtoWintheLottery?TheGreenSocksCaperIncidentConsecutiveCubesAdonisAsteroidMousterianTwoSquareQuickiesCaughtClean-HandedTheAdventureoftheCardboardBoxesTheRATSSequenceBirthdaysAreGoodforYouMathematicalDatesTheHoundoftheBasketballsDigitalCubesNarcissisticNumbersPiphilology,Piems,andPilishClueless!ABriefHistoryofSudokuHexakosioihexekontahexaphobiaOnce,Twice,ThriceConservationofLuckTheCaseoftheFace-DownAcesConfusedParents
JigsawParadoxTheCatflapofFearPancakeNumbersTheSoupPlateTrickMathematicalHaikuTheCaseoftheCrypticCartwheelTwobyTwoTheV-shapedGooseMysteryEelishMnemonicsAmazingSquaresTheThirty-SevenMysteryAverageSpeedFourCluelessPseudokuSumsofCubesThePuzzleofthePurloinedPapersMasterofAllHeSurveysAnotherNumberCuriosityTheOpaqueSquareProblemOpaquePolygonsandCirclesπr2?TheSignofOneProgressonPrimeGapsTheOddGoldbachConjecturePrimeNumberMysteriesTheOptimalPyramidTheSignofOne:PartTwoInitialConfusionEuclid’sDoodleEuclideanEfficiency123456789TimesXTheSignofOne:PartThreeTaxicabNumbersTheWaveofTranslationRiddleoftheSandsEskimoπTheSignofOne:PartFour—ConcludedSeriouslyDerangedTossingaFairCoinIsn’tFair
PlayingPokerbyPostEliminatingtheImpossibleMusselPowerProofThattheWorldisRound123456789TimesXContinuedThePriceofFameTheRiddleoftheGoldenRhombusAPowerfulArithmeticSequenceWhyDoGuinnessBubblesGoDownwards?RandomHarmonicSeriesTheDogsThatFightintheParkHowTallisThatTree?WhyDoMyFriendsHaveMoreFriendsThanIDo?Isn’tStatisticsWonderful?TheAdventureoftheSixGuestsHowtoWriteVeryBigNumbersGraham’sNumberCan’tWrapMyHeadAroundItTheAffairoftheAbove-AverageDriverTheMousetrapCubeSierpińskiNumbersJamesJosephWho?TheBafflehamBurglaryTheQuadrillionthDigitofπIsπNormal?AMathematician,aStatistician,andanEngineer…LakesofWadaFermat’sLastLimerickMalfatti’sMistakeSquareLeftoversCoinTossingoverthePhoneHowtoStopUnwantedEchoesTheEnigmaoftheVersatileTileTheThrackleConjectureBargainwiththeDevilATilingThatIsNotPeriodicTheTwoColourTheoremTheFourColourTheoreminSpace
ComicalCalculusTheErdősDiscrepancyProblemTheGreekIntegratorSumsofFourCubesWhytheLeopardGotItsSpotsPolygonsForeverTopSecretTheAdventureoftheRowingMenTheFifteenPuzzleTheTrickySixPuzzleAsDifficultasABCRingsofRegularSolidsTheSquarePegProblemTheImpossibleRouteTheFinalProblemTheReturnTheFinalSolution
TheMysteriesDemystified
Acknowledgements
page13Leftandcentrefigures:LaurentBartholdiandAndréHenriques.OrangepeelsandFresnelintegrals,MathematicalIntelligencer34No.4(2012)1–3.
page13Rightfigure:LucDevroye.page27Boxpuzzleconcept:MoloyDe.page40ExtractfromNotaWake:MikeKeith.page62Haiku:DanielMathews,JonathanAlperin,JonathanRosenberg.page67Figure:http://getyournotes.blogspot.co.uk/2011/08/why-do-some-birds-
fly-in-v-formations.htmlpage71Amazingsquares:devisedbyMoloyDeandNirmalyaChattopadhyay.page72TheThirty-SevenMystery:basedonobservationsbyStephenGledhill.page74Cluelesspseudoku:GerardButters,FrederickHenle,JamesHenleand
ColleenMcGaughey.Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.
page96Figure:EricW.Weisstein,‘Brocard’sConjecture,’fromMathWorld—AWolframWebResource:http://mathworld.wolfram.com/BrocardsConjecture.html
page101RightFigure:StevenSnape.page109Figure:CourtesyoftheUW-MadisonArchives.page123Leftfigure:[GeorgeSteinmetz,courtesyofAnastasiaPhoto].page123Rightfigure:NASA,HiRISEonMarsReconnaissanceOrbiter.page124Rightfigure:RudiPodgornik.page125Figure:VeitSchwa¨mmleandHansJ.Herrmann.Solitarywave
behaviourofsanddunes,Nature426(2003)619–620.page131Figure:PersiDiaconis,SusanHolmesandRichardMontgomery,
Dynamicalbiasinthecointoss,SIAMReview49(2007)211–223.page205Figures:JoshuaSocolarandJoanTaylor.Anaperiodichexagonaltile,JournalofCombinatorialTheorySeriesA118(2011)2207–2231;
http://link.springer.com/article/10.1007%2Fs00283-011-9255-ypages235–7Figures:MichaelElgersmaandStanWagon,ClosingaPlatonic
gap,TheMathematicalIntelligencer(2014)toappear.
ThefollowingfiguresarereproducedundertheCreativeCommonsAttribution3.0Unportedlicenseandcreditedasrequestedonthedescriptionpage:page95Krishnavedala.page101(left)RicardoLiberato.page102Tekisch.page139AndreasTrepte,www.photo-natur.de.page180Braindrain0000.page183LutzL.page296WaltersArtMuseum,Baltimore.
IntroducingSoamesandWatsup
ProfessorStewart’sCabinetofMathematicalCuriositiesappearedin2008,justbeforeChristmas.Readersseemedtolikeitsrandommixtureofquirkymathematicaltricks,games,weirdbiographies,snippetsofstrangeinformation,solvedandunsolvedproblems,oddfactoids,andtheoccasionallongerandmoreseriouspieceontopicssuchasfractals,topology,andFermat’sLastTheorem.Soin2009itwasfollowedbyProfessorStewart’sHoardofMathematicalTreasures,whichcontinuedinthesameveinwithanintermittentpiratetheme.
Theysaythatthreeisagoodnumberforatrilogy.ThelateDouglasAdamsofTheHitchhiker’sGuidetotheGalaxyfamedideventuallydecidethatfourwasbetterandfivebetterstill,butthreesoundslikeagoodplacetostart.So,afteragapoffiveyears,hereisProfessorStewart’sCasebookofMathematicalMysteries.Thistime,however,there’sanewtwist.Theshortquirkyitems,suchasHexakosioihexekontahexaphobia,theThrackleConjecture,WhatShapeisanOrangePeel?,theRATSSequence,andEuclid’sDoodle,arestillthere.Soaremoresubstantialarticlesaboutsolvedandunsolvedproblems:PancakeNumbers,theGoldbachConjecture,theErdősDiscrepancyProblem,theSquarePegConjecture,andtheABCConjecture.Soarethejokes,poems,andanecdotes.Nottomentionunusualapplicationsofmathematicstoflyinggeese,clumpsofmussels,spottyleopards,andbubblesinGuinness.ButthesemiscellaneaarenowinterspersedwithaseriesofnarrativeepisodesfeaturingaVictoriandetectiveandhismedicalsidekick—
Iknowwhatyou’rethinking.However,IdevelopedtheideaayearorsobeforeBenedictCumberbatchandMartinFreeman’sspectacularlysuccessfulmoderntakeonSirArthurConanDoyle’smuch-lovedcharactershitthesmallscreen.(Trustme.)Moretothepoint,it’snotthatpair.NotevenasportrayedinSirArthur’soriginalstories.Yes,myguysliveintheoriginaltimeperiod,butacrosstheroadatnumber222B.Fromthere,theycastenviouseyesonthestreamofrichclientsenteringthepremisesofthemorefamousduo.Andfromtimetotimeacasecomesupthattheirillustriousneighbourshaveshunnedorfailedtosolve:sucharcanemysteriesastheSignofOne,theDogsthatFightinthePark,theCatflapofFear,andtheGreekIntegrator.ThenHemlockSoames
andDrJohnWatsupputtheirbrainsingear,showtheirtruecoloursandtheirstrengthofcharacter,andtriumphoveradversityandlackofmarketpresence.
Thesearemathematicalmysteries,youappreciate.Theirsolutionsdemandaninterestinmathematicsandanabilitytothinkclearly,attributesinwhichSoamesandWatsuparebynomeansdeficient.Thesepassagesaresignalledbythesymbol .AlongthewaywelearnofWatsup’spriormilitarycareerinAl-JebraistanandSoames’sbattleswithhisarch-enemyProfessorMogiarty,inevitablyleadingtothefinalfatalconfrontationatoptheSchtickelbachFalls.Andthen—
ItisfortunatethatDrWatsuprecordedsomanyoftheirjointinvestigationsinhismemoirsandunpublishednotes.IamgratefultohisdescendantsUnderwoodandVerityWatsupforpermittingmeunprecedentedaccesstofamilydocuments,andforgenerouslygrantingmepermissiontoincludeextractshere.
Coventry,March2014
NoteonUnits
InSoamesandWatsup’sera,thestandardunitsofmeasurementinBritainwereimperial,notmetricastheymostlyaretoday,andthecurrencywasnotdecimal.Americanreaderswillhavenoproblemswithimperialunits;admittedly,thegallonisdifferentoneithersideoftheAtlantic,butthatunitofmeasurementdoesn’tappearanyway.ToavoidinconsistenciesI’veusedunitsappropriatetotheVictorianera,evenfortopicsthatarenotpartoftheSoames/Watsupcanon,exceptwhennarrativeimperativedemandsmetric.
Here’saquickguidetotherelevantunitswithmetric/decimalequivalents.Mostofthetimetheactualunitdoesn’tmatter:youcouldleavethenumbers
unchanged,butcrossout‘inch’or‘yard’andreplaceitbyanunspecified‘unit’.Orchoosewhicheverseemsconvenient(metreforyard,forexample).Lengths
1foot(ft)=12inches(in) 304·8mm1yard(yd)=3feet 0·9144m1mile(mi)=1760yards=5040feet 1·609km1league(lea)=3miles 4·827kmWeights
1pound(lb)=16ounces(oz) 453·6g1stone(st)=14pounds 6·35kg1hundredweight(cwt)=8stone=112pounds 50·8kg1ton(t)=20hundredweights=2240pounds 1·016tonnesMoney
1shilling(s)=12pence(d)[singular:penny] 5newpence1pound(£)=20shillings=240pence1sovereign=1pound(coin)
1guinea=£1.1s. £1.051crown=5s. 25newpenceThruppenybit=colloquialtermforathreepencecoin.
TheScandaloftheStolenSovereignTheprivatedetectivetookhiswalletfromhispocket,ascertainedthatitwasstillempty,andsighed.Standingatthewindowofhislodgingsatnumber222Bhestaredmoroselyacrossthestreet.ThestrainsofanIrishair,expertlyplayedonaStradivarius,werejustdiscernibleabovetheclatterofpassingcarriages.Really,themanwasinsufferable!Soamesstaredatthestreamofpeopleenteringtheportalsofhisfamouscompetitor.Mostwerewealthymembersoftheupperclasses.Thosethatappearednottobewealthymembersoftheupperclasseswere,withfewexceptions,representingwealthymembersoftheupperclasses.
Criminalsjustweren’tcommittingthekindofcrimethataffectedthesortofpeoplewhowouldengagetheservicesofHemlockSoames.
Forthepasttwoweeks,Soameshadwatchedwithenviouseyesasclientafterclientwasusheredintothepresenceofthepersontheybelievedtobethegreatestdetectiveintheworld.Or,atleast,inLondon,whichforVictorianEnglandamountedtothesamething.Meanwhilehisowndoorbellremainedmute,thebillspiledup,andMrsSoapsudswasthreateningeviction.
Hehadonlyonecaseonthebooks.LordHumphshaw-Smattering,owneroftheGlitzHotel,believedthatoneofhiswaitershadstolenagoldsovereign:value,onepoundsterling.Tobefair,Soamescoulddowithasovereignhimselfrightnow.Butitwashardlythestufftoattractthesensationalistyellowpress,uponwhom,deplorableasitmightbe,hisfuturedepended.
Soamesstudiedhiscasenotes.Threefriends,Armstrong,Bennett,andCunningham,hadpartakenofdinneratthehotel,andhadbeenpresentedwithabillfor£30.EachhadgiventhewaiterManueltengoldsovereigns.Butthenthemaîtred’noticedthattherehadbeenanerror,andthebillwasactually£25.Hegavethewaiterfivesovereignstoreturntothemen.Since£5wasn’tdivisibleby3,Manuelsuggestedthathemightkeeptwoofthecoinsasatipandgivethembackonesovereigneach,hintingthattheywerefortunatetohaveanyoftheoverpaymentreturned.
Thecustomersagreed,andallwaswelluntilthemaîtred’noticedanarithmeticaldiscrepancy.Nowthemenhadeachpaid£9,atotalof£27.Manuelhadafurther£2,makingatotalof£29.
Onepoundwasmissing.Humphshaw-SmatteringwasconvincedthatManuelhadstolenit.Although
theevidencewascircumstantial,Soamesknewthatthewaiter’slivelihood
dependedonresolvingthemystery.IfManuelweretobedismissedwithabadreference,hewouldnevergetanotherjob.
Wheredidthemissingpoundgo?
Seepage249fortheanswer.
NumberCuriosity*
Indetectivework,itisvitaltobeabletospotapattern.Soames’sunpublishedanduntitledmonographcontainingtwothousandandforty-oneinstructiveexamplesofpatternsincludesthefollowing.Workout
11×9111×909111×90909111×9090909111×9090909091
Soameswouldhaveusedpenandpaper,andmodernreadersmaydolikewiseiftheycanrememberhow.Acalculatorisalwaysanoption,buttheytendtorunoutofdigits.Thepatterncontinuesindefinitely:thiscan’tbeprovedusingacalculator,butitcanbededucedfromtheold-fashionedmethod.So,withoutdoinganyfurthercalculations,whatis
11×9090909090909091?
Aharderquestionis:whydoesitwork?
Seepage250fortheanswers.
Footnote
*Manyitemsinthiscompilationthatdonotreferdirectlytocriminalcasesareextractedfromhandwrittennotes,someofwhosecontentshavebeencollectedandpublished,withSoames’spermission,asDoctorWatsup’sVaultofForensicAnomalies,andwillbereproducedwithoutfurthernotification.Someareoflaterdate,addedbyWatsup’sliteraryexecutors,andtheassiduousreaderwillinstantlyidentifysuchanachronisms.
TrackPositionLionelPenroseinventedavariationontraditionalmazes:railwaymazes.Thesehavejunctionslikethoseonrailwaytracks,andyouhavetotakearoutethroughthemthatatraincouldfollow,onewithnosharpturns.Theyareagoodwaytocramacomplicatedmazeintoasmallspace.
Allowedandforbiddenroutesatjunctions
Hisson,themathematicianRogerPenrose,tooktheideafurther.OneofhismazesiscarvedinstoneontheLuppittMillenniumBench,inDevon,England.Thatone’sabitdifficult,sohere’sasimplerexampleforyoutotackle.
ThemapoverleafshowstherailnetworkofTardyTrains.The10.33trainstartsatstationSandmustfinishatstationF.Thetraincannotreversedirectionbyslowingdownandthengoingbackwards,butitcantravelalongalineineitherdirectionifthetrackloopsbackonitself.Atpoints,wheretwobranchesjoin,thetrainmaytakeanysmoothpath.Whatroutedoesthetraintake?
Themaze
Seepage252fortheanswer,andfurtherinformationincludingtheLuppittMillenniumMaze.
SoamesMeetsWatsupAfinedrizzle,ofthekindthatlooksinnocuousbutquicklysoaksyoutotheskin,wasfallingonthegoodcitizensofLondon,andonthebad,astheyscurriedalongBakerStreetonerrandsadmirableornefarious,dodgingthepuddles.Thenot-so-famousdetectivewasinhishabitualpositionatthewindow,staringhopelesslyintothegloom,grumblingtohimselfabouthisdirefinances,andfeelingdepressed.HisincisivesolutiontotheScandaloftheStolenSovereignhadbroughtinenoughtogetMrsSoapsudstemporarilyoffhisback,butnowthattheemotionalrushofsuccesshadsubsided,hefeltlonelyandunappreciated.
Perhapsheneededalike-mindedcompanion?Onewhocouldsharethedailycut-and-thrustofhispersonalvendettaagainstcrime,andtheintellectualchallengeofunravellingthecluesthatitsperpetratorsscatteredsocarelesslyacrossthelandscape?Butwherecouldhefindsuchaperson?Hehadnoideawheretostart.
Hisblackmoodwasinterruptedbytheappearanceofasturdyfigurestridingpurposefullytowardsthepremisesopposite.Instinctively,Soamesjudgedhimtobeamedicalman,recentlyretiredfromthearmy.Well-dressed,well-heeled:yetanotherwealthyclientforthatoverratedjackassHol—
Butno!Thefigureinspectedthehousenumber,shookhishead,andspunonhisheels.Ashecrossedtheroad,narrowlydodgingahansomcab,thebrimofhishatconcealedhisface,buthisbodylanguageshoweddetermination,perhapsvergingondesperation.Observingthemanmoreclosely,nowthathisinteresthadbeenpiqued,Soamesrealisedthathiscoatwasnotnew,ashehadfirstthought.Ithadbeenexpertlyrepaired…inOldComptonStreet,bythelookofthestitching.OnaThursday,whentheheadseamstressestookahalf-dayoff.Downatheel,notwell-heeled,hecorrectedhisinitialimpression,asthemandisappearedfromview,apparentlyheadingforthedoorwaybelow.
Apause:thenthebellrang.Soameswaited.Aknockatthedoorannouncedhislong-sufferinglandlady
MrsSoapsuds,cladinoneofherhabitualfloralprintdressesandwearingalargepinafore.“Agentlemantoseeyou,MrSoames,”shesniffed.“ShallIshowhimup?”
Soamesnodded,andMrsSoapsudsslouchedoffdownthestairs.Aminute
latersheknockedagain,andthemedicalgentlemanentered.Soameswavedathertoshutthedoorandreturntohercustomaryplacebehindthenetcurtainsinhersitting-roomonthegroundfloor,whichshedidwithevidentreluctance.
Thegentlemanlistenedforamoment,andsuddenlytuggedthedooropen,steppingbacktoallowMrsSoapsudstofallsidewaystothefloor.
“The—uh—mat.Neededdusting,”sheexplained,pickingherselfup.Soamessilentlynotedthathislandladyalsoneededdusting,gaveherathinsmile,andwavedheraway.Oncemorethedoorclosed.
“Mycard,”themansaid.Soamesplacedthevisitingcardfacedown,unread,andstudiedthenew
arrivalfromheadtotoe.Afterafewsecondshesaid,“Notmuchofnotetoidentifyyou.”
“Pardon?”“Excepttheobvious,ofcourse.YouhavebeeninAl-Jebraistanforthelast
fouryears,servingasasurgeonwiththeRoyalSixthDragoons.YounarrowlyescapedaseriouswoundattheBattleofQ’drat.Yourperiodofserviceendedsoonafter,andyoudecided—aftersomesoul-searching—toreturntoEngland,whichyoudidearlythisyear.”Soamespeeredmoreclosely,andadded,“Youkeepfourcats.”
Astheman’sjawdropped,Soamesturnedoverthecard.“DrJohnWatsup,”heread.“Surgeon,RoyalSixthDragoons,retired.”Hisfaceshowednoemotionatthisconfirmationofhisdeductions,forithadbeeninevitable.“Pleasesitdown,sir,andtellmeofthecrimethathasbeencommittedagainstyou.Icanassureyouthat—”
Watsuplaughed,afriendlychuckle.“MrSoames,Iamdelightedtohavemetyouatlast,foryourfamehasspreadfarandwide.YourdeductionsaboutmypersonprovethatyoufullydeservetheacclaimthatIhaveencountered.Yourmodestyatthefeatbecomesyou.ButIdonotcomeprimarilyasaprospectiveclient.Rather,Iamseekingapositioninyouremploy.Medicinenolongerappealstome—norwouldittoyouifyouhadseenthesightsIhavebeenforcedtoendureatthebattlefront.ButIamamanofaction,Icontinuetocraveexcitement,Istillhavemyservicerevolver,and…bytheway,howdidyoudothat?”
Soames,ignoringagrowingfeelingthathewasbeingmistakenfortheinhabitantofnumber221B,satdownfacingWatsup.“Byyourbearing,sir,Ihadyoumarkedasamilitarymanbeforeyoucrossedtheroad.Myeyesightispreternaturallykeen,andyouhavethehandsofasurgeon,strongyetlackingtheingrainedstainsofmanuallabour.LastDecembertheTimesreportedthatthefour-yearcampaigninAl-JebraistanwascomingtoacloseandtheRoyalSixth
DragoonswerereturningtoEnglandafterfightingadecisivebutcostlybattleatQ’drat.Youarewearingtheappropriateregimentalboots,andthewear-patternsonthemshowyouhavebeenbackinEnglandforsometime.Youhaveaslightscaralongyourjawbone,almosthealed,whichwasobviouslycausedbyamusket-ballofnon-Europeandesign—IhavewrittenabriefmonographonfirearminjuriesintheFarEast,Imustreadittoyousometime.Youareamanofaction,asevincedbythewayyouhandledMrsSoapsuds’ssnooping,soyouwouldnothaveretiredfrommilitaryservicevoluntarily.IfyouhadbeengivenadishonourabledischargeIwouldhaveseenitreportedinthescandalsheets,butnothingofthekindhasbeenpublishedrecently.Yourcoatbearsfourdifferenttypesofcathair—notjustfourcolours,whichmightindicateasingletabby,butdifferentlengthsandtextures…Iwillspareyoualistoftheirbreeds.”
“Astonishing!”“Tobecandid,Imustalsoadmitthatyourfaceisfamiliar.Iamsurethat
somewhere—ah,yes!Ihaveit!Asmallarticleinlastweek’sChronicle,withaphotograph…DrJohnWatsup,originatorofthewell-knownphrase‘Watsup,doc?’Yourfameexceedsmyown,Doctor.”
“Youaretookind,MrSoames.”“No,merelyrealistic.Butifwearetoworktogether,youmustconvinceme
thatyoucanthinkaswellasact.Letussee.”AndSoameswrotethedigits
49
onthebackofanenvelope.“Iwantyoutoinsertonestandardarithmeticalsymbol,toproduceawholenumberbetween1and9.”
Watsuppursedhislipsinconcentration.“Aplussign…no,13istoolarge.Aminus—notheresultisnegative.Neithermultiplicationnordivisionwilldo.Ofcourse!Asquareroot!Oh,no: ,againtoolarge.”Hescratchedhishead.“Iamstumped.Itisimpossible.”
“Iassureyouthereisasolution.”Thesilencewasbrokenonlybythetickingofaclockonthemantelpiece.
Suddenly,Watsup’sfacelitup.“Ihaveit!”Hepickeduptheenvelope,addedasinglesymbol,andhandedittoSoames.
“Youpassthefirsttest,Doctor.”
WhatdidWatsupwrite?Seepage252fortheanswer.
GeomagicSquaresAmagicsquareismadefromnumbers,whichgivethesametotalalonganyrow,column,ordiagonal.LeeSallowshasinventedageometricanalogue,thegeomagicsquare.Thisisasquarearrayofshapes,suchthattheshapesinanyrow,column,ordiagonalfittogetherlikeajigsawtomakethesameoverallshape.Thepiecescanberotatedorreflectedifnecessary.Theleft-handfigureshowshowthisgoes;theright-handoneisapuzzleforyoutosolve.Seepage253fortheanswer.
Sallowshasinventedmanyothergeomagicsquares,alongwithgeneralisationssuchasageomagictriangle.SeeTheMathematicalIntelligencer33No.4(2011)25–31andhiswebsite
http://www.GeomagicSquares.com/
TwoofLeeSallows’sgeomagicsquares.Followarow,column,ordiagonaltofindtheassembledjigsawusingthecorrespondingpieces.Left:Acompletedexample.Right:Yourtaskisfindtheassembledjigsaws
forallrows,columns,anddiagonals.
WhatShapeisanOrangePeel?Therearemanywaystopeelanorange.Someofusbreakbitsoff.Sometrytoremovetheentirepeelinasingleirregularlump.Thisusuallyproducesseverallumpsandalotofjuice.Othersaremoresystematic,peelingorangesverycarefullywithaknife,makingaspiralcutfromthetopoftheorangedowntothebottom.Ipersonallypreferamessonthetableandaquickorange,butthereyougo.
Left:Peelinganorangewithaknife.Middle:Thepeellaidoutflat.Right:Cornuspiral.
In2012LaurentBartholdiandAndréHenriqueswonderedwhatshapethepeelwouldformifitwerelaidoutflat.Usingathinknife,andbeingcarefultokeepthepeelthesamewidthallthewayalong,theyobtainedabeautifuldoublespiral.Itremindedthemofafamousmathematicaldoublespiral,variouslyknownasaCornuspiral,Eulerspiral,clothoid,orspiro.
Thiscurvehasbeenknownsince1744,whenEulerdiscoveredoneofitsbasicproperties.Itscurvature(1/rwhereristheradiusofthebest-fittingcircle)atanygivenpointisproportionaltothedistancealongthecurvetothatpoint,startingfromthemiddle.Thefurtheralongthecurveyougo,themoretightlyitcurves,whichiswhythespiralregionsbecomeevermorecloselywound.ThephysicistMarieAlfredCornucameacrossthesamecurveinthephysicsoflight,whenitdiffractsatastraightedge.Railwayengineershaveusedthecurvetoprovideasmoothtransitionbetweenastraightpieceoftrackandacircularone.
BartholdiandHenriquesprovedthattheresemblancebetweenorangepeelandthisshapeisnoaccident.Theywrotedownanequationdescribingtheorange-peelshapeforstripsofanyfixedwidth,andprovedthatwhenthewidthbecomesassmallasweplease,theshaperesemblestheCornuspiralevermoreclosely.Theyremarkedthatthisspiral“hashadmanydiscoveriesacrosshistory;oursoccurredoverbreakfast.”
Seepage253forfurtherinformation.
HowtoWintheLottery?Pleasenotethequestionmark.
TowinthejackpotintheUKNationalLottery(unimaginativelyrebrandedas‘Lotto’)youhavetochoosesixnumbersfromtherange1–49thatmatchthosedrawnonthedaybyaLottomachine.Thereareotherwaystowinsmallerprizes,butlet’ssticktothatone.Theballsaredrawninarandomorder,buttheresultsarethenconvertedtonumericalordertomakeiteasiertofindoutwhetheryou’vewon.Soadrawlike
13158484736
isreorderedas
81315364748
andinthiscasethesmallestnumberis8,thesecondsmallestis13,andsoon.Probabilitytheorytellsusthatwhenallnumbersareequallylikely,asthey
shouldbe,thenwithinachosensetofsix:
Themostlikelysmallestnumberis1.Themostlikelysecondsmallestnumberis10.Themostlikelythirdsmallestnumberis20.Themostlikelyfourthsmallestnumberis30.Themostlikelyfifthsmallestnumberis40.Themostlikelylargestnumberis49.
Thesestatementsarecorrect.Thefirstistruebecauseif1turnsup,thenitmustbethesmallest,nomatterwhatelsehappens.That’snotthecasefor2,however,becausethereisasmallchancethat1willturnupandsneakunderneathit.Thismakesitslightlylesslikelythat2willbethesmallestafterallsixballsaredrawn.
OK,thosearemathematicalfacts.Soitlooksasthoughyoucanimproveyourchanceofwinningifyoupick
11020304049becauseeachchoiceisthemostlikelynumbertooccurinthatposition.
Isthiscorrect?Seepage253fortheanswer.
TheGreenSocksCaperIncident“Youhavepassedthefirsttest,Doctor.Buttherealtestwillbetoobservehowyouhandleacriminalinvestigation.”
“Iamready,MrSoames.Whenshallwebegin?”“Notimelikethepresent.”“Iagree,wearebothmenofaction.Whichcaseshallitbe?”“Yourown.”“But—”“AmImistakeninthinkingthatalthoughyourreasonforcomingherewasto
seekemployment,youhavealsobeenthevictimofacrime?”“No,buthow—”“Whenyoufirstenteredthisroom,Iwasinstinctivelyawarethatyouwere
seekingmyassistance.Youwereattemptingtoconcealit,butIsawitinyourfaceandinyourbearing.WhenItestedmydeductionbyspeakingof‘thecrimethathasbeencommittedagainstyou’,yourreplywasevasive.Youstatedthatyouhadnotcomeprimarilyasaprospectiveclient.”
Watsupsighed,slumpinginhischair.“Iwasworriedthatmentioningmyowncasemighthaveanadverseeffectonyourdecisionaboutengagingmyservices,bysuggestingthatIwasmerelyseekingfreeadvice.Onceagainyouhaveseenthroughme,MrSoames.”
“Thatwasinevitable.Wemaydispensewithformalities.YoumaycallmeSoames.AndIshallcallyouWatsup.”
“Anhonour,Mr—er,Soames.”Watsup,clearlyupset,tookamomenttosteadyhimself.“Itisasimplematter,ofakindthatyouwillhaveencounteredmanytimesbefore.”
“Aburglary.”“Yes.Howdid—nomatter.Ithappenedearlierthisyear,andIimmediately
requestedprofessionalassistancefromyourneighbouracrosstheroad.Afteramonthinwhichhemadeabsolutelynoprogress,hedeclaredthatthematterwastootrivialtointeresthismightytalents,andshowedmethedoor.Hearingbyafortunateaccidentofyourownexploits,itoccurredtomethatyoumightsucceedwherethegreatluminaryhadfailed.”
ItwascleartoWatsupthathenowhadSoames’sfullattention.“Ivowtohelpyousolvethiscrime,toprovemyownworthtoyou,”Watsup
saidwithsomeemotion.“Ifwesucceed—nay,whenwesucceed—myhopesofamorepermanentengagementwillbeenhanced.Icanpayyounofee,butIcanoffermyownunpaidservicesfortwomonths.DuringwhichtimeIwillensureasteadyflowofclientsbysingingyourpraisestothegentry,enoughtokeepusbothfedandhousedinmoderatecomfort.”
“Iconfessthatsuchanarrangementdoeshavesomeappeal,”saidSoames.“Ihavebeenseekingwhatourtransatlanticfriendsrefertoasa‘sidekick’forsometime.Yourexposureofmylandlady’snosinessgivesmeadditionalconfidencethatyouwillfitthebilladmirably,butweshallsee.Er—speakingofbills,youdon’thappentohaveafive-poundnoteonyou,byanychance?MrsSoapsudsisalwayscomplainingabouttheunpaidrent…No,no,IseeyouareasstrappedforcashasI.Togetherweshallovercomeourmutualimpecuniosity.
“Now,tellmeofthecrime.”“AsIwassaying,itisasimplematter,”saidWatsup.“Myhousewas
burgled,andmypricelesscollectionofAl-Jebrianceremonialdaggers,representingthemajorityofmywealth,wasstolen.”
“Whenceyourpresentfinancialstate.”“Indeed.IhadplannedtohavethemauctionedatSotheby’s.”“Werethereanyclues?”“Justone.Agreensock,leftatthesceneofthecrime.”“Whatshadeofgreen?Whatmaterial?Cotton?Wool?”“Idonotknow,Soames.”“Thesethingsmatter,Watsup.Manyamanhasbeenhangedbecauseofthe
precisecolourofthedyeinasinglestrandofdarningwool.Orescapedthenooseforlackofsuchevidence.”
Watsupnodded,absorbingthelesson.“AlltheinformationIhavewasprovidedbythepolice.”
“Thatexplainsitspaucity,ofcourse.Prayproceed.”“Thepolicenarrowedresponsibilityforthecrimedowntothreemen:
GeorgeGreen,BillBrown,andWallyWhite.”Soamesnoddedthoughtfully.“The‘usualsuspects’,asIhadalready
surmised.TheyoperateintheBoswellStreetarea.”“HowdidyouknowIliveinBoswellStreet?”saidWatsupinastonishment.“Youraddressisonyourcard.”“Oh.Inanycase,oneofthosethreewasdefinitelythecriminal.Thepolice
madeenquiries,andfoundthateachmanhabituallyworeajacketandtrousers.”“Mostmendo,Watsup.Eventhelowerclasses.”“Yes.Butalso,socks.”Soamesprickeduphisears.“Afeatureperhapsofmildinterest.Itshowsthat
thesemenhaveanincomebeyondtheirmeans.”“I’msorry,Soames;Ireallydon’tsee—”“YouhaveneverencounteredMessrsBrown,Green,andWhite.”“Ah.”“Pleaseavoiddistractingremarks,Watsup,andgettothepoint.”“Apparentlyitwaseachman’sinvariablehabittodressingarmentswhose
colourswereexactlythesameonalloccasions.Subtletracesatthesceneofthecrime—”
“Yes,yes,”Soamesmutteredimpatiently.“Threadsadheringtothebrokenglass.Plainasthenoseonadonkey.”
“—uh,well,yes,asIwassaying,threads.Theseindicatedthatthethiefhadusedoneofhissockstomufflethesoundofbreakingwindow-glass,andthatthesockwasgreen.Witnessesconfirmedthatbetweenthemthethreemenworeonejacketofeachcolour,onepairoftrousersofeachcolour,andonematchingpairofsocksofeachcolour.Noneofthemworetwoormoregarmentsofthesamecolour—countingapairofsocksasasinglegarment,youappreciate,sinceevenruffianssuchasthesewouldnotwearsocksthatdonotmatch.Thatwouldbemostimproper.”
“Anddidyoudeduceanythingofconsequencefromthatinformation?”“Eachofthesuspectsmusthavewornexactlyonegarmentwiththesame
colourashisname,”saidWatsupinstantly.“Ifwededucethecolour,wefindthecriminal.”
Soamesleanedbackinhischair.“Verygood.Perhapswewillbeabletoworktogether.Anythingelse?”
“Icametotheconclusionthattheinformationtohandwasinsufficienttodeterminethecriminal.Thepoliceeventuallyadmittedasmuch,soIsuggestedtheyshouldseekfurtherevidence.”
“Anddidtheyfindany?”“AfterIhadofferedsomemorespecificadvice,theydid.”Watsuphanded
Soamesasheetofpaper.“Partofthepolicereport,”heexplained.Thedocumentread:
ExtractfromReportofInvestigationbyConstableJ.K.WugginsoftheHolbornDivisionoftheMetropolitanPoliceForce
1Brown’ssockswerethesamecolourasWhite’sjacket.2ThepersonwhosenamewasthecolourofWhite’strousersworesockswhosecolourwasthenameofthepersonwearingawhitejacket.3ThecolourofGreen’ssockswasdifferentfromthenameoftheperson
wearingthesamecolourtrousersasthejacketwornbythepersonwhosenamewasthecolourofBrown’ssocks.
“Andthereyouhaveit,”saidWatsup.“Ifwecandeterminethethief,thenthepolicewillbeabletoobtainasearchwarrant.Withlucktheywillfindmymissingdaggers,whichwouldamounttoincontrovertibleproofofguilt.Buttheyarestumped,andyouroverratedneighbourisasbaffledasI—whichiswhyhepretendsthatthecasehasnointerest.”
Soameschuckled.“Onthecontrary,mydearWatsup.Thankstoyourassiduousdevotiontopersuadingthepolicetoinvestigatethecircumstancesofthecrimeinsufficientdepth,thereisenoughinformationtodeterminetheguiltyparty.Thedeductionisofcourseelementary.”
“Howcanyoubesosure?”“Youwillcometoknowmymethods,”saidSoamesenigmatically.“Whoisthecriminal,then?”“Wewillfindoutwhenwemakethededuction.”Watsupproducedanew,fat,currentlyblank,notebook,andwrote:
MemoirsbyDrJohnWatsup(M.Chir.,R.M.C.S.,retd)
One:TheGreenSocksCaperSoames,readingthewordsupsidedown,saidquietly“Thisisnotapennydreadful,Watsup.”Watsupcrossedout‘Caper’andinserted‘Incident’.Then,pursinghislips,hebegantorecordtheirjointanalysis.Withafewhiccupsalongtheway,theidentityofthethiefsoonemerged.
Seepage254fortheanswer.“IshallsendInspectorRouladeatelegramimmediately,”Soamesdeclared.
“Hewillsendconstablestoraidtheman’spremises.Nodoubttheywillfindyourdaggersthere,sincethemanwehaveidentifiedisnotoriouslyslowtofencestolenproperty.Helikestogloat,Watsup,anerrorthathasputhimbehindbarsmorethanonce.
“Andthatwrapsupourfirstcasetogether!”Hisexcitementquicklysubsidedasheadded,“Yourassistancewasvital,butunfortunatelytheoutcomeofourdeliberationsdoesnotimproveourfinances,sinceitisyourcase.”
“Therewillbesomeimprovement.Iwillregainmydaggers.”“Ifearthepolicewillholdthemasevidenceuntilafterthetrial.Evenso,we
mayconsideritaharbingerofmoreprofitabletimestocome,eh,Watsup?”
ConsecutiveCubesThecubesofthethreeconsecutivenumbers1,2,3are1,8,27,whichaddupto36,aperfectsquare.Whatarethenextthreeconsecutivecubeswhosesumisasquare?
Seepage258fortheanswer.
AdonisAsteroidMousterian
ThreeofFarrell’smagicwordsquares
JeremiahFarrellpublishedsomeamazingmagicwordsquaresinTheJournalofRecreationalLinguistics33(May2000)83–92.Thesearesamples.Theentriesineachsquarearetwo-letterwordsappearinginstandarddictionaries.Thesamelettersappearineachrowandcolumn,andineachofthetwomaindiagonalsoftheorder4and5squares.Everyrowandcolumnisananagram(thoughnotameaningfulone)ofthesamedictionaryword,whichiswrittenunderneath.MousterianisastyleofflinttoolusedbysomeNeanderthals,bytheway.
Youmayfeelthatwordarrangementsarenotterriblymathematical.However,puzzlebuffstendtoenjoyboth,andIaminclinedtoseewordgamesascombinatorialproblemsposedwithirregularconstraints;namely,thedictionary.Butthesesquareshavemathematicalfeaturestoo.Ifnumbersaresuitablyassignedtoletters,andthenumberscorrespondingtoeachpairoflettersinagivensquareareaddedtogether,theresultingnumericalsquareisalsomagic.Thatis,thenumbersineveryrow,column,and(exceptforthe3×3square)diagonaladdtothesameamount.
Ofcourse,thispropertyholdsforanyassignmentofnumbers,exceptonthediagonalsofthe3×3square,becauseeachletteroccursexactlyonceineachrow,column,and(exceptforthe3×3square)diagonal.However,withthecorrectchoice,thenumbersrunfrom0–8,0–15,and0–24respectively.Theassignmentsaredifferentforeachmagicwordsquare.
Whichnumberscorrespondtowhichletter?Seepage258fortheanswer.
TwoSquareQuickies1Whatisthelargestperfectsquarethatuseseachdigit123456789exactlyonce?2Whatisthesmallestperfectsquarethatuseseachdigit123456789exactlyonce?
Seepage259fortheanswers.
CaughtClean-Handed
JohnNapier
JohnNapier,eighthLairdofMerchistoun(nowMerchiston,partofEdinburgh),isfamousforinventinglogarithmsin1614.Buttherewasadarkersidetohisnature:hedabbledinalchemyandnecromancy.Hewaswidelybelievedtobeamagician,andhis‘familiar’,ormagicalcompanion,wasablackcockerel.
Heusedittocatchservantswhowerestealing.Hewouldlockthesuspectinaroomwiththecockerelandtellthemtostrokeit,sayingthathismagicalbirdwouldunerringlydetecttheguilty.Allverymystical—butNapierknewexactlywhathewasdoing.Hecoatedthecockerelwithathinlayerofsoot.Aninnocentservantwouldstrokethebirdasinstructed,andgetsootontheirhands.Aguiltyone,fearingdetection,wouldavoidstrokingthebird.
Cleanhandsprovedyouwereguilty.
TheAdventureoftheCardboardBoxes*
FromtheMemoirsofDrWatsup
Withtherestorationofmyvaluableceremonialdaggers,andourpartnership’sgrowingreputationforcrackingtheuncrackable,solvingtheinsoluble,andunscrewingtheinscrutable,ourpersonalfinanceswereimprovingbytheday.TheeliteofEnglandwerevirtuallyqueuinguptoengageourservices,andmynotebookscontainmanyofmyfriend’ssuccesses:theMysteryoftheMissingMountain,theVaporisedViscount,andtheBald-HeadedLeague.Noneofthesecases,however,capturesSoames’stalentsintheirpurestform:hisabilitytodiscernsignificantaspectsofapparentlyordinaryobjectsandeventsthatfewotherswouldnotice.TheGiantBatofStAlbansnaturallyspringstomind,buttheramificationsofthecasearetooarcaneandcomplextodescribehere.
ThecuriouseventsofChristmas18—,however,suitmypurposeadmirably,anddeservewiderappreciation.(IamforcedtoconcealtheexactdateandmostofwhathappenedtoavoidembarrassmenttoafamousoperaticcontraltoandseveralCabinetMinisters.)
Iwasseatedatmywriting-desk,recordingdetailsofSoames’smostrecentcases,whilehecarriedoutaseeminglyendlessseriesofexperimentswithmyoldservicerevolverandvasesofchrysanthemums.OurseparateactivitieswereinterruptedbyMrsSoapsuds,whodepositedtwocardboardboxesofdifferentsizes,eachtiedwithribbons.“SomeChristmaspresentsforyou,MrSoames,”sheannounced.
Soameseyedthepackages.Theyborehisaddressandsomecancelledpostagestampswithillegiblepostmarks.Theywererectangularinshape…well,technicallyarectangleistwo-dimensional,sotheywereactuallyrectangularparallelepipeds.Cuboids.
Box-shaped.Hetookoutarulerandmeasuredtheirdimensions.“Remarkable,”he
muttered.“Andvery,verydisturbing.”Ihavelearnedtorespectsuchjudgements,howeverpeculiartheyfirstmay
seem.IstoppedthinkingofthepackagesasChristmaspresents,triedtodismissagrowingsuspicionthattheywerebombs,anddidmybesttoobserve.Finally,Irealisedthattheyhadbeentiedwithrathermoreribbonthanwasstrictly
necessary.
Left:Watsup’susualarrangementofribbons.Right:Thewayeachpackagewastied.
“Theribbonsformacrossoneveryfaceofthepackages,”Isaid.“WhenIwrapparcels,Inormallytietheribbonsothatitformsacrossonthetopandbottomfaces,andrunsverticallydowneachoftheotherfour.”
“Youdoindeed.”Clearlymyanalysiswasincomplete.Irackedmybrains.“Umm—thereisno
bow.”“Correct,Watsup.”Stillincomplete.Iscratchedmyhead.“ThatisallIcanobserve.”“Thatisallyoucansee,Watsup.Youhavenoticedeverythingexceptthe
crucialpattern.Ifearthatterribledeedsareafoot.”IconfessedthatIsawnothingterribleintwoChristmaspresents.Thena
thoughtstruckme.“Doyoumeanthattheboxescontainseveredbodyparts,Soames?”
Helaughed.“No,theyarealmostempty,”hesaid,pickingthemupandshakingthem.“ButsurelyyourealisethatthisparticulartypeofribboncanbeboughtonlyfromtheLadiesWilberforce?”
“Regrettably,no,butIbowtoyoursuperiorknowledge.Theestablishment,however,isfamiliartome.ItisahaberdasheryinEastcastleStreet.”Thepennydropped.“Soames!Thatiswherethatterriblemurderwascommitted!Itwas—”
“—inallthepapers.Yes,Watsup.”“Theevidencewasconvincing,butthebodyhasnotyetbeenfound.”Soamesnodded,hisfacegrim.“Itwillbe.”“When?”“ShortlyafterIopentheseboxes.”Heputonapairofglovesandsettoworkunwrappingthem.“Undoubtedly
thisistheworkoftheCartonari,Watsup.”WhenIstaredathimblankly,headded:“anItaliansecretsociety.Butitisbetterthatyouremainignorant.”Despiteallmyentreaties,herefusedtosaymore.
Heopenedbothboxes.“AsIsuspected.Oneisempty,buttheothercontainsthis.”Heheldupasmallrectangleofpaper.
“Whatisit?”Hepassedittome.“Aleft-luggageticket,”saidI.“Itmustbeamessage
fromthemurderer.Buttheserialnumberhasbeentornoff,andsohasthenameofthestation.”
“Onlytobeexpected,Watsup.He—forbythefootprintsinthebloodthecriminalwasassuredlyaman,andalargeoneatthat—istauntingus.Butweshallhavethebetterofhim.Thestationisofcourseobviousfromthearrangementoftheribbon.”
“Uh—pardon?”“Togetherwiththevalueofthestamps,whicheliminatesthepossibilityof
CharingCross.”Thismadelittlesense,soIpickedupapackageandcountedfiveone-shilling
stamps.“Anabsurdamounttopayforanemptypackage.”Isaid,puzzled.“Notifyouwishtosendamessage.Whatisanothernameforfive
shillings?”“Onecrown.”“Andacrownissymbolicof?”“Ourowndearqueen.”“Close,Watsup,butyouhavefailedtotakeaccountoftheshapeofthe
ribbon.”“Itisacross.”“Sothestampsindicate‘king’,notqueen.ThestationisKing’sCross,man!
Butthereismore.Answermethis,Watsup.Whydidthecriminalsendmetwolargeboxeswhenonewasempty?Onesmallenvelopewouldhavesufficedtosendaticket.”
AfteralongsilenceIshookmyhead.“Ihavenoidea.”“Theremustbesomethingsignificantabouttherelationshipbetweenthe
boxes.Andindeedthereis,asIrealisedassoonasImeasuredtheirsizes.”Hehandedmetheruler.“Usethis.”
Irepeatedhismeasurements.“Thelength,breadth,andheightofeachpackageisawholenumberofinches,”Isaid.“Otherwisenopatternspringstomind.”
Hesighed.“Youdidnotobservethestrangecoincidence?”“Whatstrangecoincidence?”“Bothpackageshavethesamevolume,andtheybothusethesametotal
lengthofribbon.Indeed,theirmeasurementsarethesmallestnonzerowholenumberswiththatproperty.”
“Whichleadsyoutoconclude—oh,ofcourse!Thevolumeandlengthtogethergivetheserialnumberontheticket.Therearetwodistinctwaysto
stringthemtogether,ofcourse,butwecaneasilycheckboth.”Soamesshookhishead.“No,no.Themurdererwouldhaveneededan
accompliceattheticketofficetoarrangethat,evenifaticketwiththatnumberexisted.Itismuchsimpler:hehasmarkedsomeitemofleftluggagewiththesetwonumbers.Inside,therewillbesomethingthattellsuswheretofindit.”
“Tofindwhat?”“Isitnotobvious?Thebody.”“Itakemyhatofftoyou,Soames,”Isaid.“Orwould,wereIwearingone.
Butwillfindingthebodyleadustothemurderer?”“Itwillbeusefulevidence,butinconclusive.However,thereismoretobe
gleaned.Sometimesacriminalbelieveshimselfsocleverthathedeliberatelyleavesclues,certainthattheinvestigatingauthoritieswillbetoostupidtonoticethem.TheCartonariareanarrogantbunch,andthisistypicalofthem.Now,thereisanaturalquestionthatleadsonfromtheremarkablearithmeticoftheseboxes.Whatisthesmallestsetofthreeboxeswithasimilarproperty?”
Histrainofthoughtbecameinstantlyapparenttome.“Youexpecttoreceivesuchboxesinthenearfuture!Withanothertornticket!Sothereistobeanothermurder,eh?”Istartedlookingformyrevolver.“Wemuststopit!”
“Ifearthatithasalreadybeencommitted,butwithgoodfortunewemayperhapspreventathirddeath.Tonightthemurdererwillbedepositingsomeitem—itmightbeanything—asleftluggageatoneofthemainLondonrailwaystations.Thenhewillsendustheboxes.Ifwecanworkoutthenumbersaheadoftime,wecanalertInspectorRoulade.Hewillsendpolicetoallofthemainlinestations.Theycannotcheckeverypassengerwhodepositsluggage,forthatwouldalertthecriminal,buttheycankeepawatchforanyonedepositinganitemwiththosethreenumbersmarkedonit,andarrestthem.Insidewillbethelocationofthesecondbody.Whenitisfound,theevidenceofguiltwillbeoverwhelming.”
Intheevent,itwasnotquitethatsimple,andSoamesandIhadtointerveneafterthepoliceletthemanslip.Fortunatelythethreepackages,whichdulyarrivednextdayintheafternoonpost,providednewclues,andwediscoveredthatthemurderwaspartofafarmoreextensiveplot.Thetortuouspathsdownwhichourdeductionsledus,andtheblood-curdlingsecretsthatweunearthed—Ispeakliterally—canneverbemadepublic,asIhaveexplained.Butwedideventuallycatchthecriminal.AndSoameshaspermittedmetorevealtheanswerstothetwoquestionsthatwerecentraltotheentireinvestigation.
Whatarethedimensionsofthetwoboxes?Whataretheyforthreeboxes?Seepage259fortheanswer.
Footnote
*ThisandallsubsequentcasesinvestigatedbySoamesandWatsuparereproduced(slightlyedited)fromTheMemoirsofDrWatsup:BeingaPersonalAccountoftheUnsungGeniusofanUnderratedPrivateDetective,Bromley,Thrackle&Sons,Manchester1897.
TheRATSSequence1,2,4,8,16,…Whatcomesnext?It’stemptingtoleaptoconclusionsandplumpfor32.ButsupposeItellyouthatthesequenceIhaveinmindactuallygoes
12481677145668
Nowwhatcomesnext?Thereisn’tauniqueanswer,ofcourse:byfindingsufficientlycontrivedrulesyoucanfitaformulatoanyfinitesequence.InMathematicsMadeDifficultCarlLinderholmhasanentirechapterexplainingwhyyoucanalwaysanswer‘whatcomesnextinthissequence’with‘19’.But,bearwithme:thereisasimpleruleforthesequenceabove.Thetitleofthissectionisaclue,butamuchtooobscureonetobeofanyusetoyou.
Seepage260fortheanswer.
BirthdaysAreGoodforYouStatisticsshowthatpeoplewhohavemostbirthdayslivethelongest.
LarryLorenzoni
MathematicalDatesInrecentyearsnumerousdateshavebeenassociatedwithaspectsofmathematics,basedonnumericalsimilarities,leadingtothatdatebeingdeclaredaspecialday.Nooneattachesanysignificancetosuchdays,asidefromthenumericalsimilarity.TheydonotforetelltheEndoftheWorldoranythingofthatkind—asfarasweknow.Nothingspecialhappensonthemasidefrommathematicalcelebrationsandoccasionalcommentsinthemedia.Butthey’refun—andanexcusetointerestthemediainmoresignificantmathematics.Oratleasttomentiontheword.
Amongthemarethefollowing.Manyareassociatedwithalternativedates,becauseintheAmericansystemfordates,themonthprecedestheday.IntheBritishsystem,thedaycomesfirst.Acertainamountofcalendriclicenceisallowed,suchasomittingzeros.
PiDay14March,Americansystemofdates:3/14.[π~3.14.]Quasi-officialdaysince1988inSanFrancisco.Recognisedbyanon-bindingresolutionintheUSHouseofRepresentatives.
PiMinute1.59onMarch14(Americansystem).3/141:59.[π~3.14159.]
Moreaccuratelystill:1.59and26seconds.3/141:59:26.[π~3.1415926.]
PiApproximationDay22July,Britishsystemofdates:22/7.[π~22/7.]
123456789DaySorry,youmissedit.Thisonce-in-a-lifetimeeventoccurredon7August2009(Britishsystem),or8July2009(Americansystem),shortlyafter12.34p.m.Thedateandtimewere:12:34:567/8/(0)9.Butsomeofyoumightwitness1234567890Dayin2090.
OnesdayYoumissedthistoo.Itoccurredon11November2011(eithersystem)at11minutes,11secondspast11o’clock.Thedateandtimewere:11:11:1111/11/11.
TwosdayAsIwrite,thiswillhappenafewyearsfromnow.Thisisyouropportunity!2February2022:22:22:222/2/22.
PalindromeDayApalindromereadsthesameinreverse—asin‘evilratsonnostarlive’.20February2002at8.02p.m.(Britishsystem,24-hourclock):20:0220/02/2002.
Thisrepeatsthesamepalindromethreetimes.Whenwillthenextsuchdaybe,ontheBritishsystem?Whatwasthenextdateaftertheoneabove,againusingtheBritishsystem,onwhichthewholethingwaspalindromic?
FortheanswersSeepage261.
FibonacciDay(Shortversion)3May2008(Britishsystem),5March2008(Americansystem):3/5/(0)8.
FibonacciDay(Longversion)5August2013(Britishsystem),2minutes3secondspast1o’clockon8May2013Americansystem):1:2:35/8/13.
PrimesDay2March2011(Britishsystem),3February2011(Americansystem):2:35/7/11.
TheHoundoftheBasketballsFromtheMemoirsofDrWatsup
“Aladytoseeyou,MrSoames,”saidMrsSoapsuds.SoamesandIsprangtoourfeet.Awomanofindeterminateageentered—
indeterminatebecausesheworeadarkveil.“Thereisnoneedtodisguiseyourself,LadyHyacinth,”saidSoames.Withagasp,shepulledoffherveil.“How—”“TheextraordinaryeventsatBasketHallhavebeenintheheadlinesfora
week,”saidSoames.“Ihavebeenfollowingthecaseclosely,andmycompetitoracrosstheroadhasmadenoprogress.Itwasonlyamatteroftimebeforeyousoughtmyservices.Also,Irecognisedyourdriver’shat,whichisquiteuniqueamongtheservantsofthearistocracy.”
“Basquet,notBasket,”saidLadyHyacinthwithasniff,makingthewordsoundFrench.
“Comenow,madam,”saidSoames.“ThehousehasbeenintheBasketfamilyforsevengenerations,eversinceHonoriaThumpingham-MaddelymarriedtheThirdEarl.”
“Well,yes,butthatwasthen.Thespellingandpronunciationhave…er—”“Beenmodernised,”Iinterjected,hopingtosmoothincreasinglytroubled
watersofmutualdisaffection.AtthesametimeIgaveSoamesasharplook,unobservedbyHerLadyship.Tohiscredit,hetookmypoint.
“Itwasagiganticblackhound!”shesuddenlycried,thewordssoundingasiftheywerebeingtornfromherthroat.“Withgreatslaveringjawsthatdrippedblood!”
“Yousawit?”“Well,no,buttheboywholooksafterthepiglets…Nicky,that’sthename.
OrisitRicky?Anyway,hesaidhecaughtaglimpseofthevilehorrorjustasitdisappeared.”
“Inthedark,”Soamespointedout.“Fromadistanceofonehundredandseventyyards.MichaelJenkinsisshort-sighted.Butnomatter,theevidencewillleadustothetrutheventually.Igatherthattheanimaldidnoharmtoanyhumanbeing?”
“Well,no,”sheagreed.“Notdirectly,thoughmypoorhusband…Yousee,
thehoundruinedatraditionthatgoesbackeventobeforetheThirdEarlofBask—Basquet.”
Ibelatedlyrememberedmyetiquette.“DrJohnWatsup,atyourservice,madam.IregretthatunlikemycompanionIhavenotbeenfollowingthenews.Ifyouwouldbesokindastoenlightenme?”
“Ah.Yes.Um.”Shegatheredherskirtsandherwits.“ItwasafewnightsbeforeMidwinter’sEve,andmyhusbandEdmund—that’sLordBasquet,ofcourse—hadarrangedtwelveantiquestonespheres—”
“Knownforcenturiesasbasketballs,”Soamesinterrupted.“Well,yes,butwecan’tmoderniseeverything,MrSoames.Thereare
traditions.Anyway,myhusbandhadarrangedtheballsonthelawnofourstatelyhomeinanage-oldfamilysymbol.Onlytheheirtothemalelineknowsexactlywhatthesymbolis,andnooneelseispermittedtoobservetheceremony,butovertheyearsithasbecomecommonknowledgethatitconsistsofsevenstraightrowsofballs,withfourineachrow.
“EdmundwasrehearsingforaceremonywhichmustbecarriedoutwithoutfaileveryMidwinter’sEve.Butwhenweawokethefollowingmorning,wewerehorrifiedtofindthatsomeoftheballshadbeenmoved!”
“ButyouhavesaidthatnoonesaveLordBasquetispermittedtoseethearrangement,”Soamesobjected.
“Thecircumstanceswereexceptional.HisLordshipwenttorecovertheballs,butfailedtoreturn.Eventuallyoneoftheserving-maidswassenttofindhim—Laviniaisblind,yousee,butverycapable.ShecamebackintearsscreamingthathisLordshipwaslyingontheground,notmoving.Fearinghewasdead,therestofusdisobeyedthetime-honouredinjunctionandrushedtothescene.IwasjustintimetohearEdmundgasp‘Moved!’Thenhewasstill.Hehasbeeninastuporeversince,MrSoames.Itismostdistressing.”
“Moved,”saidI.“Inwhatmanner,madam?”“Nolongerinthesameplace,DrWatsup.”“Imean,movedwhere?”“Theynowformedastar,DrWatsup.”“Yes!Onehavingonlysixstraightrowswithfourballsineachrow,”said
Soames,sketchingrapidlyonasheetofpaper.“Thisfacthasbeenwidelyreportedandhastheringoftruth,foritisunlikelytohavebeeninvented,beingtoomentallytaxingfortheyellowpress.ItalsoprovesthatwecouldhavededucedthattheballshadbeenmovedwithouthavingtorelyonhisLordship’slastgasp—”
Theballsafterthehoundhadmovedthem
“Lastgasptodate,”Isaidhurriedly,beforeSoamestriggeredafreshwaveofwails.
“Couldyounotmovethemback?”Ienquired,whenHerLadyshiphadregainedsomeofhercomposure.
“Nay!”shecried.IhavelongremarkedthattheEnglisharistocracyhaveanoticeabletendencytoresemblehorses.
“Whynot?”“AsItoldyou,onlyhisLordshipknowstheexactarrangementthattradition
requires,andthedoctorssayhemayneverrecover!”“Weretherenotmarkswheretheballshadoriginallyrested?”“Perhaps,buttheywereobscuredbythetracksoftheterriblehound!”“Iwillbringmymostpowerfulmagnifyingglass,then,”saidSoames,witha
straightface.Athoughtmusthavestruckhim,becausehesuddenlyfroze.“Yousaid‘must’.”
“Idid?When?”“Afewminutesagoyousaidthattheceremonymustbecarriedoutwithout
faileveryyear.Ithasjustoccurredtomethatyourchoiceofwordsmaybesignificant.Explainthem.”
“AccordingtoanancientTransylvanianprophecy,ifthetwelvebasketballsarenotdeployedcorrectlyonMidwinter’sEve,theHouseofBask—er,Basquet—willfall,andbeutterlydestroyed!Andwehaveonlythreedaystodoit!Oh,woe!”Shebegantosob.
“Calmyourself,madam,”saidI,waftinganopenbottleofsmelling-saltsbeneathhernose.“PleaseacceptmycondolencesforHisLordship’sunfortunatecondition,andmyassurancesasamedicalmanthatthereissomefaintchance,beiteversoslight,thatitmayundergosomemiraculousimprovementinduecourse.”Ihavelongpridedmyselfonmyimpeccablebedsidemanner,which
putsthatofmyfriendSoamestoshame,butonthisoccasion,unaccountably,HerLadyship’sresponsewastoredoublehersobbing.
Soamespacedtheroom,hisfacedrawn.“Isitonlytheformofthearrangementthatmatters,YourLadyship?Ordoestheorientationmakeasignificantdifference?”
“Ibegyourpardon?”saidshe,shakingherheadasiftoclearhermind.“Ifthearrangementwerecorrectexceptforarotation,withoutchangingthe
relativepositionsoftheballs,wouldthatbeexpectedtotriggerthedireeventspredicted?”Soamesclarified.
LadyBasquetpaused,consideringthematter.“No.Definitelynot.IrecallWillyWillikins—that’stheheadgardener—suggestingtomyhusbandthatfromtimetotimehemightcaretopointthearrangementinadifferentdirection,toavoiddamagingthelawn.Edmundraisednoobjection.”
“Thatisexcellentnews!”saidSoames.“Yes.Excellent,”Iechoed,havingnotthefaintestideawhymydetective
friendwassopleased.Or,forthatmatter,whathisquestionmeant.“Werethereanysignsofhumanintervention?”Soamesasked.“No.TheheadgardenersworethatnohumanbeingsaveEdmundhadset
footonthelawn.YoungDicky—”“Micky.”“Vickysawtheterriblehound,butevenheobtainedonlyafleetingglimpse
asitleapedovertheperimeterofthewalledgarden.Wehavesomedelightfulpeonies,MrSoames,althoughtheyarenotinbloomatthistimeof—”
“Iwilltakethecase,”saidSoames.“IfYourLadyshipcarestoreturntoBasquetHall,IandmycolleaguewillarriveonThursdaybytheearliestpossibleslowtrain.”
“Nosooner,MrSoames?ButThursdayisMidwinter’sEve!Theballsmustbeplacedcorrectlybeforethesunsets!”
“IregretthatIamdetaineduntilthenbyasmallmatterinvolvingthreeEasternpotentates,sixhundredthousandarmedwarriors,twodisputedborders,andastolencasketofemeraldsandsapphiresbelongingtoanobscureandarcaneancientreligiousorder.Andaflattenedcopperthimble,whichIbelieveholdsthekeytotheentireaffair.However,IassureyouthatIamconfidentthatyourcasewillbesettledsatisfactorilybeforesunsetonThursday.”
Herprotestationsnotwithstanding,Soamesremainedadamant,andeventuallyLadyHyacinthBasquetdeparted,snufflingquietlyintoalacekerchief.
Aftershehadleft,IaskedtowhichmatterSoameshadbeenreferring,forIhadheardnothingofthiscase.“Asmallfabricationonmypart,Watsup,”he
confessed.“Ihaveticketsfortheoperathisevening.”Wearrivedmid-afternoonontheThursday,tobemetatthestationbya
groomdrivingagovernesscart.Orpossiblyagovernessinagroomcart,mynotesbeingillegibleonthispoint.LordBasquetremainedinacoma,wewereinformed.WithinabarehalfhourwehadarrivedattheHall,andSoameswasinspectingtheextensivelawnsusinganunusuallylargemagnifyingglass,ahairbrush,andaprotractor.
“Achanceforyoutoexerciseyourdeductiveskills,Watsup,”hesaid.“Iseesomedisturbanceofthegrass,Soames.”“CorrectWatsup.Thetracksarehighlycomplex,buttheyaremainly
superimposedpaw-printsof—”heloweredhisvoicesothatIalonecouldhear—“aminiaturepoodle.”Inhisnormalvoicehecontinued:“Iamunabletodiscerntheoriginalplacingoftheballs,butunlessIamverymuchmistaken—andIneveram—itisclearthattheanimalmovedpreciselyfouroftheballs.”
“Isthatsignificant,MrSoames?”LadyBasquetaskednervously,cradlingaminiaturepoodleinherarms.
Soamesglancedmyway.“Itis…possible…”Ibegan,andsawSoamesnodimperceptibly.Well,not
totallyimperceptibly,youappreciate,sinceifithadbeenimperceptibleIwouldnothaveseenit.Takingthisasveiledencouragement,Ihazardedaguess:”…thatthisconditionwillmakeitpossibletodeducetheoriginallayout.”
“Anddoesit?”sheenquired,ahopefullookonherface.
Whatwastheoriginalarrangementofthebasketballs?Seepage261fortheanswer.
DigitalCubesThisisanoldie,butitopensupalessfamiliarquestion.Thenumber153isequaltothesumofthecubesofitsdigits:
13+53+33=1+125+27=153
Therearethreeother3-digitnumberswiththesameproperty,excludingnumberslike001withaleadingzero.Canyoufindthem?
FortheanswerSeepage262.
NarcissisticNumbersThecubespuzzlehassomenotoriety,becausein1940thefamouspuremathematicianGodfreyHaroldHardywrote,inAMathematician’sApology,thatsuchpuzzleshavenomathematicalmeritbecausetheydependonthenotationemployed(decimal)andarelittlemorethancoincidences.However,youcanlearnquitealotofusefulmathematicsbytryingtosolvethem,andgeneralisations(forexample,tonumberbasesotherthan10)avoidthenotationalissue.
Oneoftheoffshootsofthispuzzleistheconceptofanarcissisticnumber,whichisdefinedtobeanumberthatisequaltothesumofthenthpowersofitsdecimaldigitsforsomen.Thetermn-narcissisticnumberisusedifwewanttomakenexplicit.
FourthPowersofDigits(4-NarcissisticNumbers)Write[abcd]forthenumberwithdigitsa,b,c,dtodistinguishthisfromtheproductabcd.Thatis,[abcd]=1000a+100b+10c+d.Wemustsolve
[abcd]=a4+b4+c4+d4
whereallunknownsliebetween0and9.Thisisbynomeansatrivialtask.Tryit!
Seepage263fortheanswer.
FifthPowersofDigits(5-NarcissisticNumbers)Thistimetheproblemistosolve
[abcde]=a5+b5+c5+d5+e5
andasyou’dexpect,that’sevenharder.
Seepage263fortheanswer.
HigherPowersofDigits(n-NarcissisticNumbers,n≥6)Itiseasytoprovethatn-narcissisticnumbersexistonlyforn≤60,becausewhenevern>60wehaven:9n<10n–1.In1985DikWinterprovedthatthereareexactly88narcissisticnumberswithnonzerofirstdigit.Forn=1theyarethetendigits(weinclude0becauseitistheonlydigitinthiscase).Forn=2they
don’texist.Forn=3,4,5seetheanswerstoDigitalCubes(page262)andtheabovetwoproblems.Forn≥6theyare:
nn-narcissisticnumbers
6 5488347 1741725,4210818,9800817,99263158 24678050,24678051,885934779 146511208,472335975,534494836,91298515310 467930777411 32164049650,32164049651,40028394225,42678290603,44708635679,
49388550606,82693916578,9420459191414 2811644033596716 4338281769391370,433828176939137117 21897142587612075,35641594208964132,3587569906225003519 1517841543307505039,3289582984443187032,4498128791164624869,
492927388592808882620 6310542598859969391621 128468643043731391252,44917739914603869730723 21887696841122916288858,27879694893054074471405,
27907865009977052567814,28361281321319229463398,35452590104031691935943
24 174088005938065293023722,188451485447897896036875,239313664430041569350093
25 1550475334214501539088894,1553242162893771850669378,3706907995955475988644380,3706907995955475988644381,4422095118095899619457938
27 121204998563613372405438066,121270696006801314328439376,128851796696487777842012787,174650464499531377631639254,177265453171792792366489765
29 14607640612971980372614873089,19008174136254279995012734740,19008174136254279995012734741,23866716435523975980390369295
31 1145037275765491025924292050346,1927890457142960697580636236639,2309092682616190307509695338915
32 17333509997782249308725103962772
33 186709961001538790100634132976990,186709961001538790100634132976991
34 112276328532937254159282290020459335 12639369517103790328947807201478392,
1267993778027227856630388559419692237 121916721962543412156973580360996601938 1281579207836605995509977054529612936739 115132219018763992565095597973971522400,
115132219018763992565095597973971522401
Piphilology,Piems,andPilishNow,IwishIcouldrecollectpi.
‘Eureka!’criedthegreatinventor.Christmaspudding;ChristmaspieIstheproblem’sverycentre.
See,Ihavearhymeassistingmyfeeblebrain,itstaskssometimesresisting.
HowIwishIcouldenumeratepieasily,sinceallthesehorriblemnemonicspreventrecallinganyofpi’ssequencemoresimply.
Thelastonegivesthegameaway:thesearemnemonics—memoryaids—forπ.There’sevenawordforthis:piphilology.Countthelettersinsuccessivewords:3,1,4,1,5,…
SomeofthemanymnemonicsforπwerediscussedinCabinetofMathematicalCuriosities;herewerecallone(theFrenchmnemonicbelow)andlookatafewmore.Therearehundreds,inmanylanguages:see
http://en.wikipedia.org/wiki/Piphilologyhttp://uzweb.uz.ac.zw/science/maths/zimaths/pimnem.htm
OneofthemostfamousistheFrenchalexandrine(apoeticmetre),whichbegins:
Quej’aimeàfaireapprendreUnnombreutileauxsages!GlorieuxArchimède,artisteingenieux,Toi,dequiSyracuseloueencorelemérite!
andgoesonuntilitgetsto126places.IespeciallyrecommendthePortuguese:Souomedoetemorconstantedomeninovadio.
(Iamtheconstantfearandterroroflazyboys.)TheRomanian:
Asaebineascrierenumitulsiutilulnumar.(That’sthewaytowritethefamousandusefulnumber.)
hasthemeritofdirectnessandsimplicity.Thepoemsareknownaspiems.The32nddecimalplaceofπis0,anda
wordoflength0isinvisible.However,therearewaysroundthisobstacle.InPilish,thecryptographicsystemnormallyusedforπmnemonics,aten-letterwordcountsas0.MikeKeith’s‘ASelf-ReferentialStory’,whichencodes402
digitsofπ[TheMathematicalIntelligencer8(No.4)(1986)56–57]employedadifferentsetofrules.ThelongestexamplesIknowof(todate,GuinnessBookofWorldRecordsherewecome,ifIknowmyreaders)aretheshortstory‘Cadaeiccadenza’(3,834digits)andthebookNotAWake(10,000digits),alsobyKeith.Thebookbegins:
NowIfall,atiredsuburbianinliquidunderthetreesDriftingalongsideforestssimmeringredinthetwilightoverEurope.Soscreamwiththeoldmischief,askmeanotherconundrumAboutbitternessofpossiblefortunesnearalandscapeItalian.Alittlehappinessmaysometimesintervenebutusuallyfades.Amissionarycries,strivingtounderstandworthless,tediouslife.Monotony’slostamidoceanmovementsAsthebewilderedsailorshesitate.Ibecomesalt,Submergingpeopleindazzlingoceansofenshroudedunbelief.Christmasornamentsconspire.Beautyis,somewhatinevitablynow,bothFeelingsoffaithandeyesofrationalism.
Hereten-letterwordscountas0,andlongeronescountastwodigits;forexample,a13-letterwordcountsas13.
ForawealthofrelatedinformationandotherexamplesseeKeith’swebsitehttp://cadaeic.net
Clueless!FromtheMemoirsofDrWatsup
AsIflickthroughthewell-thumbedpagesofmynotebooks,mymemoriesaredrawntoinnumerablemysteriesthatSoamessolvedbyobservingcluessosubtleastoeludelesserminds,suchastheAdventureoftheSussexUmpire(aremarkablelocker-roommysterywhosepivotalfeaturewasaprematurelyscuffedcricketball),theCowwiththeCrumpledHorn,theAttemptedTripleMurderoftheDiminutiveSwine,andtheAffairoftheMissingTart.Butamongthemonestandsout:amysterywhosesolecluewastheabsenceofanyclueswhatsoever.
Itwasadamp,gloomyTuesday,andthestreetsofcentralLondonwerethickwithsmokeandfog.Wehadabandonedactivepursuitoftheperpetratorsofcrimeforaperiodofintrospectionbeforeawarmfire,accompaniedbyampleglassesofanamusinglypresumptuousclaret.
“Isay,Soames,”Iremarked.Mycolleaguewassearchingthroughathickstackofphotographicplatesof
hoofprintsinmud,producedusingEastman’snewimprovementofMaddox’sgelatineprocess.Hisresponsewasanirritated“Haveyouseenmycollectionofcarthorsephotographsanywhere,Watsup?”,butIcontinueddoggedly.
“Thispuzzlehasn’tgotaclue,Soames.”“It’snottheonlyone,”hemuttereddarkly.“No,Imean—ithasn’tgotanyclues.”IcouldseeInowhadhisattention.Hetookthenewspaperfrommy
outstretchedhand,andglancedatthediagram.
Cluelesspuzzle
“Theunstatedrulesareobvious,Watsup.”“Why?”“Theymustbesimpleenoughtomotivatethewould-besolver,yetleadtoa
sufficientlychallengingproblemtoretaintheirinterest.”“Nodoubt.Sowhataretherules,Soames?”“Clearlyeachrowandeachcolumnmustcontaineachnumber1,2,3,4once
only.”“Ah.Itisacombinatorialpuzzle,atypeofLatinsquare.”“Yes,butthereismore.Thetworegionsoutlinedinthickblacklinesare
obviouslyimportant.Iconjecturethatthenumbersineachregionmusthavethesametotal…Yes,thatleadstoauniquesolution.”
“Oh.Iwonderwhattheansweris.”“Youknowmymethods,Watsup.Usethem.”Andhereturnedtohis
photographicplates.
Seepage263fortheanswer.Formorecluelesspuzzles,Seepage74.
ABriefHistoryofSudoku
ModernreaderswillrecogniseWatsup’spuzzleasavariantofsudoku.(Ifyou’vejustcomebackfromaforty-yeartriptoProximaCentauri,thisisa9×9arraydividedintonine3×3blocks,withsomenumbersinserted.Youhavetofillintherestsothateveryrow,everycolumn,andeveryblockcontainseachdigit1–9.)
LoShu,centre,onthebackofasmallturtle,surroundedbytheChinesezodiacandthedivinatorytrigramsoftheIChing,allcarriedbyalargerturtle,whichaccordingtomythfirstrevealedthetrigrams
Similarbutsignificantlydifferentpuzzleshavealonghistory,whichgoesrightbacktotheChineseLoShu,amagicsquareallegedlyseenonthebackofaturtlearound2100BC.JacquesOzanam’s1725RécréationsMathématiquesetPhysiquesincludedapuzzleaboutplayingcardsthatgetsslightlyclosertosudoku.Takethe16courtcards(ace,king,queen,jack)andarrangetheminasquaresothateachrowandcolumncontainsallfourfacevaluesandallfoursuits.KathleenOllerenshawshowedthatthereare1152solutions,whichreducetojusttwobasicallydifferentonesifweconsidertwosolutionstobethesameifonecanbeobtainedfromtheotherbypermutingthefacevaluesandsuits.Thereare24×24=576waystodothistoanygivensolution,and1152/576=2.
Canyoufindthesetwobasicallydifferentsolutions?Seepage265fortheanswer.
In1782Eulerwroteaboutthethirty-sixofficersproblem:cansixregiments,eachhavingsixofficersofdifferentranks,bearrangedina6×6squaresothat
eachrowandcolumncontainseveryrankandeveryregiment?SucharrangementswerecalledGraeco-LatinsquaresbecauseLatinlettersA,B,C,…andGreeklettersα,β,γ,…couldbeusedtodenotetheranksandregiments.HefoundmethodsforconstructingGraeco-Latinsquareswhoseorder(thesizeofthesquare)isoddordoublyeven:amultipleoffour.
Anorder5Graeco-Latinsquare
Eulerconjecturedthatnosuchsquaresexistwhentheorderisdoublyodd:twiceanoddnumber.Thisisobviousfororder2,andGastonTarryproveditin1901fororder6.However,in1959RajChandraBoseandSharadchandraShankarShrikhandeusedacomputertofindaGraeco-Latinsquareoforder22,andErnestParkerfoundoneoforder10.ThethreeofthemthenprovedthatEuler’sconjectureisfalseforalldoublyoddordersgreaterthanorequalto10.
Squaren×narrayssuchthateachrowandcolumncontainsallofthenumbers1—n(eachnecessarilyappearingonce)becameknownasLatinsquares,andGraeco-LatinsquareswererenamedorthogonalLatinsquares.Thesetopicsarepartofthebranchofmathematicsknownascombinatorics,andtheyhaveapplicationstoexperimentaldesign,theschedulingofcompetitions,andcommunications.
AcompletedsudokugridisaLatinsquare,butthereareextraconditionsonthe3×3blocks.In1892theFrenchnewspaperLeSièclepublishedapuzzleinwhichsomeofthenumberswereremovedfromamagicsquareandreadershadtoinsertthecorrectones.LaFrancecameveryclosetoinventingsudokubyusingmagicsquarescontainingonlythedigits1–9.Inthesolutions,3×3blocksalsocontainedtheninedigits,butthiswasnotmadeexplicit.
SudokuinitsmodernformwasprobablyintroducedbyHowardGarnsandpublishedanonymouslyin1979byDellMagazinesunderthename‘numberplace’.In1986theJapanesecompanyNikolipublishedpuzzlesofthiskindinJapanunderthenotterriblyeye-catchingnamesūjiwadokushinnikagiru(‘thedigitsarelimitedtooneoccurrence’).Thenamewasthenshortenedtosūdoku.
TheTimesbeganpublishingsudokupuzzlesintheUKin2004,afterbeingcontactedbyWayneGould,whohaddevisedacomputerprogramtoproducesolutionsrapidly.In2005itbecameaworldwidecraze.
Hexakosioihexekontahexaphobia
Thisisthefearofthenumber666.In1989,whenPresidentRonaldReaganandhiswifeNancymovedhouse,
theychangedtheaddressfrom666St.CloudRoadto668St.CloudRoad.Thismaynothavebeenagenuinecaseofhexakosioihexekontahexaphobia,however,becausetheymightnothavefearedthenumberassuch—theywerejusttakingstepstoavoidobviousaccusationsandpotentialembarrassment.
Ontheotherhand…WhenDonaldRegan,ChiefofStafftoPresidentReagan,publishedhismemoirsinthe1988FortheRecord:FromWallStreettoWashington,hewrotethatNancyReaganhadregularlytakenadvicefromastrologersJeaneDixonand,later,JoanQuigley.“VirtuallyeverymajormoveanddecisiontheReagansmadeduringmytimeasWhiteHouseChiefofStaffwasclearedinadvancewithawomaninSanFranciscowhodrewuphoroscopestomakecertainthattheplanetswereinafavorablealignment.”
Thenumber666hasoccultsignificancebecauseitisstatedtobetheNumberoftheBeastintheBookofRevelation13:17–18(KingJamestranslationoftheBible):“Andthatnomanmightbuyorsell,savehethathadthemark,orthenameofthebeast,orthenumberofhisname.Hereiswisdom.Lethimthathathunderstandingcountthenumberofthebeast:foritisthenumberofaman;andhisnumberisSixhundredthreescoreandsix.”
ItisgenerallyassumedthatthereferenceistothenumerologicalsystemknownasgematriainHebrewandisopsephyinGreek,inwhichlettersofthealphabetareassociatedwithnumbers.Severalsystemsarepossible:numberthelettersofthealphabetconsecutively,ornumberthem1–9,10–90,and100-900orwherevertheprocessstops(whichistheancientGreeknumbernotation).Thenthesumofthenumbersassociatedtothelettersinaperson’snameisthenumericalvalueofthatname.
InnumerableattemptshavebeenmadetodeducewhotheBeastwas.TheyincludetheAntichrist(writteninLatinintheaccusativecaseasAntichristum),theRomanCatholicChurch(identifiedwithoneofthePope’stitles:VicariusFiliiDei),andEllenGouldWhite,founderoftheSeventhDayAdventists.Howcome?Well,countingonlytheLatinnumeralsinhername,youget
addingto666.IfyouthinktheBeastwasAdolfHitler,youcan‘prove’itby
startingthenumberingatA=100:
Basically,youchooseyourhate-figure,basedonyourownpoliticalorreligiousviews.Thenyoutwistthenumbering,andifnecessarythename,tofit.
However,allofthesedeductionsmaybebasedonamisapprehension—otherthanthebeliefthatanyofitactuallymatters—becauseithasnowbecomeapparentthat666maybeanerror.Around200ADthepriestIrenaeusknewthatseveralearlymanuscriptsstatedadifferentnumber,butheattributedthistomistakesbythescribes,assertingthat666wasfoundin“allthemostapprovedandancientcopies”.Butin2005scholarsatOxfordUniversityusedcomputerimagingtechniquestoreadpreviouslyillegibleportionsoftheearliestknownversionofRevelation,number115ofthepapyrifoundattheancientsiteofOxyrhynchus.Thisdocument,whichdatestoaboutAD300,isthoughttobethemostdefinitiveversionofthetext.ItgivestheNumberoftheBeastas616.
Once,Twice,ThriceThesquarearray
192384576
useseachoftheninedigits1–9.Thesecondrow384istwicethefirstrow192,andthethirdrow576isthreetimesthefirstrow.
Therearethreeotherwaystodothis.Canyoufindthem?
Seepage265fortheanswer.
ConservationofLuck“AfriendofminewonsevenmillionontheLotto,”saidthechapnexttomeinthegym.“That’stheendofmychances.Youcan’twinifyouknowsomeonewhohas.”
ThereareasmanyurbanmythsabouttheUKNationalLotteryastherearelegsonamillipede,butI’dnotcomeacrossthisonebefore.Itsetmewondering:whydopeoplesoreadilybelievethiskindofthing?
Thinkaboutit.Inorderformyfriend’sbelieftobetrue,theLottomachinehastosomehowbeinfluencedbyhisnetworkoffriendsandacquaintances.Ithastoknowwhetheranyofthemhaswonbefore,andthentakestepstoavoidhisparticularchoiceofnumbers,whichmeansthatitalsohastoknowwhathehaschosen.Infact,allelevenLottomachinesusedintheUKlotterymustknowthis,becausetheoneusedeachweekisitselfchosenatrandom.
SinceaLottomachineisaninanimatemechanicaldevice,thisdoesn’tmakeagreatdealofsense.
Eachweek,thechanceofanyparticularsetofsixnumberswinningthejackpotis1in13,983,816.That’sbecausetherearethatmanypossiblecombinationsofnumbers,andeachisequallylikelytooccur.Ifnot,themachinewouldbebiased,anditisdesignedtoavoidthat.Yourchancesofwinningdependonlyonwhatyou’vechosenthisweek,notonwhatsomeoneyouknowoncedid.Thelikelyamountyouwillwindoesdependonotherpeople,however:ifyouhitthejackpotandotherschosethesamenumber,youallhavetosharetheprize.Butthat’snotwhatmyfriendwasworriedabout.
Thereasonswhysomeofusbelievethiskindofmythlieinhumanpsychologyratherthanprobabilitytheory.Onepossiblereasonisanunconsciousbeliefinmagic,heremanifestingitselfasluck.Ifyouthinkthatluckisarealthingthatpeoplepossess,anditimprovestheirchances,andifyouthinkthatthereisonlyacertainamountoflucktogoround,thenperhapsyourfortunatefriendhasusedupalltheluckinyourneighbourhood.Whichinthisinstanceseemstobeyoursocialnetwork.OMG!Canyoutweetyourluckaway?PutyourluckonFacebookforyourso-calledfriendstosteal?It’sanightmare!
Ormaybetheunderlyingideaislikethepersonwhotakesabombonboardanaircraftwhenevertheytravel,onthegroundsthatthechanceoftwobombsonthesameplaneisinfinitesimal.(Thefallacyisthatyouchosetobringitonboard.Thishasnoeffectonthechanceofsomeoneelsehavingdonethesame,unknowntoyou.)
It’struethatmostLotterywinnersdonothavefriendswhoarealsowinners.Soit’seasytodeducethatifyouwanttobeawinner,youshouldavoidhavingsuchfriends.Actually,mostLotterywinnerslackwinningfriendsforthesamereasonthatmostloserslackwinningfriends:thereareveryfewwinnersbutavastnumberoflosers.
Agreed,youhavetobeinittowinit.Anacquaintanceofminewonhalfamillionpounds,andwouldnothavebeenpleasedifI’dadvisedhernottobother,andthenherusualnumberscameup.
Knowingthattheoddsarefirmlyagainstme,andnotfindingtheallegedthrillofgamblingworththevirtualcertaintyofthrowingmyhard-earnedmoneydownthedrain,IneverbetontheLottery.ButIhave,overtheyears,beeninadvertentlybettingonalotteryofmyown:writingabestseller.Ihaven’twonthejackpot,butI’vedefinitelycomeoutahead.AfewyearsagotheauthorJ.K.Rowling(youknowwhatshe’swritten)becameBritain’sfirstself-madefemalebillionaire.That’sfivehundredtimesthesizeofatypicalLottojackpot.Andtherearealotfewerthan14millionwritersinBritain.
ForgettheLottery.Writeabook.
TheCaseoftheFace-DownAcesFromtheMemoirsofDrWatsup
Mydetectivefriendsuddenlystoppedfiringhisrevolveratthechimneybreast,havinginscribeddottedversionsofthelettersVIGTOintheplaster.“Whatisit,Watsup?”saidheinanirritatedtone.
Isurfacedfrommyreverie.“I’msorry,Soames.WasIdisturbingyou?”“Icouldseeyouthinking,Watsup.Thewayyoupurseyourlips,andtugat
yourearwhenyoubelievenooneiswatching.Itismostdistracting.Onebulletwentawry,andnowtheClookslikeaG.”
“Iwasthinkingaboutthatnewstagemagicchap,”Isaid.“Um—”“TheGreatWhodunni.”“That’sthecove,yes.Cleverblighter.Wenttoseehisshowlastweek.Did
themostamazingcardtrick,beenwonderingaboutiteversince.Firsthetookapackofcardsanddealtthetopsixteenfacedowninfourrowsoffour.Thenheturnedfourofthecardsfaceup.Hecalledforavolunteerfromtheaudience,soofcourseIputmyhandup,butforsomereasonhechoseanattractiveyoungladyinstead.NameofHelena…well,anywayhetoldhertorepeatedly‘foldup’thesquareofcards,likefoldingasheetofstampsalongtheperforations,untilitendedupasasinglepileofsixteencards.”
“Shewasanaccomplice,”Soamesmuttered.“It’selementary.”“Idon’tthinkso,Soames.Thatwouldn’thavehelped.Theaudiencedecided
wherethefoldsoccurred.Forinstance,thefirstfoldcouldbealonganyofthethreehorizontallinesbetweenthecards,orthethreeverticallines—buttheaudiencecalledoutwhich.”
“Theaudiencewasanaccomplice,then.”Icouldtellhewasgettingintooneofhismoods.“Ichoseoneofthefolds
myself,Soames.”Thegreatmannoddeddistractedly.“Thenperhapsthetrickwasgenuine.In
whichcase—ah,yes,theConundrumoftheConcealedCupcakesspringstomind…Tell,me,Watsup:whenthecardshadbeenfoldedintoasinglepile,wasHelenainstructedtospreadthemoutonthetable?Withoutturninganyover?”
“Yes.”“Anddiditmiraculouslytranspirethateithertwelveofthecardswereface
downandfourfaceup,orfourwerefacedownandtwelvefaceup?”“Yes.Theformer.Andtheface-upcardswere—”“Thefouraces.Whatelse?Thewholethingisutterlytransparent.”“Butitcouldhavebeentheotherwayround,Soames,”Iprotested.“InwhichcaseHelenawouldhavebeentoldtotakethefourface-down
cards,andturnthemovertoreveal…”“Ah.Thefouraces.Isee.Butevenso,it’sanamazingpieceofmagic.Think
ofallthedifferentplacestheacescouldbe,andallofthewaystheaudiencecouldhavechosentofoldthecards!”
“Anamazingpieceofchicanery,Watsup.”Iallowedmyastonishmenttoshow.“Youmean—hefixedtheaudience’s
votes?Cleverpsychologicaltrickery?”“No,Watsup:hefixedthecards.GetmethatpackMrsSoapsudskeeps
underthehatstanddownstairsforherbridgeevening,andIwilldemonstrate.”Ihastenedtocarryouthisinstructions.
WhenIreappeared,puffingslightlyfrommyexertions,forIwasoutofcondition,Soamestookthecards.Hesortedoutthefouraces,andslidthembackintothepack,apparentlyatrandom.Afterdealingoutfourrowsoffourcardseach,heturnedfourofthecardsover,likethis:
Whodunni’sinitialarrangement
Thenheinstructedmetofoldthepackintoapile,followingthesameinstructionsthatWhodunnihadgiventoHelena.Thisdone,Ispreadthecardsout,andloandbehold,fourweretheoppositewayuptotheothertwelve.Andthosefourwere…theaces!
“Soames,”Icried,“thatistrulythemostamazingcardtrickIhaveeverseen!Idonowrealisethatyoumusthavechosenwheretoplacethefouraces,but
evenso,thenumberofdifferentwaysIcouldhavechosentofoldthecardsishuge!”
Soamesreloadedhisgun.“MydearWatsup,howmanytimeshaveItoldyounottoleaptounwarrantedconclusions.”
“Buttherereallyarethousandsofways,Soames!”Thedetectivegaveacurtnod.“Thatwasn’ttheconclusionIhadinmind,
Watsup.Doyoureallythinkthechoiceoffoldsmakestheslightestdifference?”Istruckmybrowwiththepalmofmyhand.“Youmean…itdoesn’t?”But
inreply,Soamesmerelyresumedhisattackonthechimneybreast.
HowdoesWhodunni’strickwork?Seepage265fortheanswer.
ConfusedParentsOneofthestrangestnamesforamathematicianisthatofHermannCa¨sarHannibalSchubert(1848-1911)whopioneeredenumerativegeometry,whichcountshowmanylinesorcurvesdefinedbyalgebraicequationssatisfyparticularconditions.Presumablyhisparentsexpectedgreatthingsoftheirson,butcouldn’tworkoutwhosesidetheywereon.
HermannCa¨sarHannibalSchubert
JigsawParadoxThetwotrianglesappeartohavethesamearea,namely13×5/2=32.5.Butonehasaholeinitsedge,sothediagramprovesthat31.5=32.5.What(ifanything)iswrong?
Seepage267fortheanswer.
Jigsawparadox
TheCatflapofFearFromtheMemoirsofDrWatsup
Hoovesskatedonthemuddyroad.Thecabscreechedroundacorner,narrowlymissingabarrowloadedwithpotatoes.Thecabbiewipedhisbrowwithadirtyrag.
“Cor,guv!ForamomentthereIthoughtwe’d’adourchips!”*“Driveon,man!There’saguineainitforyouifyouproceedwithallundue
haste!”Wearrivedatourdestination.Ileapedfromthecab,flingingcoinsatthe
driver,andrushedpastastartledMrsSoapsuds,upthestairs,andintoSoames’slodgings.Withoutknocking.
“Soames!It’sterrible!”Ipanted.“My—”“Yourcatshavebeenstolen.”“Catnipped,Soames!”“Surelyyoumean‘catnapped’?”“No,theywereluredawayusingabundleofcatniptiedtoapieceofstring.”“Howdoyouknowthat?”“Thecatnipperleftitbehind.”Soamesgavemeasharpstare.“Unusual.Notlikehim.Notlikehimatall.”“Him?”“Yes.He’sback.”Iwenttothewindow.“Soheis.Butthisishardlythetimeforroasted
chestnuts,Soames.”“Watsup,areyououtofyourmind?”“Theoldmanwhorunstheroastedchestnutstallacrossthestreet,”I
explained.“Hewasn’tthereyesterday,buttodayheis.Iassumethatistowhomyouarereferring.”
“Youassume,”Soamessaidscathingly.“Donotassume,Watsup.Examinetheevidenceanddeduce.”Irealisedthathewasnotmerelyspeakingingeneralities.Theremustbesomethingspecificthathewishedmetodeduce.
IpridemyselfinbeingunusuallysensitivetoSoames’smoods,andaftersomereflectionIrecalledthatafewdaysagoIhadcomeuponhimassemblingasmallarsenalofpistols,rifles,andhandgrenades.Nowitstruckmethatperhapsallwasnotwell.
Iputthishypothesistohim,andhenodded.“Itisasthoughaghostfromthepasthasrisenfromthegraveandissuckingthelifeoutoftheassembledmultitudesofhumanity,”hesaid.
“Isit?”Iasked.“Whatis,Soames?”“Afoulanddangerousfiend,theWellingtonofcrime.”“Doyounotmean‘Napoleon’?Itwouldseemmoreapt.TheDukewas
entirely—”“Hewearsrubberboots,”Soamesexplained.“Withanextremelycommon
treadpattern,todisguisehisfootprints.Hewearsgloves,toleavenofingerprints.Heisamasterofdisguise.Hecomesandgoeswithoutimpedimentthroughlockeddoors.Hehastheearofeverypolitician,theeyesofalltheirwives,andlongbeforethatfateddaywhenourpathsfirstcrossedhehadafingerineveryillicitpieinEngland.ButwithsuperhumaneffortItrackedhimdown,securedconvincingevidence,andbrokeuphisnetworkofcriminalgangs.Hefledthecountry,andIfoolishlythoughtthatwastheendofhim.Butnow,Ifindthathewasmerelylyinglow.Hehasreturned,andhehasresumedhisnefariousactivities.Andnowithasbecomepersonal.”
“Ofwhomdoyouspeak?”“Why,Mogiarty!ProfessorJimMogiarty,abrilliantbutflawed
mathematicianwhoturnedtotheDarkSide.Hebeganasamerecatburglar,beforeturninghisevilattentionstomoreprofitablecommodities.Notonlywillhestealanythingnotnaileddown:hewillalsostealnails,hammer,andfloorboards.Hehasdoggedmycareersince—”
“Soames:howcanacatburglardoganything?”“LikeIsaid,heisamasterofdisguise,Watsup.Dolisten.”“Andhowhashemanifestedhimself?”“Extortion,theft,murder,andkidnap.Andnow:catnip.Mogiartyis
revertingtohisoldways.”Hisexpressionwentgrimwithdetermination.“Neverfear,Watsup.Wewillrescueyourpets—”Iglaredathim—“yourfurryfelinecompanions.Youhavemyword.”
Ifinallythoughttoaskavitalquestion.“Soames?Howdidyouknowmycatsweremissing?”Silentlyheshowedmeatornenvelope.Insidewasascrapofpaperandasoggycatnipmouse.
“That’sDysplasia’smouse!”Istifledamanlysob.“What’sthepieceofpaperabout?”
Heshowedittome.Itread:
CSNSGISTCSTEEVTAOOHAGIAIEITNRETET
“Itisabitofajumble,Soames,butIseethewordsSTEEVandHAGIA.Er…doyouknowanyonecalledStepheninConstantinople?”
“No,Watsup!Itisacode.Ihavedecipheredit.”“How?”“Iobservedthatthereare33letters.Whatdoesthatsuggesttoyou,
Watsup?”“Er—therewasn’tmuchroomonthepaper.”“Watsup:33isequalto3×11,aproductoftwoprimes.Iinstantlythought
ofMogiarty’smathematicalpast.Anditoccurredtometorearrangethelettersina3×11rectangle.Likeso.”
C S N S G I S T C S TE E V T A O O H A G IA I E I T N R E T E T
Hebeamedwithpride;Icouldnotunderstandwhy.Itwasstillgibberish.“Readdownthecolumnsinorder,Watsup!”“CEASEINVESTIGATIONSORTHECATSGETIT.Ohdear!”Iwas
tremblingnowfromheadtotoe.“WhyisMogiartydoingsuchaterriblethingtoinnocentcreatures?”
“Heissendingusamessage.”“Thatmuchisclear.”“No,Imeantitmetaphorically.”“Ah.Hashedemandedaransom?”“No.Ibelieveittobeatest.Isuspectthatthiscrimeismerelyatrialrunfor
moreoutrageousones.Heistoyingwithuslikeacatwithamouse.”Istifledanothersob.“Whatcanwedo?”“Thegameisafoot,andwemustgetaheadofthegame,soasnottobetaken
aback.Alreadymytrustedinformantshavelocatedyourcatsinaperfectlyordinarylookinghouse—ironically,inBarking.Actuallyitisequippedwithconcealedmantraps,steeldoors,bulletproofwindows,andalarmsofseveralkinds.Thereisnopossibilityofusmountingaclandestinebreak-in.”
Iputmyservicerevolverbackinmypocket.“Apity.”“However,Mogiartyhasmadeamistake.Thereisaboarded-upcatflap.We
maybeabletorestoreitsfunctionandenticeyourcatsthroughit.”“Yes!”Icried.“Ihaveit!Wecantemptthemwiththeirfavouritetreats.
Aneurysmlikesartichokes,Borborygmusiscrazyaboutbananabread,Cirrhosiscanneverresistacreambun,andDysplasia’sdownfallisdumplings!”
“Dumplings,”saidhe.“Nevermind.Alittlebrainwork,somecrucialinformation,andyousee?Weprogress.Wecanemploytheseitemstoenticethecatstocomeoutthroughthecatflap.”
“Ihavesubstantialstocksofthenecessarycomestiblesathome,”Itoldhim.“Iwillgetthem.”
“Thatwillindeedbeuseful,Watsup,whenthetimecomes.Butthereisaproblem.Wemustpresentthedelicaciesinthecorrectorder,becausethecatsmustnotbepermittedtofight.”
“Ofcourse.Itcouldcauseinjury.”“No,becauseMogiartyhasfilledthebasementwithhighexplosive,and
riggedittodetonateiftheanimalsfight.”“What!Why?”“Becausehehasreasontobelievethatanyattempttorescuethecreatures
willprecipitateafelinealtercation.Heisusingtheanimalsthemselvesasawarningsystem.Typically,heignoresthevileconsequencesofhisbloodymachinations.AsIsaid,heissendingusamessage:thathewillstopatnothing.”
“Isee.”“Yousee,Watsup,butyoudonotobserve.Observationbeginswith
enquiries,whichprovideacontextfordeduction.Inowenquire.Inwhichcircumstancesdoyourcatsfight?Beprecise,forsuccessorfailuredependsonit.”
“Onlywhentheyareindoors,”Ireplied,afterduereflection.“Thenthehousemaygoupatanymoment!”“No:mycatsareentirelypeaceableprovidedcertaincombinationscanbe
prevented.”Iwrotedownalistofconditions:
IfCirrhosisandAneurysmarebothindoors,theyfightunlessDysplasiaispresent.IfDysplasiaandBorborygmusarebothindoors,theyfightunlessAneurysmispresent.IfAneurysmandDysplasiaarebothindoors,theyfightunlessBorborygmusorCirrhosis(orboth)arepresent.IfCirrhosisandDysplasiaarebothindoors,theyfightunlessBorborygmusorAneurysm(orboth)arepresent.IfAneurysmorBorborygmusareindoorsalone,theywon’tgooutatall.
HowcanSoamesandWatsupenticethecatsoutwithoutcausinganexplosion?Onlyonecatatatimecanusethecatflap.Ignoretrivialmoveswhereacatgoesoutandissentstraightbackin.Ifnecessary,acatcanbeshovedbackthrough
thecatflapaspartoftheprocess.
Seepage267fortheanswer.
Footnote
*Thisisnotananachronism:JosephMalinopenedthefirstfishandchipshopinLondonin1860.ThedelicacywasintroducedtoBritaininthesixteenthcenturybyJewishrefugeesofSpanishandPortugueseorigin,underthenamepescatofrito:deep-friedfish.Thechipswereaddedlater.
PancakeNumbersHereisagenuinemathematicalmystery—asimpleproblemwhoseansweriscurrentlyaselusiveasthecriminalmastermindMogiarty.
Youaregivenastackofcircularpancakes,allofdifferentsizes.Yourtaskistorearrangetheminorder,fromthelargestatthebottomtothesmallestatthetop.Theonlychangeyouareallowedtomakeistoinsertaspatulaunderneathsomepancakeinthestack,anduseittopickupthepancakesaboveitandfliptheentirepileover.Youcanrepeatthisoperationasmanytimesasyoulike,choosingwheretoplacethespatulaasyouwish.
Here’sanexamplewithfourpancakes.Ittakesthreeflipstogettheminorder.
Flippingastack
Herearesomequestionsforyou.1Caneverystackoffourpancakesbeputinorderusingatmostthreeflips?2Ifnot,whatisthesmallestnumberofflipsthatwillputanystackoffourpancakesinorder?3DefinethenthpancakenumberPntobethesmallestnumberofflipsthatwillputanystackofnpancakesinorder.ProvethatPnisalwaysfinite.Thatis,everystackcanbeputintotherightorderusingafinitenumberofflips.4FindPnforn=1,2,3,4,5.I’vestoppedatn=5becausetherearealready120differentstackstoconsider,andtobehonestthat’sanawfullotofwork.
Seepage269fortheanswers,andwhatelseisknown.
TheSoupPlateTrickContinuingthecookerytheme,thereisacurioustrickthatyoucanperformusingasoupplate,orsimilarobject.Startbybalancingtheplateonyourhand,inthemannerofawaiterservingdinner.Nowexplainthatyouwillaccomplishtheamazingfeatoftwistingyourarmthroughacompleteturnwhilekeepingtheplatehorizontalthroughout.
Todothis,firstrotateyourarminwards,tuckingtheplateunderyourarmpit.Continuetomovetheplateinacircle,butnowraiseyourarmaboveyourhead.Everythingreturnsnaturallytothestartingposition,andtheplatedoesn’tfalloffeventhoughyouarenotgrippingit.
Youcanfindvideosofthe(soup)platetrickontheInternet,forexampleat
http://www.youtube.com/watch?v=Rzt_byhgujg
whereitisreferredtoastheBalinesecuptrick,inreferencetoBalinesedanceusingacupfullofliquidinsteadofaplate.AsimilarPhilippinedanceusingwineglasses(twoperdancer,oneineachhand)canbeseenonYouTubeat
http://www.youtube.com/watch?v=mOO_IQznZCQ
Thismayseemafairlytrivialmanoeuvre,butithasdeepmathematicalconnections.Inparticular,ithelpsparticlephysicistsunderstandoneofthecuriousfeaturesofthequantumpropertyknownasspin.Quantumparticlesdon’treallyspin,likeaballbeingtwirledonajuggler’sfinger,butthereisanumber,calledspin,whichinacertainsensehasasimilareffect.Spinscanbepositiveornegative,analogoustoclockwiseandanticlockwise.Someparticleshavewholenumberspins:thesearecalledbosons(rememberthediscoveryoftheHiggsboson?).Others,morebizarrely,havehalf-integerspinslike or .Thesearecalledfermions.
Thehalvesarisebecauseofaverystrangephenomenon.Ifyoutakeaparticlewithspin1(oranyinteger)androtateitinspacethrough360°,itendsupinthesamestate.Butifyoutakeaparticlewithspin androtateitinspacethrough360°,itendsupwithspin– .Youhavetorotateitthrough720°,twofullturns,togetthespinbackwhereitbegan.
Themathematicalpointhereisthatthereisa‘transformationgroup’calledSU(2),whichdescribesspinandactsbytransformingquantumstates,andadifferentgroupSO(3)thatdescribesrotationsinspace.Thesearecloselyrelated,butnotidentical:everyrotationinSO(3)correspondstotwodistinct
transformationsinSU(2),oneofthembeingminustheother.Thisiscalledadoublecovering.It’sasifSU(2)wrapsroundSO(3),butitgoesroundtwice.Abitlikewrappingarubberbandtwiceroundabroomstick.
PhysicistsillustratethisideausingtheDiracstringtrick,namedforthegreatquantumphysicistPaulDirac.Theideatakesmanyforms;aparticularlysimpleoneusesaribbonwithoneendfixedandtheotherattachedtoarotor,whichfloatsinmid-air.Theribbonisshapedlikeaquestionmark.Afterarotationof360°theribbonhasnotreturnedtoitsoriginalposition,buttothatpositionrotatedthrough180°.Asecondfullrotationoftherotorto720°doesnottwisttheribbon,butputsitbackwhereitstarted.Thewaytheribbonmovesisessentiallythesameasthatofthearmholdingthesoupplate,exceptthattheplatemovesaroundabit.Anastronautfloatinginzerogravitycoulddothesamemovementswithafixedplate,whilekeepinghisbodypointingthesamedirectionatalltimes.
TheDiracstringtrickusingaribbon.Numbersshowtheangleindegreesthroughwhichtherotorhasturned.
Acomputer-generatedmovieAironDiracStringsbyGeorgeFrancis,LouKauffmanandDanielSandin(graphicsbyChrisHartmanandJohnHart)at
http://www.evl.uic.edu/hypercomplex/html/dirac.html
showstherelationshipbetweentheDiracstringtrickandthePhilippinewinedance.
Thesameideacanbeusedtoconnectelectricalcurrenttoarotatingdevicesuchasawheel.Atfirstsightthere’saproblem:thewheelhastohoverunsupportedinmid-air,toallowtheribbontodisentangle.However,in1975D.A.Adamsdesignedandpatentedadevicethatusesgearstoallowtheribbontorotatecompletelyroundthewheelonallsides.It’stoocomplicatedtoexplainhere,butseeC.L.Stong,Theamateurscientist,ScientificAmerican(December1975)120–125.
MathematicalHaikuThehaikuisashortJapaneseverseform,traditionallycomprisingthreeseparatephrases(lines)thatuseatotalof17syllables.TheactualJapaneseworddoesn’tcorrespondexactlytotheEnglishconceptofasyllable,butthatworkswellenoughforhaikuinEnglish.Thestricttraditionalpatternusesfivesyllablesinthefirstandthirdphrasesandseveninthemiddleone.Asanexample,here’sahaikubyMatsuoBashō,(1644–1694),inwhichboththeoriginal(omitted)andthetranslationhavethatformat:
Attheageoldpondafrogleapsintowateradeepresonance.
Inthesedecadentmoderntimes,the5-7-5patternisoftenrelaxed,withvariationssuchas6-5-6beingpermitted.Infact,thetotalof17syllablescanalsobechanged.Themostimportantfeatureisnotthepreciseform,buttheemotionalcontent,whichrequirespresentingtwodistinctbutlinkedimages.
Thesimpleformatofthehaikuhasadefinitemathematical‘feel’,andthereareinnumerablemathematicalhaiku.Forexample:
DanielMathews
RulerandcompassDegreeoffieldextensionMustbepoweroftwo.
JonathanAlperin
BeautifultheoremThebasiclemmaisfalseRejectthepaper.
JonathanRosenberg
Colloquiumtime.Lightsout,somebody’ssnoring.Mathissuchhardwork.
Accidentalhaikuoccurwhenwritersunintentionallyproduceasentencein
haikuformat.Forexample:
Andinthewestwardsky,Isawacurvedpalelinelikeavastnewmoon.
inTheTimeMachinebyH.G.Wells.OnethatAngelaBrettnoticed(amongmanyothers)inthePrincetonCompaniontoMathematicsis
Iseveryevennumbergreaterthanfourthesumoftwooddprimes?
TimPostonandIdedicatedour1977CatastropheTheoryandItsApplicationswithahaiku:
ToChristopherZeemanAtwhosefeetwesitOnwhoseshoulderswestand.
TheCaseoftheCrypticCartwheelFromtheMemoirsofDrWatsup
Soameswasrifflingthroughastackofnewspapers,seekingacrimethatwouldexercisehistalentssufficientlytobeworthinvestigating.Atthatmoment,Ihappenedtoglanceoutofthewindowandsawafamiliarfigurealightingfromahansomcab.“Why,Soames!”Icried.“Itis—”
“InspectorRoulade.Hewillbeheretorequestourassistance.”Therecameaknockonthedoor.IopenedittoseeMrsSoapsudsandthe
Inspector.“Soames!I’vecomeabout—”“TheDowninghamkidnappingcase.Yes,itdoeshavesomefeaturesof
interest.”HepassedRouladethenewspaper.“Asensationalistreport,MrSoames.Ill-informedspeculationsaboutthe
likelyfateoftheEarlofDowningham,andthesizeoftheransombeingdemanded.”
“Predictabilityofthepress,”saidSoames.“Yes.Althoughinthisinstanceitplaysintoourhandsbynotrevealing
certainkeyfactsthatmighthelpustoidentify—”“Thecriminal.Suchastheabsenceofanyransomdemand.”“HowonEarth—?”“Iftherehadbeenademand,itwouldbynowbepublicknowledge.Itisnot.
Evidentlythisisnoordinarykidnapping.WeshouldproceedwithallhastetoDowninghamHall.Which,ifmymemoryfailsmenot—anditneverdoes—isonUppinghamDown.”
“ThereisatrainfromKing’sCrosstoUppinghaminelevenminutes’time,”saidI,havinganticipatedhisdecisionandpulledacopyofBradshawfromthebookcase.
“Ifwepaythecabbieaguinea,wecanjustcatchit!”criedSoames.“Wecandiscussthecasewhilewetravel.”
UponourarrivalatDowninghamHall,theDukeofSouthmoreland—whoaccordingtothetime-honouredrulesofthenobilitywasthefatheroftheEarlofDowningham,whotookoneofhisfather’slessertitles—greetedusinperson,andquicklyconductedustothesceneoftheabduction,amuddypaddockoutsideabarn.
“Mysonvanishedsometimeduringthenight,”hestated,visiblyshaken.Soamesproducedhismagnifyingglassandcrawledaroundinthemudfor
severalminutes.Fromtimetotimehemutteredtohimself.Hetookoutatapemeasureandmadesomemeasurementsinacornerofthebarn.Thenherosetohisfeet.
“IhavealmostalloftheevidenceIneed,”hesaid.“WemustreturntoLondontofindthelastmissingpiece.”LeavingabaffledDukestandinginhisowndoorway,nexttoanequallybaffledInspector,wedidjustthat.
“But,Soames—”Ibeganwhenwehadboardedthetrain.“Didyounotnoticetheimpressionmadebythewheels?”hechallengedme.“Wheels?”“Thepolicehadtrampledallovertheevidence,asusual,butafewtraces
remained.EnoughformetodeterminethattheEarldepartedinafarmyardcart,oneofwhosewheelsfittedtightlyagainsttheendofthebarnwhereitmeetsahighwall.Atraceofmudonthewalltellsmethataspotontherimis8inchesfromthegroundand9inchesfromtheendofthebarn.Ifwecandeducethediameterofthewheel,thecasemaybeclosetoitssolution.”
Dataforwheel
“Maybe?”“Thatdependsontheanswer.Wemustalsobearinmindthatnocartwheel
hasadiameteroflessthan20inches.Letmesee…Ah,yes,itisasIsuspected.”UponourarrivalatKing’sCrossstationhecalledforaBakerStreetIrreducible—therewasalwaysoneofthelittlescampsnearby—anddispatchedhimtosendatelegramtoRoulade.
“Whatdoesitsay?”
“IttellshimwherethemissingEarlcanbefound.”“But—”“IknowofonlyonefarmintheneighbourhoodofDowninghamHallthat
hasacartwithwheelsoftheexactdiameterthatIhavecalculated,whichisdistinctivelylarge.IamconvincedthattheEarllefttheHallvoluntarilyundercoverofdarkness,usingahumblecarttoavoidattractingattention.Hewillbeattheplacewherethecartishabituallykept.”
NextmorningMrsSoapsudsbroughtatelegramfromtheInspector:EARLOFDSAFEANDWELLCONGRATULATIONSROULADE.
“SowheredidtheEarlgo?”Iaskedeagerly.“That,Watsup,isasecretwhosedisclosurewoulddestroythereputationsof
severalofthemostreveredfamiliesinEurope.ButIcantellyouthesizeofthewheel.”
Whatwasthewheel’sdiameter?Seepage271fortheanswer.
TwobyTwoTherearethousandsofNoah’sarkcartoons.Myfavouritehasabiologicaltheme.Thelastfewpairsofanimals—elephants,giraffes,monkeys—arebeingloadeduptherampintotheark.Noahisgrubbingaroundonthegroundonhandsandknees.Hiswifeisleaningoverthesideoftheark,yelling“Noah!Forgettheotheramoeba!”
There’samathematicalNoah’sarkjoke,wellwornbutperfectlycrafted.Asthefloodrecedes,Noahletsalltheanimalsloose,tellingthemtogoforth
andmultiply.Afterayearorso,hedecidestocheckuponthem.Therearebabyelephants,rabbits,goats,crocodiles,giraffes,hippos,andcassowarieseverywhere.Butthenhecomesacrossalonepairofsnakes,lookingdejected.
“What’stheproblem?”asksNoah.“Can’tmultiply,”saysoneofthesnakes.(BearinmindthatNoahisasortof
DrDolittlefigureandcantalktotheanimals.)Apassingchimpanzeeoverhearstheconversation.“Cutdownsometrees,
Noah.”Noahispuzzledbutdoesasthechimpsays.Afewmonths
laterhevisitsthesnakes,andnowtherearelittlesnakeseverywhereandeveryoneishappy.
“OK,howdidthathappen?”Noahasksthesnakes.“We’readders.Wecanonlymultiplyusinglogs.”
TheV-shapedGooseMysteryMigratingflocksofbirdsoftenflyinaV-shapedformation.V-shapedskeinsofgeeseareespeciallyfamiliar,andtheyoftencontaindozensorevenhundredsofbirds.Whatmakesthemadoptthisshape?
Researchershavelongsuggestedthatthisformationsavesenergybyavoidingbirdsgettingcaughtupintheturbulentwakeofthoseinfront,andrecentexperimentalandtheoreticalstudieshaveconfirmedthisgeneralviewpoint.Butthistheoryreliesonthebirdsbeingabletosensetheaircurrentsandadjusttheirflightaccordingly,anduntilrecentlyit’snotbeenclearthattheycandothis.
Analternativeexplanationisthattheflockhasaleader—theoneinfront—andeveryoneelsefollowstheleader.Perhapstheleaderisthebestnavigator,theonewhoknowswheretogo.Perhapsit’sjustwhicheverbirdfindsitselfatthefront.
BirdsinV-formation,flyingfromrighttoleft.Mostly.Where’sthebirdattoprightgoing?There’salwaysone…
Beforeproceedingtotheanswer,weneedtounderstandafewbasicfeaturesofbirdflight.Insteadyflight,abirdflapsitswingsinarepetitivecycle,down-beatfollowedbyup-beat.Itgainsliftfromthedown-beatasvorticesofairspinofffromtheedgesofthewings,anditusestheup-beattoreturnthewingtoitsoriginalposition,sothatthecyclecanrepeat.Thelengthofthecycleiscalledtheperiod.
Supposethattwobirdsareflyingusingcyclesofthesameperiod,whichisprettymuchwhathappensinamigratingflock.Althoughtheymoveinthesameway,theyneednotmakethesamemotionsatthesametime.Forexample,when
onebirdisproducingadown-beat,theothermaybeonanup-beat.Therelationbetweentheirtimingiscalledtherelativephase,anditisthefractionofacyclebetweenonebirdstartingitsdown-beatandtheotherbirdstartingitsowndown-beat.
ThankstosomeremarkabledetectiveworkbyStevenPortugalandhisteam,wenowknowthattheenergy-savingtheoryiscorrect,andthatthebirdscanindeedsensetheinvisibleaircurrentswellenoughtocarryitout.Thebigproblemforexperimentalstudiesisthatthebirdsyouaretryingtoobserverapidlydisappearfromview,alongwithanyattachedequipment.
Enterthebaldibis.OncethereweresomanybaldibisesthattheancientEgyptiansuseda
stylisedpictureastheirhieroglyphforakh,meaning‘toshine’.Todayonlyafewhundredsurvive,mainlyinMorocco.SoacaptivebreedingprogrammehasbeensetupatazooinVienna.Alotofeffortgoesintoteachingthebirdstofollowthecorrectmigrationroutes.Thisisdonebytrainingthemtofollowamicrolightaircraft,whichisflownalongpartsoftheroute—butalsoreturnstobase,alongwiththebirds.
Portugalrealisedthatitwouldbepossibletomakeextensivemeasurementsofthepositionsofthebirds,andhowtheymovetheirwings,fromtheaircraft.Insteadofthebirdsdisappearingoverthehorizon,theystayclosetotheequipment.Whathisteamfoundwasamazingandelegant.Eachbirdpositionsitselfbehindandslightlytothesideoftheoneinfront,anditadjuststherelativephaseofitswingflapssothatitridesontheupdraughtcreatedbythevortexspunoffbythebirdinfront.Thesecondbirdmustnotonlygetitswingtipintotherightplace,whichisrelativelytiny,butitmustalsoadjustthephaseofitsflaptoexploittheupdraughtefficiently.
Wingtipplacementandphaseadjustment.Greycurves:vorticesspunofffromwingtips.Arrowsshowrotationofvortex.
Atfirstsighttheseconsiderationswouldalsopermitazigzagformation,inwhicheachbirdfliestoonesideoftheoneinfront,butnotformingasingleVshape.(Ithasachoiceofflyingtotheleftortotheright.)However,thefirstbirdtobreaktheVshape(saybyflyingtotherightofthebirdaheadratherthanontheoutsideoftheVtoitsleft)wouldbedirectlybehindthebirdtwoplacesinfrontofit.Theairtherewouldbeturbulent,disturbedbythebirddirectlyinfront,soitwouldbemuchhardertoextractliftbyplacingthewingtipcorrectly.ThisproblemisavoidedbyflyingontheoutsideoftheVwheretheairisundisturbed.
Itwouldalsobepossibleforthebirdstoformasinglediagonalline,likeonearmoftheV.Butthiswouldleaveroomforbirdstojointheotherarm,closertotheleader.However,itiscommonforonearmoftheVtobelongerthantheother.
Whynotflyinamorecomplexzigzagformation,likethisorsomethingsimilarbutmorewiggly?
Inexperimentswithibises,ittookquiteawhileforjuvenilebirdstolearnhowtopositionthemselves.Inpracticeafewbirdsmaygetitwrong,andtheVshapeisseldomperfect.Nonetheless,thedetailedexperimentsshowconclusivelythatibisesarecapableofsensingtheairflowwellenoughtopositionthemselvesinorclosetothemostenergy-efficientlocationrelativetothebirdinfront.
Seepage272forfurtherinformation.
EelishMnemonicsThereareinnumerablemnemonicsforπ(seepages39–40).Mnemonicsforthatotherfamousmathematicalconstant,thebaseofnaturallogarithms
e=2.718281828459045235360287471352662497757…
arerarer.Amongthemaretwothatgivetenfigures:
Todisruptaplayroomiscommonlyapracticeofchildren.Itenablesanumskulltomemoriseaquantityofnumerals.
There’salsoa40-digitself-referentialmnemonicdevisedbyZeevBarel[Amnemonicfore,MathematicsMagazine68(1995)253],whichyoushouldcomparewiththedecimalexpansionabove.Itusesanexclamationmark‘!’inquotestorepresent0,anditgoeslikethis:
WepresentamnemonictomemoriseaconstantsoexcitingthatEulerexclaimed:‘!’whenfirstitwasfound,yes,loudly‘!’.Mystudentsperhapswillcomputee,usepowerorTaylorseries,aneasysummationformula,obvious,clear,elegant.
The‘easysummationformula’is
goingonforever.Now!denotesthefactorial
n!=n×(n–1)×…×3×2×1
SinceπmnemonicsarewritteninPilish[page39],areemnemonicswritteninEelish?
AmazingSquaresThereareinfinitelymanynaturalnumbersthatcanbeexpressedassumsofthreesquaresintwodifferentways:a2+b2+c2=d2+e2+f2.Butmoreisachievable.Anamazingexampleis
1237892+5619452+6428642=2428682+7619432+3237872
Therelationshipispreservedifwerepeatedlydeletetheleftmostdigit:
237892+619452+428642=428682+619432+23787237892+19452+28642=28682+19432+378727892+9452+8642=8682+9432+7872892+452+642=682+432+87292+52+42=82+32+72
Itisalsopreservedifwerepeatedlydeletetherightmostdigit:
123782+561942+642862=242862+761942+32378212372+56192+64282=24282+76192+323721232+5612+6422=2422+7612+3232122+562+642=242+762+32212+52+62=22+72+32
orifwedeletedigitsfrombothends:
23782+61942+42862=42862+61942+23782372+192+282=282+192+372
ThismathematicalmysterywassenttomebyMoloyDeandNirmalyaChattopadhyay,whoalsoexplainedthesimplebutcleverideainvolved.CanyouemulateHemlockSoamesandferretoutthesecret?
Seepage273fortheanswer.
TheThirty-SevenMysteryFromtheMemoirsofDrWatsup
“Howcurious!”Iremarked,thinkingaloud.“Manythingsarecurious,Watsup,”saidSoames,whomIhadthoughtto
havefallenasleepinhischair.“Whichcuriositydoyouhaveinmind?”“Itookthenumber123andrepeateditsixtimes,”Iexplained.“Getting123123123123123123,”Soamessaiddismissively.“Ah,yes,butIhaven’tfinished.”“Youmultipliedthatby37,nodoubt,”saidthegreatdetective,onceagain
confoundingmyexpectationthatIcouldtellhimanythingthathedidnotknowalready.
“Yes!Idid!AndwhatIgotwas—no,Soames,pleasedonotinterrupt—theanswer
4555555555555555551
withagreatmanyrepetitionsofthedigit5.”“Andthatiscurious?”“Undoubtedly.Whileonesuchcalculationmightbemerecoincidence,
somethingsimilarhappensifIuse234or345or456insteadof123.Look!”AndIshowedhimmyarithmetic:
234234234234234234×37=8666666666666666658345345345345345345×37=12777777777777777765456456456456456456×37=16888888888888888872
“Notonlythat:ifIrepeat123or234or345or456adifferentnumberoftimes,andmultiplyby37,Iagainobtainagreatmanyrepetitionsofthesamedigit,exceptneartheends.”
“Iaminclinedtothink,”murmuredSoames,“thatthepatterns123,234,345,andsoon,areirrelevant.Haveyoutriedothernumbers?”
“Itried124,andthatdidn’twork.Look!”
124124124124124124×37=4592592592592592588
“Digitsrepeatinblocksofthree,butIdon’tfindthatsurprising,sincetheydothesameinthefirstnumber.”
“Didyoutry486?”“No—well,since124fails,Ireallydon’tthink…Oh,verywell.”Ireturned
tomynotebookandwrotedownthecalculation.“Howcurious!”Isaidagain,havingdiscoveredtheanswer:
486486486486486486×37=17999999999999999982
Inspired,Itriedvariousthree-digitnumbersatrandom,writingthemdownsixtimesinarowandmultiplyingby37.Sometimestheresultincludedmanyrepetitionsofthesamedigit,butmoreoften,itdidnot.IshowedSoamesmyworking,andconfessed:“Iamflummoxed.”
“Themysterywillnodoubtresolveitself,”Soamesreplied,“ifyouconsiderthenumber111.”
Iwrotedown
111111111111111111×37=4111111111111111107
andstaredatit.Aftertwentyminuteshadpassed,Soamesgotupandglancedovermyshoulder.Heshookhisheadinamusement.“No,no,Watsup!Iwasnotsuggestingyoutryyourmethodonthenumber111!”
“Oh.Iassumed—”“HowmanytimeshaveItoldyou,Watsup:donotassumeanything!
Althoughthemysteryseemstoinvolvethenumber37,thatissomethingofasideissue.Iwassuggestingthatyoushouldcontemplatehowthenumber111relatesto37.”
Seepage273fortheanswer.
AverageSpeedBecauseofheavytraffic,abusdrivesfromEdinburghtoLondon,adistanceof400miles,in10hours,aspeedof40mph.Itmakesthereturnjourneyin8hours,aspeedof50mph.Whatisitsaveragespeedforthewholejourney?
Theobviousansweris45mph,thearithmeticmeanof40and50,obtainedbyaddingthemanddividingby2.However,thebusperformsthetotalroundtripof800milesin18hours,anaveragespeedof800/18= mph.
Howcome?
Seepage275fortheanswer.
FourCluelessPseudokuThecluelesspuzzleonpage41wascreatedbyGerardButters,FrederickHenle,JamesHenle,andColleenMcGaughey.ItisavariantofsudokuwhichIliketocallcluelesspseudoku.Herearefourmorecluelesspseudokumysteriestosolve.Therulesare:
Eachrowandeachcolumnmustcontaineachnumber1,2,3,…,nonceonly,wherenisthesizeofthesquare.Thenumbersineachoftheregionsoutlinedinthickblacklinesmusthavethesametotal.I’vewrittenitabovethepuzzletosaveyouthebotherofworkingitout.Eachsolutionisuniqueexceptforthefourthpuzzle,wheretherearetwosolutions,symmetricallyrelated.
Fourcluelesspseudokumysteries
Seepage276fortheanswersandfurtherreading.
SumsofCubesTriangularnumbers1,3,6,10,15,andsoon,aredefinedbyaddingconsecutivenumberstogether,startingfrom1:
1=11+2=31+2+3=61+2+3+4=101+2+3+4+5=15
andsoon.Thereisaformula
1+2+3+…+n=n(n+1)/2
andonewaytoproveitistowritethesumtwice,likethis:
1+2+3+4+55+4+3+2+1
andobservethatthenumbersinverticalcolumnsalladdtothesamething,here6.Sotwicethesumis6×5=30,andthesumis15.Ifyoudidthiswiththenumbersfrom1to100,itwouldworkinmuchthesameway:therewouldbe100columnseachaddingto101,sothesumofthefirst100numbersmustbehalfof100×101,whichis5,050.Moregenerally,ifweaddthefirstnnumberswegethalfofn(n+1),andthat’stheformula.
There’saformulaforsumsofsquares,butit’sabitmorecomplicated:
1+4+9+…+n2=n(n+1)(2n+1)/6
Butwhathappensforcubesisverystriking:
13=113+23=913+23+33=3613+23+33+43=10013+23+33+43+53=225
Theresultsarethesquaresofthecorrespondingtriangularnumbers.Whyshouldsumsofcubesgivesquares?Wecanfindtheformulaandprove
itthatway,butthere’saveryneatpictorialproofthat
13+23+33+…+n3=(1+2+3+…+n)2
withoutusinganyformulas.
Visualisingsumsofcubes
Thepictureshows1squareofside1,2ofside2(makinga2×2×2cube),3ofside3(a3×3×3cube),andsoon.Sothetotalareaisthesumofconsecutivecubes.Readingalongthetopedgewefind1+2+3+4+5,thesumofconsecutivenumbers.Buttheareaofasquareisthesquareofthelengthofitsside.Done!
Ifyouwantaformula,weknowthat(1+2+3+…+n)=(n(n+1)/2,sosquaringgives13+23+33+…+n3=n2(n+1)2/4.
ThePuzzleofthePurloinedPapersFromtheMemoirsofDrWatsup
Soamespassedmeanenvelope,andhelduptheletterthatithadenclosed.“Atestofyourpowersofobservation,Watsup.Whodoyouthinksentme
this?”Iheldittothelight,lookedatthepostmarkandstamp,sniffed,examinedthe
gluewhereithadbeensealed.“Thesenderisawoman,”Isaid.“Unmarriedbutnotyetcondemnedtospinsterhood,andactivelyseekingahusband.Sheisfrightened,butcourageous.”Ipaused,andfurtherinspirationstruck.“Herfinancesarestrained,butnotyetdisastrous.”
“Verygood,”saidhe.“Iseeyouhaveabsorbedsomeofmymethods.”“Idomybest,”saidI.“Explainwhatledyoutothesedeductions.”Iorderedmythoughts.“Theenvelopeispink,anditbearsdistincttracesof
perfume.NuitsdePlaisir,ifIamnotmistaken,formyfriendBeatrixoftenwearsthesame.Itistoodaringforamarriedwoman,notdaringenoughforayoungone.Thatshewearsperfumeatallimpliessheisactivelyseekingmaleattention.Tracesofcosmeticsontheflapconfirmthis.Butthegluehadbeenlickedonlypartially,suggestingthathermouthwasdrywhenshesealedtheenvelope,andadrymouthisasignoffear.Sincesheneverthelesscompletedthetaskandpostedthelettertoyou,sheisstillabletofunctionrationally,underseverestress,asignofcourage.
“Finally,thestampshowssignsofhavingbeensteamedoffanotherenvelopeandbeenreused—abentcorner,tracesofapreviouspostmark.Thisindicatesafrugalattitude.However,shecanaffordperfume,sosheisnotyetonpoverty’sdoorstep.”
Henoddedthoughtfully,andImentallypreenedmyself.“Thereareafewsignsyoumissed,”hesaidquietly,“whichputthematterin
afreshlight.Theshapeandsizeoftheenveloperevealitasgovernmentissue,nottobefoundinanyhigh-streetstationer’s.Ireferyoutomymonographonstationeryanditscharacteristicdimensions.Theinkusedtowritetheaddressisastrangeshadeofdarkbrown,unavailablecommerciallybutprovidedinbulktocertaindepartmentsinWhitehall.”
“Ah!Thenhercurrentbeauisacivilservantandsheborrowedboth
envelopeandinkfromhim.”“Asensibletheory,”saidhe.“Totallyincorrect,ofcourse,buteminently
sensibleandconsistentwithmuchoftheevidence.However,infactthisletterisfrommybrotherSpycraft.”
Iwasshockedrigid.“Youhaveabrother?”Soameshadnevertalkedofhisfamily.
“Oh,haveInotmentionedhim?Howveryremissofme.”“Howdoyouknowhewrotetheletter?”“Hesignedhisnametoit.”“Oh.Butwhatoftheotherclues?”“Spycraft’slittlejoke.Butwemustmakehaste,forwearetomeethim
forthwithattheDiophantusClub.Giveanurchinsixpencetocallahansom,andIwilltellyoumoreaswemakeourwaythere.”
AsweclatteredalongPortlandPlace,heexplainedthathisbrotherwasaretiredexpertonprimenumbers,whodidoccasionalfreelanceworkforHerMajesty’sGovernment.Herefusedtobedrawnonthenatureofthework,sayingonlythatitwashighlyconfidentialandpoliticallysensitive.
UponarrivingattheDiophantusClub,wewereusheredintotheVisitor’sLounge,whereagentlemanwaswaitinginacomfortablearmchair.Myimmediateimpressionwasoneoflanguidcorpulence,butitconcealedasharpnessofmindandalertnessofbodythatbeliedthisinitialassessment.
Soamesintroducedus.“Youoftenfindmyowndeductiveabilitiesastonishing,Watsup,”hesaid,
“butSpycraftputsmetoshame.”“Thereisoneareawhereyourabilitiesexceedmyown,”hisbrother
contradicted.“Namely,logicalconundrumsinwhichthepreciseconditionsarefluid.IfindthatIhavenobasisfromwhichtoattackthequestion.Whencemynote.”
“ItakeitthatyouhavenoobjectiontoDrWatsupbeingtoldall?”“HisservicerecordinAl-Jebraistanisimpeccable.Hemustbeswornto
secrecy,buthiswordwillbesufficient.”Soamesgavehisbrotherasharplook.“It’snotlikeyoutoacceptsomeone’s
word.”“ItwillbesufficientwhenIinformhimoftheconsequencesofbreakingit.”Idulyswore,andwegotdowntobusiness.“Animportantdocumentwasaccidentallymislaid,andthenstolen,”Spycraft
said.“ItisessentialtothesecurityoftheBritishEmpirethatitberecoveredwithoutdelay.Ifitgetsintothehandsofourenemies,careerswillberuinedandpartsoftheEmpiremayfall.Fortunately,alocalconstablecaughtaglimpseof
thethief,enoughtonarrowitdowntopreciselyoneoffourmen.”“Pettythieves?”“No,allfouraregentlemenofhighrepute.AdmiralArbuthnot,Bishop
Burlington,CaptainCharlesworth,andDoctorDashingham.”Soamessatboltupright.“Mogiartyhasahandinthis,then.”Notfollowinghisreasoning,Iaskedhimtoexplain.“Allfourarespies,Watsup.WorkingforMogiarty.”“Then…Spycraftmustbeengagedincounter-espionage!”Icried.“Yes.”Heglancedathisbrother.“Butyoudidnothearthatfromme.”“Havethesetraitorsbeenquestioned?”Iasked.Spycrafthandedmeadossier,andIreaditaloudforSoames’sbenefit.
“UnderinterrogationArbuthnotsaid‘Burlingtondidit.’Burlingtonsaid‘Arbuthnotislying.’Charlesworthsaid‘ItwasnotI.’Dashinghamsaid‘Arbuthnotdidit.’Thatisall.”
“Notquiteall.Weknowfromanothersourcethatexactlyoneofthemwastellingthetruth.”
“YouhaveaninformerinMogiarty’sinnercircle,Spycraft?”“Wehadaninformer,Hemlock.Hewasgarrottedwithhisownnecktie
beforehecouldtellustheactualname.Verysad—itwasanOldEtoniantie,totallyruined.However,allisnotlost.Ifwecandeducewhowasthethief,wecanobtainasearchwarrantandrecoverthedocument.Allfourmenarebeingwatched;theywillhavenoopportunitytopassthedocumenttoMogiarty.Butourhandsaretied;wemuststicktotheletterofthelaw.Moreover,ifweraidthewrongpremises,Mogiarty’slawyerswillpublicisethemistakeandcauseirreparabledamage.”
Whichmanwasthethief?Seepage276fortheanswer.
MasterofAllHeSurveysAfarmerwantedtoencloseaslargeanareaoffieldaspossible,usingtheshortestpossiblefence.Perhapsunwisely,hecalledthelocaluniversity,whosentanengineer,aphysicist,andamathematiciantoadvisehim.
Theengineerbuiltacircularfence,sayingitwasthemostefficientshape.Thephysicistbuiltastraightlinesolongthatyoucouldn’t
seetheends,andtoldthefarmerthattoallintentsandpurposesitwentrightroundtheEarth,sohehadfencedinhalftheplanet.
Themathematicianbuiltatinycircularfencearoundhimselfandsaid“Ideclaremyselftobeontheoutside.”
AnotherNumberCuriosity1×8+1=912×8+2=98123×8+3=9871234×8+4=987612345×8+5=98765
So,allyoubuddingHemlockSoameses:whatcomesnextandwhendoesthepatternstop?
Seepage278fortheanswers.
TheOpaqueSquareProblemSpeakingoffences:whatistheshortestfencethatwillblockeverylineofsightacrossasquarefield?Thatis,afencethatmeetseverystraightlinethatintersectsthefield.ThisistheOpaqueSquareProblem:thenameindicatesthatyoucan’tseethroughit.ThequestiongoesbacktoStefanMazurkiewiczin1916,whoaskeditforanyshape,notjustasquare.Itremainsbaffling,butsomeprogresshasbeenmade.
Supposethesideofthefieldisoneunit.Thenafenceroundallfoursideswouldcertainlywork,andthelengthwouldbe4.However,wecouldcutoutoneofthesidesandstillhaveanopaquefence,reducingtheanswerto3.Thisistheshortestfenceformedbyasinglepolygonalline.Butifweallowfencesusingseverallines,ashorterpossibilityquicklyspringstomind:thetwodiagonalsofthefield,totallength ,approximately.
Canwedobetterstill?Onegeneralfactisclear:anopaquefencecontainedentirelywithinthefieldmustincludeallfourcornersofthesquare.Ifsomecornerwerenotincluded,therewouldbealineintersectingthesquareonlyatthatcorner(cuttingdiagonallythroughitfromoutside)andthatwouldmissthefence.Evenonesuchlinebreakstheconditionsoftheproblem.
Anyfencethatincludesallfourcornersandjoinsthemtogethermustbeopaque,becauseanylinethatintersectsthesquaremusteitherpassthroughacornerorseparatetwoofthem.Thenanyconnectingfencemustcrossthatline.Isthepairofdiagonalstheshortestsuchfence?No.Theshortestfencethatconnectsallfourcorners,calledaSteinertree,haslength =2.732,approximately.Thelinesmeetatanglesof120°.
Itturnsoutthateventhisfenceisnottheshortestopaqueone.Thereisadisconnectedfence,inwhichonepartblockslinesofsightthroughthegap,oflength .Itiswidelybelieved,butnotyetproved,thatthisistheshortestopaquefence.BerndKawohlhasprovedthatthisistheshortestfencethathasexactlytwoconnectedpieces.OneistheSteinertreelinkingthreecorners,threelinesthatmeetat120°theotheristheshorteststraightlinebetweenthecentreandthefourthcorner.
Opaquefencesforasquare.Fromlefttoright:lengthsare4,3,2.828,2.732,and2.639.
Wedon’tevenknowforsurethatthereisashortestopaquefence.Orthatifthereisone,itmustbeentirelywithinthesquare.VanceFaberandJanMycielskihaveprovedthatforanygivenfinitenumberofpieces,atleastoneshortestopaquefenceexists.(Forallweknow,theremightbeseveral.)Thetechnicalproblemhere,whichiscurrentlyunsolved,isthepossibilitythatthemorepiecesyoupermit,theshorterthefencecanbe.Itwouldthenbepossibletofindaseriesoffenceswithever-shorterlengths,butnofencethatisshorterthanallofthese.Alternatively,afencecomposedofinfinitelymanydisconnectedpiecesmightbetheshortest.
OpaquePolygonsandCirclesAstandardmathematician’strick,whenyoucan’tsolveaproblem,istogeneraliseit:considerarangeofsimilarbutmorecomplicatedproblems.Thismightseemastupididea:howcanmakingthequestionharderhelpyoutosolveit?Butthemoreexamplesyouhavetothinkabout,thebetteryourchancesofspottingsomeinterestingcommonfeaturethatcrackstheproblem.Itdoesn’talwayswork,andsofarithasn’tdonehere,butoccasionallyitdoes.
OnewaytogeneralisetheOpaqueSquareProblemistochangetheshape.Replacethesquarebyarectangle,orapolygonwithmoresides,acircle,anellipse—thepossibilitiesareendless.
Mathematicianshavemainlyconcentratedontwogeneralisations:regularpolygonsandcircles.TheshortestknownopaquefencefortheequilateraltriangleisaSteinertreejoiningeachcornertothecentrebyastraightline.Thereisageneralconstructionthatgivestheshortestknownopaquefencesforregularpolygonswithanoddnumberofsides,andasimilarbutdifferentoneforanevennumberofsides.
Shortestknownopaquefencesforregularpolygons.Fromlefttoright:equilateraltriangle;odd-sidedregularpolygon;even-sidedregularpolygon.
Whataboutanopaquecircle?Ifthefencehastostaywithintheshape,theobviousansweristheperimeterofthecircle.Ifit’saunitcircle,thishaslength2π=6.282.Ifpartoftheperimeterismissing,youneedextrabitsoffenceinsidethecircletoblockpathsthatcrossthismissingsegment,anditgetscomplicated.Intuitively,acirclecanbethoughtofasaregularpolygonwithinfinitelymanyinfinitelyshortsides.Basedonthisidea,Kawohlhasprovedthataconstructionlikethatforregularpolygons,butusinganinfinitenumberofpieces,givesanopaquefenceoftotallengthπ+2=5.141,whichissmallerthan2π.ButthereisashorterU-shapedopaquefenceifsomeofitcanlieoutsidethecircle.Thisalsohaslengthπ+2,andisconjecturedtobetheshortestpossible,andithasbeenprovedtobesoforfencesformedbyasinglecurvewithnobranchpoints.
Opaquefencesforacircle.Left:Obviousbutnotthebest.Right:Ashorteropaquefencethatgoesoutsidethecircle.
Theproblemhasalsobeenextendedtothreedimensions:nowthefencehastobeasurface,orsomethingmorecomplicated.Thebestknownopaquefenceforacubeisformedfromseveralcurvedpieces.
Thebestknownopaquefenceforacube.
πr2?No,pieareround.Chocolatearesquared.
TheSignofOneFromtheMemoirsofDrWatsup
“Soames!Here’saprettypuzzle.Itmightinterestyou.”HemlockSoamesputdownhisclarinet,uponwhichhehadbeenplayinga
Bolivianfuneraldirge.“Idoubtit,Watsup.”Hehadbeeninthismelancholymoodforsomeweeks,andIwasdeterminedtokickhimoutofit.
“Theproblemistoexpresstheintegers1,2,3,andsoon,usingatmost—”“Four4’s,”saidSoames.“Iknowitwell,Watsup.*”Idecidednottolethislackofenthusiasmdauntme.“Withbasicarithmetical
symbolsitispossibletoreach22.Squarerootsincreasethislimitto30.Factorialsraiseitto112;powersto156—”
“Andsubfactorialsto877,”Soamesfinished.“Itisanoldpuzzle,andonethathasbeenwrungdry.”
“What’sasubfactorial,Soames?”Iasked,buthehadalreadyburiedhisnoseinyesterday’sDailyWail.
Afteramoment,hereappeared.“Mindyou,therearemanypossiblevariations.Theuseof4allowsconsiderablefreedom,andseveralusefulnumberscanbecreatedusingjustone4.Suchas =2and4!=24.”
“Whatdoestheexclamationmarkmean,”Iasked.“Factorial.Forinstance,4!=4×3×2×1,andsoon.Which,asIsaid,is
24.”“Oh.”“Theseextranumberscomefreeofcharge,andsomakethepuzzleeasier.
ButIwonder…”Hisvoicetrailedoff.
“Wonderwhat,Soames?”“Iwonderhowfaronecangetusingfourones.”SilentlyIrejoiced,forhisinterestwasclearlypiqued.“Yes,Isee,”Isaid.
“Now =1and1!=1,sononewnumbersarise‘forfree’.Whichmakestheproblemharder,andperhapsmoreworthyofourattention.”
Hegrunted,andIhastenedtopresshomemyslightadvantage.ThebestwaytointerestSoamesinaproblemwastoattemptit,andfail.
“Iseethat
1=1×1×1×1
and
2=(1+1)×1×13=(1+1+1)×14=1+1+1+1
Butanexpressionfor5escapesme.”Soamesraisedoneeyebrow.“Youmightconsider
5=(1/.1)/(1+1)
wherethedotisthedecimalpoint.”“Oh,that’sratherclever!”Icried,butSoamesmerelysnorted.“Sowhat
about6?”Icontinued.“Icanseehowtodoitusingfactorials:
6=(1+1+1)!×1
Ireallyonlyneedthree1’s,butanysparescanalwaysbegotridofbymultiplyingby1.”
“Elementary,”hemuttered.“Haveyouconsidered
Watsup?Or,forthatmatter,
ifyouinsistonemployingfactorials.Youmaymultiplyby1×1or1/1,oradd1–1,touseallfour1’s,ofcourse.”
Istaredattheformula.“Irecognisethedecimalpoint,Soames,butwhatistheextra ?”
“Recurring,”Soamesrepliedwearily.“Noughtpointonerecurringisequalto0.111111…goingonforever.Ofcoursetheinitialzeromaybeomitted.Theinfiniterecurringdecimalequals1/9exactly.Dividingthatinto1gives9,whosesquarerootis3—”
“Andthen3+3=6,”Iyelledexcitedly.“And,ofcourse,
7=(1+1+1)!+1
iseasywithoutsquareroots.But8isakettleofwormsofadifferentcolour—”“Dopayattention,”saidSoames.
8=1/. –1×19=1/. +1–1
“Ah!Yes!Andthen
10=1/. +1×111=1/. +1+1
and…”“Youareusingup1’sprodigiously,”saidSoames.“Itisbesttosavethem
forlater.”Andhewrotedown
10=1/.111=11
adding,“Notetheabsenceofthe‘recurring’symbol,Watsup.Thistimeitisjusttheordinarydecimal.1.Oh,andyoumustmultiplybothby1×1touseupthespare1’s,ordosoinoneoftheotherwaysIremarkedupon.Butlater,youcanomitthosetwo1’sandputthemtogoodpurpose.”
“Yes!Youmean,like
andsoon?”AflickerofasmilecrossedSoames’scountenance.“Youhaveit,Watsup!”“Butwhatabout15?”Iasked.“Trivial,”hesighed,andwrotedown
TowhichItriumphantlyappended
andSoamesnoddedapprovingly.“Nowitstartstobecomeinteresting,”heremarked.“Whatof23,Iwonder?”
“I’vegotit,Soames!”Icried:
Bearinginmind,”Iclarified,“that4!=24,asyousosagelyremarked.Thisisfun,Soames!ThoughforthelifeofmeIcan’tmanagetoexpress26.”
“Well…”hebegan,andpaused.“Stumped,you,hasit?”“Notintheleast.Iwasmerelywonderingwhetheritisnecessaryto
introduceanewsymbol.Itwillcertainlymakelifeeasier.Watsup,areyouawareofthefloorandceilingfunctions?”
Mygazeinadvertentlywenttomyfeet,andthenupabovemyhead,butnoinspirationstruck.
“Iseeyoudonot,”saidSoames.HowdoesheknowwhatI’mthinking?Ithought.It’s—
“Uncanny…yes,isitnot?Ireadyoulikeabook,Watsup.PossiblyMotherGoose.Now,thesefunctionsare
[x]=thegreatestintegerlessthanorequaltox(floor)[x]=thesmallestintegergreaterthanorequaltox(ceiling)
andyouwillfindthemindispensableinallpuzzlesofthissort.”“Excellent,Soames.ThoughIadmit,Ifailtosee…““Theidea,Watsup,isthatbytheirmeanswecanexpressusefulsmall
numbersusingonlytwo1’s.Forexample,
isanotherwaytorepresent3withtwo1’s,and
isnew.”Seeingmybewilderment,headded:“Youappreciatethat,whoseflooris3andceilingis4.”
“Yes…”Isaiddoubtfully.“Thenweadvance,because
Nottomentionvariousalternatives.”Thousandsofincoherentthoughtssurgedthroughmybrain.Onestoodout.
“Why,Soames,I’vejustrealisedthat
because ,whoseceilingis5.SoIcannowmake29and30!”BywhichImeant30,notfactorial30,youunderstand.Punctuationissuchapain.
WatsupandSoamesinvestigatedthepuzzlemuchfurther,andwewillseewhattheydiscoveredlater.Butbeforecontinuingthestory,youmightliketoseehowfaryoucangetonyourown.Startingwith31.
TheSignofOnecontinuesonpage105.
Footnote
*W.W.RouseBall,MathematicalRecreationsandEssays(11thedition),Macmillan,London1939.
ProgressonPrimeGapsRecallthatawholenumberiscompositeifitcanbeobtainedbymultiplyingtwosmallerwholenumberstogether,andprimeifitcan’tbeobtainedbymultiplyingtwosmallerwholenumberstogetheranditisgreaterthan1.Thenumber1isexceptional:afewcenturiesagoitwasconsideredtobeprime,butthisconventionstopsprimefactorsbeingunique.Forexample,6=2×3=1×2×3=1×1×2×3,andsoon.Nowadays,forthisreasonandothers,1isconsideredtobespecial.Itisneitherprimenorcomposite,butaunit:awholenumberxsuchthat1/xisalsoawholenumber.Indeed,itistheonlypositiveunit.
Thefirstfewprimesare
23571113171923293137
Thereareinfinitelymanyofthem,scatteredinanirregularmannerthroughoutthewholenumbers.Theprimenumbershavelongbeenahugesourceofinspiration,andmanyoftheirmysterieshavebeensolvedovertheyears.Butmanyothersremainasopaqueastheyhaveeverbeen.
In2013numbertheoristsmadesuddenandunexpectedprogressontwoofthegreatmysteriesaboutprimenumbers.Thefirstconcernedthegapsbetweensuccessiveprimes,andI’lldescribethatnow.Thesecondfollowsimmediatelyafter.
Allprimesotherthan2areodd(sinceallevennumbersaremultiplesof2),soitisnotpossiblefortwoconsecutivenumbersexcept(2,3)bothtobeprime.However,itispossiblefortwonumbersthatdifferby2tobeprime:examplesare(3,5),(5,7),(11,13),(17,19),anditiseasytofindmore.Suchpairsarecalledtwinprimes.
Ithaslongbeenconjecturedthatthereareinfinitelymanytwinprimepairs,butthishasnotyetbeenproved.Untilrecentlyprogressonthisquestionwasminimal,butin2013YitangZhangstunnedthemathematicalworldbyannouncingthathecouldprovethatinfinitelymanypairsofprimesdifferbyatmost70million.HispaperhassincebeenacceptedforpublicationbytheleadingpuremathematicsjournalAnnalsofMathematics.Thismaysoundfeeblecomparedtothetwinprimeconjecture,butitwasthefirsttimeanyonehadshownthatinfinitelymanyprimesdifferbysomefixedamount.If70millioncouldsomehowbereducedto2,thiswouldresolvethetwinprimesconjecture.
Today’smathematiciansincreasinglyusetheInternettojoinforcesonproblems,andTerenceTaoorchestratedacollaborativeefforttoreducethefigureof70milliontosomethingmuchsmaller.HedidthiswithintheframeworkofthePolymathproject,asystemsetuptofacilitatethiskindofwork.AsmathematiciansgainedabetterunderstandingofZhang’smethods,thenumbertumbled.JamesMaynardreducedthefigureof70millionto600(indeedto12ifanotherconjecturecalledtheElliott–HalberstamConjectureisassumed,Seepage278).Bytheendof2013newideasbyMaynardhadreduceditto270.
Notyet2,butalotcloserthan70million.
TheOddGoldbachConjectureThesecondprimemysterytoberesolved(probably!)goesbackto1742,whentheGermanamateurmathematicianChristianGoldbachwrotealettertoLeonhardEulercontainingseveralobservationsaboutprimes.Onewas:“Everyintegergreaterthan2canbewrittenasthesumofthreeprimes.”EulerrecalledapreviousconversationinwhichGoldbachhadmadearelatedconjecture:“Everyevenintegeristhesumoftwoprimes.”
Withtheconventionthenprevailing,that1isprime,thisstatementimpliesthefirstone,becauseanynumbercanbewrittenaseithern+1orn+2whereniseven.Ifnisthesumoftwoprimes,theoriginalnumberisthesumofthreeprimes.Eulersaid“Iregard[thesecondstatement]asacompletelycertaintheorem,althoughIcannotproveit.”Thatprettymuchsumsupitsstatustoday.
Goldbach’slettertoEuler,statingthatifanumberisasumoftwoprimesthenitisasumofanynumberofprimes(uptothesizeofthenumberconcerned).Inthemarginistheconjecturethateverynumbergreaterthan2isasumofthreeprimes.Goldbachdefined1tobeprime,whichisnotthemodernconvention.
However,wenolongerdeem1tobeprime,asdiscussedonpage90.So
nowadayswesplitGoldbach’sconjecturesintotwodifferentones:
TheevenGoldbachconjecturestates:
Everyevenintegergreaterthan2isthesumoftwoprimes.
TheoddGoldbachconjectureis:
Everyoddintegergreaterthan5isthesumofthreeprimes.
Theevenconjectureimpliestheoddone,butnotconversely.Overtheyears,variousmathematiciansmadeprogressonthesequestions.
PerhapsthestrongestresultontheevenGoldbachconjectureisthatofChenJing-Run,whoprovedin1973thateverysufficientlylargeevenintegeristhesumofaprimeandasemiprime(eitheraprimeoraproductoftwoprimes).
In1995theFrenchmathematicianOlivierRamaréprovedthateveryevennumberisasumofatmostsixprimes,andeveryoddnumberisasumofatmostsevenprimes.TherewasagrowingbeliefamongtheexpertsthattheoddGoldbachconjecturewaswithinreach,andtheywereright:in2013HaraldHelfgottclaimedaproofusingrelatedmethods.Hisresultisstillbeingcheckedbyexperts,butseemstobeholdingupwellunderscrutiny.Itimpliesthateveryevennumberisthesumofatmostfourprimes(ifniseventhenn–3isodd,henceasumofthreeprimesq+r+s,son=3+q+r+sisasumoffourprimes).ThisisclosetotheevenGoldbachconjecture,butitseemsunlikelythatthiscanbeprovedinfullusingcurrentmethods.Sothere’sstillsomewaytogo.
PrimeNumberMysteriesMathematicshasmysteriesofitsown,andmathematicianswhotrytoresolvethemarelikedetectives.Theyseekoutclues,makelogicaldeductions,andlookforproofthattheyarecorrect.AsinSoames’scases,themostimportantstepisknowinghowtogetstarted—whatlineofthinkingmightleadtoprogress.Inmanycases,westilldon’tknow.Thismaysoundlikeanadmissionofignorance,andindeeditis.Butitisalsoastatementthatnewmathematicsstillremainstobefound,sothesubjecthasnotrundry.Theprimesarearichsourceofplausiblethingsthatwedon’tactuallyknowaretrue.Hereareafewofthem.Inallcasespndenotesthenthprime.
Agoh–GiugaConjectureAnumberpisprimeifandonlyifthenumeratorofpBp–1+1isdivisiblebyp,whereBkisthekthBernoullinumber[TakashiAgoh1990].LookthoseupontheInternetifyoureallywanttoknow:thefirstfewareB0=1,B1= ,B2= ,B3=0,B4=– ,B5=0,B6= ,B7=0,B8=– .
Equivalently:Anumberpisprimeifandonlyif
[1p–1+2p–1+3p–1+…+(p–1)p–1]+1
isdivisiblebyp[GiuseppeGiuca1950].Acounterexample,ifitexists,musthaveatleast13,800digits[David
Borwein,JonathanBorwein,PeterBorwein,andRolandGirgensohn1996].
Andrica’sConjectureIfpnisthenthprime,then
[DorinAndrica1986].ImranGhoryhasuseddataonthelargestprimegapstoconfirmthe
conjecturefornupto1.3002×1016.Thefigureplots againstnforthefirst500primes.Thenumber1isatthetopoftheverticalaxis,andallofthespikesshownarelowerthanthat.Theyseemtoshrinkasthengetsbigger,butforallweknow,theremightbeahugespikepokingoutabove1forsomeverylargen.Inorderfortheconjecturetobefalse,therewouldhavetobean
extremelylargegapbetweentwoverylargeconsecutiveprimes.Thisseemshighlyunlikely,butcan’tyetberuledout.
Plotof againstnforthefirst200primes
Artin’sConjectureonPrimitiveRootsAnyintegera,otherthan–1oraperfectsquare,isaprimitiverootmoduloinfinitelymanyprimes.Thatis,everynumberbetween1andp–1isapowerofaminusamultipleofp.Therearespecificformulasfortheproportionofsuchprimesastheirsizebecomeslarge[EmilArtin1927].
Brocard’sConjectureWhenn>1thereareatleastfourprimesbetween and [HenriBrocard1904].Thisisexpectedtobetrue;indeedmuchstrongerstatementsoughttobetrue.
Numberofprimesbetween and plottedagainstn[EricW.Weisstein,‘Brocard’sConjecture’,fromMathWorld—AWolframWebResource:http://mathworld.wolfram.com/BrocardsConjecture.html]
Cramér’sConjectureThegappn+1–pnbetweenconsecutiveprimesisnogreaterthanaconstantmultipleof(logpn)2whennbecomeslarge[HaraldCramér1936].
Cramérprovedasimilarstatementreplacing(logpn)2by ,assuming
theRiemannHypothesis—perhapsthemostimportantunsolvedprobleminmathematics,seeCabinetpage215.
Firoozbakht’sConjectureThevalueof isstrictlydecreasing[FaridehFiroozbakht1982].Thatis,
foralln.Itistrueforallprimesupto4×1018.
FirstHardy–LittlewoodConjectureLetπ2(x)denotethenumberofprimesp≤xsuchthatp+2isalsoprime.Definethetwinprimeconstant
(wherethePsymbolindicatesaproductextendingoverallprimenumbersp≥3).Thentheconjectureisthat
where~meansthattheratiotendsto1asnbecomesarbitrarilylarge[GodfreyHaroldHardyandJohnEdensorLittlewood1923].
ThereisasecondHardy–Littlewoodconjecture(below).
Gilbreath’sConjectureStartwiththeprimes
2,3,5,7,11,13,17,19,23,29,31,…
Computethedifferencesbetweenconsecutiveterms:
1,2,2,4,2,4,2,4,6,2,…
Repeatthesamecalculationforthenewsequence,ignoringsigns,andcontinue.Thefirstfivesequencesare
1,0,2,2,2,2,2,2,4,…
1,2,0,0,0,0,0,2,…
1,2,0,0,0,0,2,…
1,2,0,0,0,2,…
1,2,0,0,2,…
GilbreathandProthconjecturedthatthefirsttermineachsequenceisalways1,nomatterhowmanytimestheprocessiscontinued[NormanGilbreath1958,FrançoisProth1878].
AndrewOdlyzkoverifiedtheconjectureforthefirst3.4×1011sequencesin1993.
Goldbach’sConjectureforEvenNumbersEveryevenintegergreaterthan2canbeexpressedasthesumoftwoprimes[ChristianGoldbach1742].
T.OliveiraeSilvahasverifiedtheconjecturebycomputerforn≤1.609×1018.
Grimm’sConjectureToeachelementofasetofconsecutivecompositenumbersonecanassignadistinctprimethatdividesit[C.A.Grimm1969].
Forexample,ifthecompositenumbersare32,33,34,35,36,thenonesuchassignmentis2,11,17,5,3.
Landau’sFourthProblemIn1912EdmundLandaulistedfourbasicproblemsaboutprimes,nowknownasLandau’sproblems.ThefirstthreeareGoldbach’sconjecture(above),thetwinprimeconjecture(below),andLegendre’sconjecture(below).Thefourthis:arethereinfinitelymanyprimespsuchthatp–1isaperfectsquare?Thatis,p=x2+1forintegerx.
Thefirstfewsuchprimesare2,5,17,37,101,197,257,401,577,677,1297,1601,2917,3137,4357,5477,7057,8101,8837,12101,13457,14401,and15377.Alargerexample,bynomeansthelargest,is
p=1,524,157,875,323,883,675,049,535,156,256,668,194,500,533,455,762,536,198,787,501,905,199,875,019,052,101
x=1,234,567,890,123456789,012,345,678,901,234,567,890
In1997JohnFriedlanderandHenrykIwaniecprovedthatinfinitelymany
primesareoftheformx2+y4forintegerx,y.Thefirstfeware2,5,17,37,41,97,101,137,181,197,241,257,277,281,337,401,and457.Iwaniechasprovedthatinfinitelymanynumbersoftheformx2+1haveatmosttwoprimefactors.Close,butnobanana.
Legendre’sConjectureAdrien-MarieLegendreconjecturedthatthereisaprimebetweenn2and(n+1)2foreverypositiven.ThiswouldfollowfromAndrica’sconjecture(above)andOppermann’sconjecture(below).Cramér’sconjecture(above)impliesthatLegendre’sconjectureistrueforallsufficientlylargenumbers.Itisknowntobetrueupto1018.
Lemoine’sConjectureorLevy’sConjectureAlloddintegersgreaterthan5canberepresentedasthesumofanoddprimeandtwiceaprime[ÉmileLemoine1894,HymanLevy1963].
Theconjecturehasbeenverifiedupto109byD.Corbitt.
MersenneConjecturesIn1644MarinMersennestatedthatthenumbers2n–1areprimeforn=2,3,5,7,13,17,19,31,67,127and257,andcompositeforallotherpositiveintegersn<257.EventuallyitwasshownthatMersennehadmadefiveerrors:n=67and257givecompositenumbersandn=61,89,107giveprimes.Mersenne’sconjectureledtotheNewMersenneConjectureandtheLenstra–Pomerance–WagstaffConjecture,whichfollow.
NewMersenneConjectureorBateman-Selfridge-WagstaffConjectureForanyoddp,ifanytwoofthefollowingconditionshold,thensodoesthethird:
1p+2k 1orp=4k 3forsomenaturalnumberk.
22p–1isprime(aMersenneprime).
3(2p+1)/3isprime(aWagstaffprime).
[PaulBateman,JohnSelfridge,andSamuelWagstaffJr1989]
Lenstra–Pomerance–WagstaffConjectureThereisaninfinitenumberofMersenneprimes,andthenumberofMersenneprimeslessthanxisapproximatelyeγloglogx/log2whereγisEuler’s
constant,roughly0.577[HendrikLenstra,CarlPomerance,andWagstaff,unpublished].
Oppermann’sConjectureForanyintegern>1,thereisatleastoneprimebetweenn(n–1)andn2,andatleastanotherprimebetweenn2andn(n+1)[LudvigHenrikFerdinandOppermann1882].
Polignac’sConjectureForanypositiveevenn,thereareinfinitelymanycasesoftwoconsecutiveprimeswithdifferencen[AlphonsedePolignac1849].
Forn=2,thisisthetwinprimeconjecture(seebelow).Forn=4,itsaysthereareinfinitelymanycousinprimes(p,p+4).Forn=6,itsaysthereareinfinitelymanysexyprimes(p,p+6)withnoprimebetweenpandp+6.
Redmond–SunConjectureEveryinterval[xm,yn](thatis,thesetofnumbersrunningfromxmtoyn)containsatleastoneprime,exceptfor[23,32],[52,33],[25,62],[112,53],[37,133],[55,562],[1812,215],[433,2822],[463,3122],[224342,555][StephenRedmondandZhi-WeiSun2006].
Theconjecturehasbeenverifiedforallintervals[xm,yn]below1012.
SecondHardy–LittlewoodConjectureIfπ(x)isthenumberofprimesuptoandincludingxthen
π(x+y)≤π(x)+π(y)
forx,y≥2[GodfreyHaroldHardyandJohnLittlewood1923].Therearetechnicalreasonsforexpectingthistobefalse,butthefirst
violationislikelytooccurforverylargevaluesofx,probablygreaterthan1.5×10174butlessthan2.2×101198.
TwinPrimeConjectureThereareinfinitelymanyprimespsuchthatp+2isalsoprime.
On25December2011,PrimeGrid,a‘distributedcomputingproject’thatmakesuseofsparetimeonvolunteers’computers,announcedthelargestpairoftwinprimescurrentlyknown:
3,756,801,695,685×2666,669 1
Thesenumbershave200,700digits.Thereare808,675,888,577,436twinprimepairsbelow1018.
TheOptimalPyramidThinkofancientEgyptandyouthinkofpyramids.EspeciallytheGreatPyramidofKhufuatGiza,thelargestofthemall,flankedbytheslightlysmallerpyramidofKhafreandtherelativeminnowofMenkaure.Theremainsofover36majorEgyptianpyramidsandhundredsofsmalleronesareknown;theyrangefromhugealmostcompleteonestoholesinthegroundcontainingafewbitsofstonefromtheburialchamber—orless.
Left:TheGizapyramids:Frombacktofront:GreatPyramidofKhufu,Khafre,Menkaure,andthreeQueens’pyramids.Perspectivemakesthosebehindlooksmallerthantheyreallyare.Right:Thebent
pyramid.
Anenormousamounthasbeenwrittenabouttheshapes,sizes,andorientationsofpyramids.Mostofitisspeculative,usingnumericalrelationshipstoconstructambitiouschainsofargument.TheGreatPyramidisespeciallysusceptibletothistreatment,andithasbeenvariouslylinkedtothegoldennumber,π,andeventhespeedoflight.Therearesomanyproblemswiththiskindofreasoningthatit’shardtotakeitseriously:thedataareofteninaccurateanyway,andwithsomanymeasurementstoplaywithit’seasytocomeupwithanythingyouwant.
OneofthebestsourcesfordataisMarkLehner’sTheCompletePyramids.Amongotherthings,itliststheslopesofthefaces:theanglesbetweentheplanesformedbyatriangularfaceandthesquarebase.Afewexamples:
pyramid angle
Khufu 51°50’40”Khafre 53°10’Menkaure 51°20’25”Bent 54°27’44”(lower),43°22’(upper)
Red 43°22’Black 57°15’50”
Youcanfindmoreextensivedataat
http://en.wikipedia.org/wiki/List_of_Egyptian_pyramids
Twoobservationsspringtomind.Thefirstisthatstatingsomeoftheseanglestothenearestsecondofarc(andotherstothenearestminute)isunwise.TheblackpyramidofAmenemhatIIIatDashurhasabaseof105metresandaheightof75metres.Adifferenceof1secondofarcintheslopecorrespondstoachangeinheightof1millimetre.Admittedly,therearetracesoftheedgesofthebase,andsomefragmentsofcasingstonesmayhavesurvived,butgivenwhatremainsofthepyramid,you’dbehardputtoestimatetheoriginalslopetowithin5°ofthetruefigure.
WhatremainsoftheblackpyramidofAmenemhatIII
Theotheristhatalthoughtheslopesvarysomewhat—withinasinglemonumentinthecaseofthebentpyramid—theyhaveatendencytoclusteraround54°orso.Why?
In1979R.H.Macmillan[Pyramidsandpavements:somethoughtsfromCairo,MathematicalGazette63(December1979)251–255]startedfromthewell-attestedfactthatthepyramidbuildersusedexpensivecasingstonesontheoutsideoftheirpyramids,suchaswhiteTuralimestoneorgranite.Insidetheyusedcheapermaterials:low-gradeMokattamlimestone,mudbrick,andrubble.Soitmakessensetoreducetheamountofstonecasing.Whatshapeshouldapyramidbeifthepharaohwantsthebiggestpossiblemonumentforagivencostofcasingstones?Thatis:whichangleofslopemaximisesthevolumeforafixedareaofthefourtriangularfaces?
Left:Slicingapyramid.Right:Maximisingtheareaofanisoscelestriangle,orequivalentlyarhombuswithgivenside.
Thisisaniceexerciseincalculus,butitcanalsobesolvedgeometricallyusingaclevertrick.Slicethepyramidinhalfverticallythroughadiagonalofthebase(shadedtriangle).Thisyieldsanisoscelestriangle.Thevolumeofthehalf-pyramidisproportionaltotheareaofthistriangle,andtheareasoftheslopingfacesofthehalf-pyramidareproportionaltothelengthsofthecorrespondingsidesofthetriangle.Sotheproblemisequivalenttofindingtheisoscelestriangleofmaximalareawhenthelengthsofthetwoequalsidesarefixed.
Reflectingthetriangleinitsbase,thisisequivalenttofindingtherhombusofmaximalareawithagivenlengthofside.Theanswerisasquare(orientedinadiamondposition).Sotheanglesofeachtriangularsectionofthiskindare90°atthetop,and45°forthetwobaseangles.Itfollowsbybasictrigonometrythattheangleofslopeofafaceofthepyramidis
whichisclosetotheaveragefigureforactualpyramids.Macmillanmakesnoclaimsaboutwhatthistellsusaboutbuildingpyramids;
hismainpointisthatit’saneatexerciseingeometry.However,theMoscowmathematicalpapyruscontainsaruleforfindingthevolumeofatruncatedpyramid(onewiththetopcutoff),andaproblemshowingthattheEgyptiansunderstoodsimilarity.Italsoexplainshowtofindtheheightofapyramidfromitsbaseandslope.Moreover,boththispapyrusandtheRhindmathematicalpapyrusexplainhowtofindtheareaofatriangle.SotheEgyptianmathematicianscouldhavesolvedMacmillan’sproblem.
MoscowmathematicalpapyrusProblem14:findingthevolumeofatruncatedpyramid
Intheabsenceofapapyruscontainingthatexactcalculation,there’snoconvincingreasontosupposethattheydid.Wedon’thaveanyevidencethattheywereinterestedinoptimisingtheshapeoftheirpyramids.Eveniftheywere,theycouldhaveestimatedtheshapeusingclaymodels.Ormadeaneducatedguess.Ortheshapecouldhaveevolvedtowardstheleastexpensiveone:buildersandpharaohsarelikethat.Alternatively,theangleofslopemighthavebeendeterminedbyengineeringconsiderations:thebentpyramidiswidelybelievedtobetheshapeitisbecausehalfwayupitstartedtocollapse,sotheslopewasreduced.Thatsaid,thislittlepieceofpyramidmathematicshasmoregoingforitthanconnectionstothespeedoflight.
TheSignofOne:PartTwoFromtheMemoirsofDrWatsup
Soamesbeganstridingupanddowntheroomlikeamanpossessed.InwardlyIgavealoud“huzzah!”—forIcouldseethathewashooked.NowIwouldreelhiminoutofthedarkdepressionintowhichhehadfallen,andridmyselfofBolivianfuneraldirgestoboot.
“Wemustbemoresystematic,Watsup!”hedeclared.“Inwhatway,Soames?”“Inamoresystematicway,Watsup.”Theensuingsilenceledhimtobeless
obscure.“Wemustlistsmallnumbersthatcanbederivedfromjusttwo1’s.Byputtingthosetogetherwecan—well,you’llseeinaminute,I’msure.”
SoSoameswrotedown:
atwhichpointSoamesgroundtoahalt.“Iadmitthat7and8constitutetemporarylacunae,”hesaid.“Nevermind,
allowmetocontinue:
9=1/.10=1/.111=11”
“IconfessIdonotyet—”“Beassured,Watsup,youwill.Suppose,forargument’ssake,thatwecan
express7and8usingtwo1’s.Thenwewouldhavecontrolofallnumbersfrom0to11.So,givenanynumbernexpressibleusingtwo1’s,wewouldbeableto
expresseverythingbetweenn–11andn+11usingfour1’s—merelybysubtractingoraddingtheexpressionsonmysystematiclist.”
“Ah,Iseenow,”Isaid.“Youusuallydo,onceI’vetoldyou,”herepliedacerbically.“ThenletmecontributeanewthoughttoshowthatIhaveunderstood!Since
weknowhowtoexpress24usingtwo1’s,forexampleas !,weimmediatelyexpresseverynumberfrom24–11to24+11usingfour1’s.Thatis,therangefrom13to35,inclusive.”
“Exactly!Ithinkweneednotwritethoseexpressionsdown.”“No,Aha!Wecangetfurther!Look:
“Yes,”hereplied.“Butbeforeyourexcitementcarriesyouintorealmsuntold,Iwillremindyouthatwedonotyethaveexpressionsfor7and8usingonlytwo1’s.”
Ilookedsuitablycrestfallen.Butthenawildthoughtstruckme.“Soames?”Iaskedtentatively.
“Yes?”“Factorialsmakenumbersbigger?”Henodded,irritably.“Andsquarerootsmakethemsmaller?”“Agreed.Gettothepoint,man!”“Andfloorsandceilingsroundthingsofftowholenumbers?”Icouldseetherealisationdawningonhisface.“Stoutfellow,Watsup!Yes,I
seeitnow.Weknow,forexample,howtoexpress24usingtwo1’s.Thereforewecanalsoexpress24!usingtwo1’s,andthatis”—hiseyebrowsnarrowed—“620,448,401,733,239,439,360,000.Whosesquarerootis”—hisfacereddenedashecarriedoutthementalarithmetic—“887,516.46,whosesquarerootis942.08,whosesquarerootis30.69.”
“Sowecanexpress30and31usingonlytwo1’s,”Isaid.“Namely:
Noneofwhichhelpsusexpress7and8withtwo1’s,ofcourse,butifwecoulddothat,wecouldextendtherangeofnumbersto31+11,whichis42.Allofwhichargues,asyousocogentlyputit,Soames,thatweshouldbesystematic.I
proposethatwenowinvestigaterepeatedsquarerootsoffactorialsofnumbersthatwecanexpresswithtwo1’s.”
“Agreed!Anditisimmediatelyobvious,”saidSoames,“thatsuchanexpressionfor7immediatelyyieldsonefor8.”
“Uh—isit?”“Naturally.Since7!=5040,whosesquarerootis70.99,whosesquarerootis
8.42,wededucethat
So,notforthefirsttimeinhumanhistory,thekeytothemysteryisthenumber7!”Bywhich,dearreader,hewasplacinganemphasisonthenumber7,notreferringtoitsfactorial.Pleasepayattention,Ihaveexplainedthisbefore.
Soames’sbrowfurrowed.“Icandoitusingadoublefactorial.”“Youmeanafactorialofafactorial?”“No.”“Asubfactorial?You’venotyetexplained—”“No.Thedoublefactorialisatrifleobscure;itis
n!!=n×(n–2)×(n–4)×…×4×2
whenniseven,and
n!!=n×(n–2)×(n–4)×…×3×1
whennisodd.So,forexample,
6!!=6×4×2=48
whosesquarerootis6.92,whoseceilingis7.”Imeeklywrotedown
ButSoamesremaineddissatisfied.“Theproblem,Watsup,isthatbyintroducingevermoreobscurearithmetical
functions,wecouldexpressanynumberwhatsoeverwithease.WemightusePeano,forinstance.”
Iobjectedvociferously.“Soames,youknowthatourlandladycomplainsincessantlyaboutyourclarinet.Shewouldneverpermitapiano!”
“GiuseppePeanowasanItalianlogician,Watsup.”“Tobehonest,thatmightnotmakemuchdifference.I’mnotsurethatMrs
Soapsudswould—”“Quiet!InPeano’saxiomatisiationofarithmetic,thesuccessorofanyinteger
nis
s(n)=n+1
SoPeanocouldwrite
1=12=s(1)3=s(s(1))4=s(s(s(1)))5=s(s(s(s(1))))
andthepatterncontinuesindefinitely.Everyintegerwouldbeexpressibleusingjustone1.Or,forthatmatter,justone0,since1=s(0).Itistootrivial,Watsup.”
Canyoufindawaytowrite7usingonlytwo1’sandnotusinganythingmoreesotericthanthefunctionsSoamesandWatsupemployedbeforetheystartedarguingaboutdoublefactorialsandsuccessors?Seepage279fortheanswer.
SoamesandWatsuphavenotfinishedyet.TheSignofOnecontinuesonpage115.
InitialConfusion
R.H.Bing
R.H.BingwasanAmericanmathematician,borninTexas,whospecialisedinwhatbecameknownasTexantopology.WhatdoestheR.H.standfor?Well,hisfatherwasRupertHenry,buthismotherfeltthatthissoundedtooBritishforTexas,sowhenhewaschristenedshecutitdowntothebareinitials.SoR.H.standsforR.H.,nothingmore.Thiscausedacertainamountofpuzzlement,butnothingtooserious,untilBingappliedforavisatovisitsomewhereorother.Whenaskedhisname,anticipatingtheusualreaction,hesaiditwas“R-onlyH-onlyBing”.
HereceivedavisamadeouttoRonlyHonlyBing.
Euclid’sDoodleThisisamathematicalmysterythatwassolvedovertwothousandyearsago,andusedtobetaughtinschools,butnotanymore—forsensiblereasons.However,it’sworthknowing,becauseit’sfarmoreefficientthanthemethodthat’susuallytaughtinstead.Anditlinksuptoallsortsofimportantbitsofmathematicsathigherlevels.
Peopleliketodoodle.Youseethemonthephone,engagedinconversation,idlyfillinginalltheo’sonapageofthenewspaperwithaballpointpen.Ordrawingwigglylinesthatgoroundandroundlikeirregularspirals.Thissenseoftheword‘doodle’,whichoriginallymeantafool,seemstohavebeenintroducedbythescreenwriterRobertRiskininthe1936comedymovieMrDeedsGoestoTown;MrDeedsrefersto‘doodle’asscribblesthatcanhelppeoplethink.
Ifamathematiciandoodled—andmostdo—theymightwellgetroundtodrawingarectangle.Whatcanyoudotoarectangle?Youcanfillitin,youcandrawspiral-likecurvesroundtheedge…oryoucancutoffasquarefromoneendtomakeasmallerrectangle.Thenit’sonlynatural,andtypicalofthedoodlingmentality,todothesameagain.
Whathappens?Youmightcaretotryafewrectanglesbeforereadingon.OK,herewego.Istartedwithalong,thinrectangle,andhere’swhat
happened.
Mydoodle
EventuallyIgottoasmallsquare,andranoutofrectangle.Doesthisalwayshappen?Doeseveryrectangleeventuallygetgobbledup?
Nowthat’sagoodquestionforamathematiciantothinkabout.Whatsizewasmyrectangle?Well,thefinalpictureshowsthat:
Thecombinedsidesoftwolittlesquaresequalthesideofamediumsquare.Thecombinedsidesoftwomediumsquaresandonelittleonemakethesideofabigone,andalsogiveonesideoftherectangle.Thecombinedsidesofthreebigsquaresandamediumonemaketheothersideoftherectangle.
Ifthesmallsquarehasside1unit,thenthemediumonehasside2andthelargeonehasside2×2+1=5.Theshortsideoftherectangleis5,andthelongsideis3×5+1=17.SoIstartedwitha17×5rectangle.
That’sinteresting:bylookingatthewaythesquaresfittogether,Icanworkouttheshapeofmyrectangle.Amoresubtleimplicationis:iftheprocessstops,thesidesoftheoriginalrectanglearebothintegermultiplesofthesamething:thesideofthelastsquareremoved.Inotherwords,theratioofthetwosidesisoftheformp/qforintegerspandq.Whichmakesitarationalnumber.
Thisideaiscompletelygeneral:ifthedoodlestops,theratioofthesidesoftherectangleisarationalnumber.Infact,theconverseisalsotrue:iftheratioofthesidesoftherectangleisarationalnumber,thedoodlestops.Sodoodlesthatstopcorrespondpreciselyto‘rationalrectangles’.
Toseewhy,let’slookmorecloselyatthenumbers.Thepicturesineffecttellusthis:
17–5=1212–5=77–5=2
Nowwehavea5×2rectangleleftandwehavetomovetothemediumsquare
5–2=33–2=1
Nowwehavea2×1rectangleleftandwehavetomovetoasmallsquare
2–1=11–1=0
Stop!Anditmuststop,becausetheintegersinvolvedarepositive,andtheyaregettingsmallerateachstage.Theymustdo,we’resubtractingfromthemorleavingthemalone.Now,asequenceofpositiveintegerscan’tdecreaseforever.Forexample,ifyoustartwithamillionandthendecrease,youhavetostopafteratmostamillionsubtractions.
Morecompactly,thedoodletellsusthat
17dividedby5gives3withremainder25dividedby2gives2withremainder12dividedby1goesexactlywithzeroremainder
andtheprocessstopsoncetheremainderiszero.Euclidusedhisdoodletosolveaprobleminarithmetic:giventwointegers,
calculatetheirhighestcommonfactor.Thisisthelargestintegerthatdividesbothexactly;itisusuallyabbreviatedtohcf.Anotherphraseisgreatestcommondivisor,gcd.Forinstance,ifthenumbersare4,500and840,thehcfis120.
ThewayIwastaughttodothisatschoolistofactorisethetwonumbersintoprimes,andseewhatfactorstheyhaveincommon.Forinstance,supposewewantthehcfof68and20.Factoriseintoprimes:
68=22×1720=22×5
Thehcfis22=4.Thismethodislimitedtonumbersthataresmallenoughtobefactorised
quicklyintoprimes.It’shopelesslyinefficientforlargernumbers.TheancientGreeksknewamoreefficientmethod,aprocedurethattheygavethefancynameanthyphairesis.Inthiscaseitgoeslikethis:
68dividedby20gives3withremainder820dividedby8gives2withremainder48dividedby4goesexactlywithzeroremainderStop!
Thisisthesameasthepreviouscalculationwith17and5,butnowallthenumbersarefourtimesthesize(exceptforhowmanytimeseachdividesintotheother,whichstaysthesame).Ifyoudothedoodlewitha68×20rectangleyougetthesamepicturesasbefore,butthefinalsmallsquareis4×4,not1×1.
ThetechnicalnameisEuclid’salgorithm.Analgorithmisarecipeforacalculation.EuclidputthisoneinhisElements,andheuseditasthebasisofhistheoryofprimenumbers.Insymbols,thedoodlegoeslikethis.Taketwopositiveintegersm≤n.Startwiththepair(m,n)andreplaceitby(m,n–m)innumericalorder,smallestfirst:thatis,transform
(m,n) (min(m,n–m),max(m,n–m))
whereminandmaxaretheminimumandmaximumrespectively.Repeat.At
eachstagethelargestnumberinthepairgetssmaller,soeventuallytheprocessstopswithapair(0,h),say.Thenhistherequiredhcf.Theproofiseasy:anyfactorofbothmandnisalsoafactorofbothmandn–m,andconversely.Soateachstepthehcfremainsthesame.
Thismethodisgenuinelyefficient:youcanuseitbyhandforreallybignumbers.Toproveit,here’saquestionforyou.Findthehcfof44,758,272,401and13,164,197,765.
Seepage279fortheanswer.
EuclideanEfficiencyHowefficientisEuclid’salgorithm?
Choppingoffonesquareatatimeissimplerfortheoreticalpurposes,butthemorecompactform,intermsofdivisionwithremainder,isthebestonetouseinpractice.Thistelescopesallcutsusingafixedsizeofsquareintoasingleoperation.
Mostofthecomputationaleffortoccursinthedivisionstep,sowecanestimatetheefficiencyofthealgorithmbycountinghowmanytimesthisstepgetsused.ThefirstpersontoinvestigatethisquestionwasA.A.-L.Reynaud,andin1811heprovedthatthenumberofdivisionstepsisatmostm,thesmallerofthetwonumbers.Thisisaverypoorestimate,andhelatergotitdowntom/2+2,notmuchbetter.In1841P.-J.-E.Finckreducedtheestimateto2log2m+1,whichisproportionaltothenumberofdecimaldigitsinm.In1844GabrielLaméprovedthatthenumberofdivisionstepsisatmostfivetimesthenumberofdecimaldigitsofm.Soevenfortwonumberswith100digits,thealgorithmgetstheanswerinnomorethan500steps.Ingeneral,youcan’tdoitthatquicklyusingprimefactors.
What’stheworst-casescenario?LaméprovedthatthealgorithmrunsmostslowlywhenmandnareconsecutivemembersoftheFibonaccisequence
1123581321345589…
inwhicheachnumberisthesumoftheprevioustwo.Forthesenumbers,exactlyonesquaregetschoppedoffeachtime.Forinstance,withm=34,n=55,weget
55dividedby34gives1withremainder2134dividedby21gives1withremainder1321dividedby13gives1withremainder813dividedby8gives1withremainder58dividedby5gives1withremainder35dividedby3gives1withremainder23dividedby2gives1withremainder12dividedby1goesexactly.
It’sanunusuallylongcalculationforsuchsmallnumbers.Mathematicianshavealsoanalysedtheaveragenumberofdivisionsteps.
Withfixedn,thenumberofdivisionstepsaveragedoverallsmallermisapproximately
whereC,calledPorter’sconstant,is
Here ’(2)isthederivativeofRiemann’szetafunctionevaluatedat2,andγisEuler’sconstant,about0.577.Itwouldbehardtofindasensibleproblemleadingtoamorecomprehensiveselectionofmathematicalconstantsinoneformula.Theratioofthisformulatotheexactanswertendsto1asngetslarger.
123456789TimesXSometimesverysimpleideasleadtomysteriousresults.Trymultiplying123456789by1,2,3,4,5,6,7,8,and9.Whatdoyounotice?Whendoesitgowrong?
Seepage279fortheanswers.Foranextension,Seepage145.
TheSignofOne:PartThreeFromtheMemoirsofDrWatsup
HeapsofpapercoveredinarcanescribblesweresproutinglikemushroomsfromeveryflatsurfaceinSoames’slodgings.This,youunderstand,wasnotunusual;MrsSoapsudsoftenberatedhimabouthisdeeplitterfilingsystem,tonoavail.Butonthisoccasionthescribblesweresums.
“Icanobtain8usingtwo1’s,withoutinvolvingahypotheticalexpressionfor7,”Iannounced.“Infact,
ButforthelifeofmeIcannotderive7.”“Thatonedoesseemtricky,”Soamesagreed.“Butyourresultleadsto
progressinotherways:
whereofcoursewesubstituteyourexpressionfor8wherenecessary.Icouldwriteitoutinfull—”
“No,no,Soames,Iamalreadyconvinced!”“Butnowwehavetwomorelacunaeat12and13.However,Watsup,I
suspecttheseproblemsarerelated.Letmesee…Well,
andwealreadyhave15usingonlytwo1’s.Then
andfurther
andfinally
whichresolvestheissueentirelysatisfactorily.Sosubstitutinginturnforthevariousnumbers,wefindthat
Iammortifiednottohaveseenthatimmediately.”“Isthatthesimplestsolution,Soames?”saidI,gulping.“Ihopenot!”“Ihavenoidea.Perhapssomeingeniouspersoncoulddobetter.Itishardto
besurewiththesematters.Iamsuretheywillinformusbytelegramiftheycanbetterourownfeebleefforts.”
“Anway,”Isaid,“ifwecanexpressanyintegernwithtwo1’s,wecannowexpresstheentirerangefromn–17ton+17.”
“Exactly,Watsup.Ourquestbecomessimplerbythemoment.Allweneedisaseriesofnumbers,eachexceedingthepreviousonebynomorethan35,sothattheirrangesabutoroverlap.Thatwillenableustoreachthelargestsuchnumberplus17.”
“Whichmeans—”Ibegan—“Thatweshouldbesystematic!”“Quite.”“Wehadalreadyreached…remindme,Watsup.Consultthoseextensive
notesofyours.”Idelvedintoseveralpilesofdocuments,eventuallyfindingmynotebook
beneathastuffedskunk.“Wehadreached32,Soames,ifweincludeyourincidentalremarkearlierwhenseeking7.”
“And,ofcourse,
saidhe.“Verygood.Soideallyweneedtoexpress78,103,138,andsoon,withtwo1’s.Butwecanusesmallernumbersifthosearemoreconvenient.Providedthattheincreaseisatmost35fromonetothenext.”
Severalhoursofintensecomputation,andmoreheapsofpaper,ledtoashortbutvitallist:
butlittlemore.“PerhapsIwastoohastyindismissingdoublefactorials,Watsup.”
“Verylikely,Soames.”Soamesnoddedandwrote
105=7!!
Then,inasuddenfitofinspiration,headded
exclaiming“Ifwecanfindawaytowrite18withtwo1’s,thenweextendtherangesurroundinganyintegernexpressibleusingtwo1’s:wecanthendealwithn–20ton+20.”Hepausedforbreath,andadded“Failingthat,theonlygapswouldben–18andn+18,whichperhapswecouldderiveinotherways.”
“Ithinkitistimewetookstock,”Isaid.Iperusedourscribblednotes.“Itseemstomethatwehadexpressedallnumbersfrom1to33usingfour1’s.Then
needonlytwo1’s,soweimmediatelyfillineverythingbetween26and61.Thereisagapat62(becausethatis44+18andwearestuckon18ifonlytwo1’sareallowed)butwecando63and64.Now,basedon80,wecancontinueto97.Thenwearestuckon98,but99and100canbeachieved.”
“Moreeasily,inpointoffact,”saidSoames:
“Sowearecompleteupto100,”Isaid,“withtheexceptionsof62and98.”“But98istakencareofby105,alongwithallnumbersupto122,”said
Soames.“Oh,Ihadforgottenwecoulddo105withtwo1’s.”“Andsince120=5!,alsoexpressibleusingtwo1’s,wecanreach137.
Indeed,wealsoget139and140.”“Sotheonlygapsupto140are62and138,”Isaid.“Soitseems,”saidSoames.“Iwonderifthosegapscanbefilledbysome
othermethod?”Canyoufindawaytowrite62and138withfour1’s,notusinganything
moreesotericthanthefunctionsSoamesandWatsuphaveusedsofar?Seepage280fortheanswer.
SoamesandWatsupstillhaven’tfinished.Buttheendisnigh:TheSignofOneconcludesonpage126.
TaxicabNumbers
SrinivasaRamanujan
SrinivasaRamanujanwasaself-taughtIndianmathematicianwithanamazingtalentforformulas—usuallyverystrangeones,yethavingtheirownkindofbeauty.HewasbroughttoEnglandbyCambridgemathematiciansGodfreyHaroldHardyandJohnEdensorLittlewoodin1914.By1919hewasterminallyillwithlungdisease,andhediedinIndiain1920.Hardywrote:
IrememberoncegoingtoseehimwhenhewaslyingillatPutney.Ihadriddenintaxi-cabNo.1729,andremarkedthatthenumberseemedtoberatheradullone,andthatIhopeditwasnotanunfavourableomen.“No,”hereplied,“itisaveryinterestingnumber;itisthesmallestnumberexpressibleasthesumoftwo[positive]cubesintwodifferentways.”
Theobservationthat
1729=13+123=93+103
wasfirstpublishedbyBernardFrénicledeBessyin1657.Ifnegativecubesarepermittedthenthesmallestsuchnumberis
91=63+(–5)3=43+33
Numbertheoristshavegeneralisedtheconcept.Thenthtaxicabnumber,Ta(n),isthesmallestnumberthatcanbeexpressedasasumoftwopositivecubesinnormoredistinctways.
In1979HardyandE.M.Wrightprovedthatsomenumberscanbeexpressed
asthesumofanarbitrarilylargenumberofpositivecubes,soTa(n)existsforalln.However,todateonlythefirstsixareknown:
Ta(1)=2=13+13Ta(2)=1729=13+123=93+103Ta(3)=87539319=1673+4363=2283+4233
=2553+4143Ta(4)=6963472309248
=24213+190833=54363+189483=102003+180723=133223+1663083
Ta(5)=48988659276962496=387873+3657573=1078393+3627533=2052923+3429523=2214243+3365883=2315183+3319543
Ta(6)=24153319581254312065344=5821623+289062063=30641733+288948033=85192813+286574873=162180683+270932083=174924963+265904523=182899223+262243663
Ta(3)wasdiscoveredbyJohnLeechin1957.Ta(4)wasfoundbyE.Rosenstiel,J.A.DardisandC.R.Rosenstielin1991.Ta(5)wasfoundbyJ.A.Dardisin1994andconfirmedbyDavidWilsonin1999.In2003C.S.Calude,E.Calude,andM.J.DinneenestablishedthatthenumberstatedaboveisprobablyTa(6),andin2008UweHollerbachannouncedaproof.
TheWaveofTranslation
JohnScottRussell
Mathematicalresearchonhorseback?Whynot?Inspirationstrikeswheneveritdoes.Youdon’tgettochoose.In1834,JohnScottRussell,aScottishcivilengineerandnavalarchitect,was
ridinghishorsealongsideacanal,whenhenoticedsomethingremarkable:
Iwasobservingthemotionofaboatwhichwasrapidlydrawnalonganarrowchannelbyapairofhorses,whentheboatsuddenlystopped—notsothemassofwaterinthechannelwhichithadputinmotion;itaccumulatedroundtheprowofthevesselinastateofviolentagitation,thensuddenlyleavingitbehind,rolledforwardwithgreatvelocity,assumingtheformofalargesolitaryelevation,arounded,smoothandwell-definedheapofwater,whichcontinueditscoursealongthechannelapparentlywithoutchangeofformordiminutionofspeed.Ifolloweditonhorseback,andovertookitstillrollingonatarateofsomeeightorninemilesanhour,preservingitsoriginalfiguresomethirtyfeetlongandafoottoafootandahalfinheight.Itsheightgraduallydiminished,andafterachaseofoneortwomilesIlostitinthewindingsofthechannel.Such,inthemonthofAugust1834,wasmyfirstchanceinterviewwiththatsingularandbeautiful
phenomenonwhichIhavecalledtheWaveofTranslation.
Hewasintriguedbythisdiscovery,becausenormallyindividualwavesspreadoutastheytravel,orbreaklikesurfonabeach.Heconstructedawavetankathishome,andcarriedoutaseriesofexperiments.Theserevealedthatthiskindofwaveisverystable,anditcantravelalongwaywithoutchangingshape.Wavesofdifferentsizestravelatdifferentspeeds.Ifonesuchwavecatchesupwithanotherone,itemergesinfrontafteramorecomplicatedinteraction.Andabigwaveinshallowwaterwilldivideintotwo:amediumoneandasmallone.
Thesediscoveriespuzzledthephysicistsofthetime,becausetheircurrentunderstandingoffluidflowcouldnotexplainit.Infact,GeorgeAiry,adistinguishedastronomer,andGeorgeStokes,theleadingauthorityonfluiddynamics,hadtroublebelievingit.WenowknowthatRussellwasright.Inappropriatecircumstances,nonlineareffects,beyondthescopeofthemathematicsofhisera,counteractthetendencyofawavetospreadoutbecausethespeedofthewavedependsonitsfrequency.Theseeffectswerefirstunderstoodaround1870byLordRayleighandJosephBoussinesq.
In1895DiederikKortewegandGustavdeVriescameupwiththeKorteweg–deVriesequation,whichincludedsucheffects,andshowedthatithassolitarywavesolutions.Similarresultswerederivedforotherequationsofmathematicalphysics,andthephenomenonacquiredanewname:solitons.AseriesofmajordiscoveriesledPeterLaxtoformulateverygeneralconditionsforequationstohavesolitonsolutions,andexplainedthe‘tunnelling’effect.Itismathematicallyquitedifferentfromthewaythatshallowwavesinteractbysuperposingtheirshapes,liketwosetsofripplescrossingeachotheronapond,whichisadirectconsequenceofthemathematicalformofthewaveequation.Soliton-likebehaviouroccursinmanyareasofscience,fromDNAtofibreoptics.Thishasledtoawiderangeofnewphenomenawithnameslikebreathers,kinks,andoscillons.
There’salsoatantalisingideathathasnotyetbeenmadetowork.Fundamentalparticlesinquantummechanicssomehowcombinetwoapparentlydifferentcharacteristics.Likemostthingsataquantumlevel,theyarewaves,yettheyhangtogetherinaparticle-likeclump.Physicistshavetriedtofindequationsthatrespectthestructureofquantummechanicsbutallowsolitonstoexist.Theclosesttheyhavecomesofarisanequationthatproducesaninstanton:thiscanbeinterpretedasaparticlethathasaveryshortlifetime,winkingintoexistencefromnowhereanddisappearingimmediately.
RiddleoftheSands
Barchandunes.Left:ParacasNationalPark,Peru.Right:Hellespontusregion,fromMarsReconnaissanceOrbiter.
Sanddunesformavarietyofpatterns:linear,transverse,parabolic,…Oneofthemostintriguingisthebarchan,orcrescent,dune.ThenamecomesfromTurkestan,andissaidtohavebeenintroducedintogeologyin1881bytheRussiannaturalistAlexandervonMiddendorf.BarchanscanbefoundinEgypt,Namibia,Peru,…andevenonMars.Theyarecrescent-shaped,comeinarangeofsizes,andtheymove.Theyformswarms,interactwitheachother,breakupandjointogether.Inrecentyearsmathematicalmodellinghasprovidedalotofinsightintotheirshapesandbehaviour,butmanymysteriesremain.
Dunesareformedbytheinteractionofwindandsandgrains.Theroundedendofabarchanfacesintotheprevailingwind,whichpushesthesandupthefrontoftheduneandroundthesides,whereitformstwotrailingarmsthatgiveititscharacteristiccrescentshape.Atthetopofthedune,thesandtumblesoverandissuckeddownthe‘slipface’betweenthearms.Alargevortexofrotatingaircalledaseparationbubblescoursoutthespacebetweenthearms.
Left:Schematicofabarchanandtheseparationbubble.Right:SimulationbyBarbaraHorvatofthemotionofsandgrains,computedfromamathematicalmodel.
Barchansbehavelikesolitons(seepreviousitem),althoughtechnicallythey
differinsomerespects.Asthewindblowsthemalong,thesmalldunestravelfasterthanthebigones.Ifasmalldunecatchesupwithabigone,itappearstobeabsorbed,butafteratimethebigbarchanspitsoutasmallone,almostasthroughthesmallonehadtunnelledthroughthebigone.Thesmallonethenheadsoff,fasterthanthelumberingbeastbehindit.
Intheirpaper,VeitSchwa¨mmleandHansHerrmanncommentonthesimilaritiesanddifferencesbetweenbarchancollisionsandsolitons.Thefigureshowswhathappensifthetwodunesareofsimilarsizes.Initially(a)thesmallerduneisbehindthelargeronebutmovingfaster.Ithitsthebackofthelargerone(b)andclimbsupitswindwardface,butgetsstuckpartway(c).Thenthefrontseparatestoformasmallerdune(d).
SimulatedcollisionofasmallbarchanandalargeronebyVeitSchwa¨mmleandHansHerrmann.(a)Time0:smalldunebehindbigone.(b)After0.48years:smalldunecatchesbigoneandtheycollide.(c)After
0.63years:dunesmixedtogether.(d)After1.42years:smallduneinfrontofbigone.
Forsomecombinationsofheights,theemergingduneisbiggerthanthesmalleronewasoriginally,whereasforothersitissmaller.Thisbehaviourdiffersfromthatofsolitons,wherebothwavesendupthesamesizeastheybegan.However,thereisanintermediaterangeofheightcombinationsforwhichthedunesretaintheirsizesandvolumesexactly.Inthesecasestheybehavelikesolitons.
Ifthesmallduneisalotsmallerthanthebigone,itjustgetsgobbledupandasingle,larger,barchanforms.Ifthedifferenceinheightsismoderate,thecollisioncanresultin‘breeding’:twosmallbarchansappearatthetipsofthehornsofthelargerone,andheadoffinfrontofit.Realbarchansdoallthesethings.Thedynamicsofbarchandunesisricherthanthatofconventionalsolitons.
EskimoπWhyisπonly3intheArctic?
Everythingshrinksinthecold.
TheSignofOne:PartFour—ConcludedFromtheMemoirsofDrWatsup
“Well,thisisaprettypickle,”Imuttered.“Agherkin,Ibelieve,”saidSoames,pluckingthevinegar-soakedvegetable
fromthejarandconsumingitwithrelish.Iputtherelishbackinthepantry,alongwiththepicklejar.“Wedohavetheoption,”Soamesremarked,“ofmultiplyingnumbersby3,
9,or10usingonlyoneextra1.Wemerelydivideby ,.i,or.”“ThenIhaveit!”Icried:
62=63–1=7×9–1=7/. –1
recallingthatwecanexpress7usingonlytwo1’s—indeed,inatleasttwodifferentways.”
“Leavingonly138toperplexus.”“Itis3×46,”Imused.“Canweachieve46usingjustthree1’s?Thenwe
coulddividethatby asyousuggested.”Asystematicsearchoffloorsandceilingsofrepeatedsquarerootsof
factorialsledustoanunexpecteddiscovery:itispossibletoachieve46withonlytwo1’s.Ipresentonlythesolution:theroutetoitsdiscoveryinvolvedmanydeadendsandfailures.Startwitharepresentationof7usingtwo1’s,forexample,
Thenobservethat
Workingbackwardsandsubstitutingtheformulasforthenumbers,theendresultexpresses138withjustthree1’s.
“ShallIwriteitoutexplicitly,Soames?”“Goodheavensno!Anyonewhowishestoseethefullformulacandothat
forthemselves.”Buoyedbythisunanticipatedsuccess,Iwantedtocontinueevenfurtherwith
ourlist.ButSoamesmerelyshrugged.“Perhapstheproblemmeritsfurthercomputation.Perhapsnot.”
Athoughtstruckme.“Mightweprovethateverynumbercanbeobtainedwithfour1’s—perhapsevenfewer—byiteratingfloorsandceilingsofrepeatedsquarerootsoffactorials?”
“Itisaplausibleconjecture,Watsup,buttobefrankIseenoroutetoaproof,andthestrainofsomuchmentalarithmeticisbeginningtotell.”
Hewassinkingbackintodepression.DesperatelyIsuggested“Youcouldtrylogarithms,Soames.”
“Ithoughtofthoserightatthestart,Watsup.Youmaybesurprisedtohearthatusingnothingmorethanlogarithmstheexponentialfunction,andtheceilingfunction,anypositiveintegercanbeexpressedusingonlyone1.”
“No,no,Imeantusinglogarithmsasacomputationalaid,notintheformula—”ButSoamesignoredmyprotestations.
“Recallthattheexponentialfunctionis
exp(x)=exwheree=2.71828…
anditsinversefunctionisthenaturallogarithm
log(x)=whicheverysatisfiesexp(y)=x
Isthatnotso,Watsup?”Iaffirmedthattothebestofmyknowledgeitwas.“Thenwemerelyobservethat
n+1=[log([exp(n)])]
whoseproofisstraightforward.”Igawpedathim,butmanagedtogetoutastrangled“Ofcourse,Soames.”“Thenweiterate:
1=12=[log([exp(1)])]3=[log([exp([log([exp(1)])])])]
4=[log([exp([log([exp([log([exp(1)])])])])])]
and—”Ihastilygraspedhiswritinghand.“Yes,Soames,Iunderstand.Itisathinly
disguisedvariationonthePeanomethod,whichwerejectedearlierbecauseofitstriviality.”
“Sothegameisnolongerafoot,Watsup,ifexponentialsandlogarithmsarepermitted.”
Iconcurred,withsomesadness,forimmediatelyhetookuphisclarinetandcommencedtoplayarhythmlessatonalcompositionbysomeobscureEasternEuropeancomposer.Itsoundedlikeacatcaughtinamangle.Atone-deafcat.Withasorethroat.
Hisblackmoodwouldnowbeunshakeable.HereEndsTheSignofOne.ExceptthatIstillhaven’ttoldyouwhatasubfactorialis.Thatcomesnext.
SeriouslyDerangedTimetoexplainsubfactorials.
Supposethatnpeopleeachownahat.Theyallpickuponehatandputiton.Inhowmanydifferentwayscanthisbedonesothatnoonegetstheirownhat?Suchanassignmentiscalledaderangement.
Forexample,iftherearethreepeople—Alexandra,Bethany,andCharlotte,say—thentheirhatscanbeassignedinsixways:
ABCACBBACBCACABCBA
ForABCandACB,Alexandragetsherownhat,sothesearenotderangements.ForBAC,Charlottegetsherownhat.ForCBA,Bethanygetsherownhat.Thatleavestwoderangements:BCAandCAB.
Withfourpeople—supposeDeirdrejoinsthegroup—thereare24arrangements:
but15ofthem(crossedout)assignsomeonetheirownhat.(RemoveanythingwithAinthefirstposition,Binthesecond,Cinthethird,orDinthefourth.)Sothereare9derangements.
Thenumberofderangementsofnobjectsisthesubfactorial(denotedby!norn¡).Thishasanumberofdefinitions.Thesimplestisprobably
Itsvaluesbegin
!1=0!2=1!3=2!4=9!5=44
!6=265!7=1,854!8=14,833!9=133,496!10=1,334,961
TossingaFairCoinIsn’tFair
Tossingacoin
Thefaircoinisastapleofprobabilitytheory,equallylikelytolandheadsortails.Itisgenerallyconsideredtobetheepitomeofrandomness.Ontheotherhand,acoincanbemodelledasasimplemechanicalsystem,andassuchitsmotioniscompletelydeterminedbytheinitialconditionswhenitistossed—mainlytheverticalvelocity,theinitialrateatwhichitspins,andtheaxisofspin.Thismakesthemotionnon-random.Sowheredoestherandomnessinacointosscomefrom?I’llcomebacktothatafterdescribingarelateddiscovery.
PersiDiaconis,SusanHolmes,andRichardMontgomeryhaveshownthattossinga‘fair’coinisn’tactuallyfair.Thereisasmallbutdefinitebias:whenacoinistossed,itisslightlymorelikelytolandthesamewayupasitsinitialorientationonyourthumb.Infact,thechanceofitdoingthatisapproximately51%.Theiranalysisassumesthatthecoindoesn’tbouncewhenithitstheground,whichisreasonableforgrass,orwhenitiscaughtinthehand,butnotifitlandsonawoodentabletop.
The51%biasbecomesstatisticallysignificantonlyafterabout250,000flips.Itarisesbecausetheaxisaboutwhichthecoinisspinningmightnotbehorizontal.Asanextremecase,supposethattheaxisisatrightanglestothecoin,sothecoinalwaysremainshorizontalasitspins,likeapotter’swheel.Inthiscase,itwillalwayslandthesamewayupasitwastobeginwith,a100%chanceofnotflippingover.Theotherextremecaseiswhenthespinaxisishorizontal,andthecoinflipsendoverend.Althoughinprinciplethefinalstateisthendeterminedbytheupwardvelocityandrateofspinwhenthecoinleavesthehand,evensmallerrorsinspecifyingthesenumbersimplythatthecoinlandsthesamewayupasitstartedonly50%ofthetime.Withthistypeoftoss,thepredeterminedstateofamechanicalcoinisrandomisedbysmallerrors.
Usually,thespinaxisisinneitheroftheseextremelocations,butsomewhereinbetween,andclosetothehorizontal.Sothereisaslightbiasinfavouroflandingthesamewayup.Detailedcalculationsleadtothe51%figure.Experimentswithacoin-tossingmachineconfirmthisfigurereasonablywell.
Inpracticearealcointossisrandom,witha50%chanceofheadsortails,fornoneofthesereasons.Itisrandombecausetheinitialorientationofthecoin,whenit’ssittingonthethumb,israndom.Inthelongrun,thecoinstartsheadsuphalfthetime,andtailsuphalfthetime.Thisremovesthe51%biasbecausetheinitialstateisnotknownwhenthecoinistossed.
Howheads(white)andtails(shaded)varywiththeinitialrateofrotation(verticalaxis)andthetimespentintheair(horizontalaxis)whenthespinaxisishorizontal.Thehead/tailstripesbecomeverycloselyspaced
whenthespinrateishigh.
Seepage280forfurtherinformation.
PlayingPokerbyPostSupposethatAliceandBob—thetraditionalparticipantsinanycryptographicexchange—wanttoplaypoker,specifically,five-cardstud.ButAliceisinAliceSprings,Australia,whileBobisinBobbington,atowninStaffordshire,England.Cantheyperhapsmaileachotherthecards?Themainproblemisdealingthecards,a‘hand’offivetoeachplayer.Howcanbothplayersbesurethateachhasahandfromthesamepack,withouttheotherknowingtheirhand?
IfBobjustmailsAlicefivecards,shecan’tbesurethathehasn’tseenhercards;moreover,whenBobplayscardsfromwhatallegedlyishishand,shecan’tbesurewhetherhereallyhasonlyfivecardstoworkwith,orwhethertheremainderofthepackisavailabletohim,andheisonlypretendingtouseafixedhandoffivecards,dealtbeforethegamestarted.
Ifthishandarrivedinthepost,youcouldprobablybesurethedealerwasn’tcheating.Butformosthands,howcanyoutell?
Surprisingly,itispossibletoplayacardgamelikepokerbymail,oroverthephone,orovertheInternet,withoutanydangerthateitherplayerischeating.AliceandBobcanusenumbertheorytocreatecodes,andresorttoacomplicatedseriesofexchanges.Theirmethodisknownasazeroknowledgeprotocol,awaytoconvincesomeonethatyoupossessaspecificitemofknowledgewithouttellingthemwhatitis.Forexample,youcouldconvinceanonlinebankingsystemthatyouknowthesecuritycodeonthebackofyourcreditcard,withoutconveyinganyusefulinformationaboutthecodeitself.
Hotelsoftenlockguests’valuablesinasafetydepositboxinthereceptionarea.Toensuresecurity,eachboxhastwokeys:onekeptbythemanagerandtheotherbytheguest.Bothkeysareneededtoopenthebox.AliceandBobcan
useasimilaridea:
1Alicelocksacardineachof52boxes,usingpadlockswhosecodesonlysheknows.ShemailsthelottoBob.
2Bob(whocannotunlocktheboxestoseewhatcardsareinside)picksfiveboxesandmailsthembacktoAlice.Sheunlocksthemandreceivesherfivecards.
3Bobchoosesanotherfiveboxes,andputsanextrapadlockoneach.Heknowsthecodesforunlockingthese,butAlicedoesnot.HemailstheboxestoAlice.
4Aliceremovesherpadlocksfromtheseboxes,andmailsthembacktoBob.Hecannowopenthemtoreceivehisfivecards.
Afterthesepreliminaries,thegamecanstart.Cardsarerevealedbymailingthemtotheotherplayer.Toprovenoonecheated,theycanunlockalltheboxesaftertheendofthegame.
AliceandBobconvertthisideatomathematicsbyextractingtheessentialfeatures.Theyrepresentthecardsbyanagreedsetof52numbers.Alice’spadlockscorrespondtoacodeA,knownonlytoAlice.Thisisafunction,amathematicalrule,thatchangesacardnumbercintoanothernumberAc.(I’mtakinglibertieswiththenotationbynotwritingA(c),inordertoavoidtalkingabout‘composing’functions.)AlicealsoknowstheinversecodeA–1,whichdecodesAcbackintoc.Thatis,
A–1Ac=c
Bobdoesn’tknowAorA–1.Similarly,Bob’spadlockscorrespondtocodesBandB–1,knownonlyto
Bob,suchthat
B=1Bc=c
Withthesepreliminaries,themethodcorrespondstothepadlockprocedurelikethis:
1Alicesendsall52numbersAc1,…,Ac52toBob.Hehasnoideawhichcardsthesecorrespondto;ineffect,Alicehasshuffledthepack.
2Bob‘deals’fivecardstoAliceandfivetohimself.HesendsAlicehercards.Tosimplifythenotation,considerjustoneofthese,andcallitAc.AlicecanfindcbyapplyingA–1,sosheknowswhatcardsareinherhand.
3Bobneedstofindoutwhathisfivecardsare,butonlyAliceknowshowtoworkthatout.Buthecan’tsendhiscardstoAlicebecausethenshewillknowwhattheyare.SoforeachcardAdinhishand,heapplieshisowncodeBtogetBAd,andsendsthattoAlice.
4AlicecanagainapplyA–1to‘removeherpadlock’,butthistimethere’sasnag:theresultis
A–1BAd
InordinaryalgebrawecouldswapA–1andBroundtoget
BA–1Ad
whichequals
Bd
ThenAlicecouldsendthatbacktoBob,whowouldthenapplyB–1tofindd.However,functionscan’tbeswappedlikethis.Forexample,ifAc=c+1(so
A–1c=c–1)andBc=c2,then
A–1Bc=Bc–1=c2–1
whereas
BA–1c=(A–1c)2=(c–1)2=c2–2c+1
whichisdifferent.Thewayroundthisobstacleistoavoidthatsortoffunction,andsetupthe
codessothatA–1B=BA–1.Inthiscase,AandBaresaidtocommute,becausealittlealgebraturnsthisintotheequivalentconditionAB=BA.Noticethatinthephysicalmethod,Alice’sandBob’spadlocksdoindeedcommute.Theycanbeappliedineitherorder,andtheresultisthesame:aboxwithtwopadlocks.
AliceandBobcanthereforeplaypokerbypostiftheycansetuptwocommutingcodesAandB,sothatthedecodingalgorithmA–1isknownonlyto
Alice,andB–1isknownonlytoBob.BobandAliceagreeonalargeprimenumberp,whichcanbepublic
knowledge.Theyagree52numbersc1,…,c52(modp)torepresentthecards.Alicepickssomenumberabetween1andp–2,anddefineshercode
functionAby
Ac=ca(modp)
Usingbasicnumbertheory,theinverse(decoding)functionisoftheform
A–1c=ca’(modp)
foranumbera’thatshecancompute.Alicekeepsbothaanda’secret.BobsimilarlychoosesanumberbanddefineshiscodefunctionBby
Bc=cb(modp)
withinverse
B–1c=cb’(modp)
foranumberb’thathecancompute.Hekeepsbothbandb’secret.ThecodefunctionsAandBcommute,because
ABc=A(cb)=(cb)a=cba=cab=(ca)b=B(ca)=BAc
whereallequationshold(modp).SoAliceandBobcanuseAandBasdescribed.
EliminatingtheImpossibleFromtheMemoirsofDrWatsup
“Watsup!”“Uh—I’mnotsure,Soames.Whatisup?”“I’mcallingyourname,man,notaskingaquestion!HowmanytimeshaveI
toldyounottobringcopiesofTheStrandmagazineintothishouse?”“But—how—”“Youknowmymethods.Youweretappingyourfingersimpatiently,asyou
dowhenwaitingformetogoout.Andyoureyeskeptflickingtowardsthenewspaperrolledupinyourcoatpocket.WhichistoothickfortheDailyReporter,despitewhatitsaysonthefrontpage,soitmustcontainamagazine.Sinceyouhabituallyconcealonlyonesuchfromme,itsidentitywasneverindoubt.”
“I’msorry,Soames.Iwasjusthopingtogainsomecomparativeinsightsintoinvestigativemethodsfromthewritingsofthecompanionofthe,er,charlatanlivingacrossthestreetfromus.”
“Pah!Themanisafraud!Amountebankwhocallshimselfadetective!”TherearetimeswhenSoamescanbeoverbearing.Indeed,fewwhenheis
not,nowthatIcometothinkofit.“Ihaveoccasionallysiftedsomeusefulhintsfrommyexploitedcounterpart’soutpouringsofvapidity,Soames,”Iobjected.
“Suchas?”heasked,inanaggressivetone.“Iamimpressedbyhisargument:‘Whenyouhaveeliminatedthe
impossible,thenwhateverremains,howeverimprobable,mustbe—’““—wrong,”Soamesbrusquelycompletedmysentence.“Ifwhatremainsis
trulyimprobable,thenyouhavealmostsurelymadesomeunstatedassumptionwhendeclaringotherexplanationstobeimpossible.”
ConsistencyisnotoneofSoames’svirtues.“Well,perhaps,but—”“Nobuts,Watsup!”“Butonotheroccasionsyouhaveagreed—”“Pah!Realityisnotimprobable,Watsup.Itmaylookthatway,butits
probabilityis100%,becauseithashappened.”“Yes,technically,but—”“Acaseinpoint.Thismorning,Watsup,whileyouwereoutbuyingthat
scurrilousrag,Ireceivedanunexpectedvisitor.TheDukeofBumbleforth.”
“ThetoastofLondon,”Isaid.“Anoblemanofunimpeachableprobity,arolemodelforusall.”
“Indeed.Yetheinformedme…Well,therehad,hesaid,beenadinnerpartyatBumbleforthHall,atwhichtheEarlofMaunderingattemptedtoentertaintheguestsbyarrangingtenwineglassesinarowandfillingthefirstfiveofthem—likethis.”
Hedemonstratedwithourownglasses,fillingthemwitharatheracidMadeirathatwehaddecidedtothrowout.“Hethenchallengedthegueststorearrangetheglassessothattheyalternatedfullandempty.”
“Butthatiseasy,”Ibegan.“Ifyoumovefourglasses,yes.Interchangethesecondwiththeseventhand
thefourthwiththeninth.Likeso.”(Seethefigurebelow.)“However,theEarl’schallengewastoobtainthesameresultbymovingonlytwooftheglasses.”
Howtosolvethepuzzleinfourmoves
Ipressedmyfingerstogetherinanattitudeofdeepthought,andafteramomentIdrewaroughsketchoftheoriginalandfinalarrangements.“ButSoames:thefourglassesyouhavenamedmustallendupindifferentlocations!Soallfourmustmove!”
Henodded.“So,Watsup,youhavenoweliminatedtheimpossible.”“ByJove,yesIhave,Soames!Incontrovertibly.”Hebeganstuffingtobaccointohispipe.“SowhatdoyouconcludeifItell
youthataccordingtotheDukeofBumbleforth,afteralloftheguestshadexpressedsimilaropinions,theEarlofMaunderingthendemonstratedasolution.”
“I—uh—”“YouareforcedtoconcludethatthehonourableDuke,ascionoftheBritish
Empireandamanofhighnobility…isactuallyabaseliar.Fornosolutionexists,asyouhaveproved.”
Myfacefell.“Itdoesseem—no,wait,perhapsyouarenottellingmethe—”“Mydeardoctor,IfreelyconfessthatIdissemblefromtimetotime,always
withyourownbestinterestsatheart,butnotonthisoccasion.Youhavemyword.”
“Then…IamshockedattheDuke’sbehaviour.”“Comenow,Watsup,HavefaithinBritishcharacter.”“TheEarlcheated?”“No,no,no.Nothingofthekind.Youcandobetterthanthis.Theremaybe
anotherperfectlyprosaicexplanationthatyouhaveoverlooked.Infact,Ipredictthatyouwillshortlybetellingmehowchildishlysimpletheansweris.”
SoamesthentoldmewhatMaunderinghaddone.“Why,howchildishlysim—”Ibegan.Istoppedabruptly,andamforcedon
groundsofcandidnesstoadmitthatIflushedadeepcrimson.
WhatwasMaundering’ssolution?Seepage281fortheanswer.
MusselPowerIt’sanidyllicseasidescene:aquietbaywithwavesbreakingovertherocks,whicharefestoonedwithclumpsofshellfishandseaweed.Butthosesedatestaticmusselbedsareactuallyahiveofactivity.Toseeit,youjusthavetospeeduptheflowoftime.Intime-lapsephotography,themusselsareconstantlyonthemove.Theytetherthemselvestotherocksusingspecialthreads,secretedbytheirfoot.Bydetachingsomethreadsandaddingothersinnewlocations,themusselscancontroltheirpositionsontherocks.Ontheonehand(foot?),theyliketostayclosetoothermusselsbecausethatwaytheyarenotsolikelytoberippedofftherockbythewaves.Ontheother,theygetmorefoodiftherearen’tothermusselsnearbytocompetewiththem.Facedwiththisdilemma,musselsdowhatanysensibleorganismwould:theycompromise.Theyarrangethemselvessothattheyhavealotofnearneighboursbutfewdistantones.Thatis,theycongregateinpatches.Youcanseethepatcheswiththenakedeye,butnothowtheyform.
Clumpofbluemussels
In2011MoniquedeJagerandco-workersappliedthemathematicsofrandomwalkstodeducehowthemussels’clumpingstrategymighthaveevolved.Arandomwalkisoftencomparedtoadrunkardmovingalongapath:sometimesforward,sometimesback,withnoclearpattern.Goingupadimension,arandomwalkintheplaneisaseriesofsteps,whoselengthsanddirectionsarechosenrandomly.Differentrulesforthechoices—differentprobabilitydistributionsforthelengthsanddirections—leadtorandomwalks
withdifferentproperties.InBrownianmotion,thelengthisdistributedinabellcurve,closetoonespecificaveragestepsize.InaLévywalk,theprobabilityofmakingastepisproportionaltosomefixedpowerofitssize,somanyshortstepsareoccasionallyinterruptedbyamuchlongerone.
StatisticalanalysisofobservedstepsizesshowveryclearlythataLévywalkfitswhatmusselsonintertidalmudflatsactuallydo,whereasBrownianmotiondoesn’t.Thisagreeswithecologicalmodels,whichdemonstratemathematicallythataLévywalkdispersesthemusselsfaster,opensupmorenewsites,andavoidscompetitionwithotherspeciesofshellfish.Thisinturnsuggestswhythisparticularstrategymayhaveevolved.Naturalselectionprovidesafeedbackloopbetweenmovementstrategiesandthegeneticinstructionsthatcausethemtobeused.Individualmusselsaremorelikelytosurviveiftheyemploystrategiesthatincreasetheirchancesofobtainingfood,anddecreasetheirchancesofbeingsweptawaybyawave.
DeJager’steamusedfieldobservationsofwhatmusselsdoandsimulationsofmathematicalmodelsoftheevolutionaryprocess.ThesimulationsshowedthatLévywalksarelikelytoevolveasaresultofthispopulation-levelfeedback,andtheconditionsforthatstrategytobeevolutionarilystable—thatis,notsusceptibletoinvasionbyamutantwithadifferentstrategy—predictthatthepower-lawexponentshouldbe2.Thefielddataindicateavalueof2.06.
Thenovelfeatureofmusselbeds,inthiscontext,isthattheeffectivenessofanindividual’smovementstrategydependsonwhatalltheothermusselsaredoing.Eachmussel’sstrategyisdeterminedbyitsowngenetics,butthesurvivalvalueofthatstrategydependsonthecollectivebehaviouroftheentirelocalpopulation.Sohereweseehowtheenvironment—intheformoftheothermussels—interactswithindividualgenetic‘choices’toproducepatternformationonthepopulationlevel.
Seepage281forfurtherinformation.
ProofThattheWorldisRoundMostofusareawarethatourplanetisround—notanexactsphere,though:abitflattenedatthepoles.Ithasenoughbumpstoturnitintoapotatoifyouexaggeratethediscrepancyfromaspheroidbyafactorofabout10,000.Afew—veryfew—hardysoulspersistinthebeliefthattheworldisflat,eventhoughtheancientGreeks,2,500yearsago,amassedevidenceofitsrotunditythatconvincedevenmedievalclerics,andmoreevidencehasbeenpilingupeversince.BeliefinaflatEarthalmostdiedout,butitwasrevivedaround1883withthefoundingoftheZeteticSociety.ThisbecametheFlatEarthSocietyin1956,anditisstillactivetoday.YoucanfinditontheInternetandfollowitonFacebookandTwitter.
There’saneasyandvirtuallyfoolproofwaytocheckforyourselfthatourplanetcan’tbeflatiftheusualgeometryofEuclidapplies.ItrequiresInternetaccessoratoleranttravelagent,butnootherspecialapparatus,andit’snotlookinguptheshapeonWikipedia.Themethoddoesn’tofitselfshowthattheEarthisround,butasystematicandcarefulextensionwouldbeabletodojustthat.I’lldiscusspotentialwaystodenythisevidenceinamoment.Idon’tclaimthere’snowayout—ifyou’reaflat-Eartherthere’salwaysawayout.Butinthisinstance,thestandardploysareevenlessconvincingthanusual.Inanycase,theargumentmakesarefreshingchangefromtheusualscientificevidenceforaroundworld.
I’mnotthinkingofsatellitephotosofaroundplanet—thoseare,ofcourse,fakes.WeallknowthatNASAneverwenttotheMoon,itwasalldoneinHollywood,whichprovesthey’refakes,sothere.Nothingthatreliesonscientificmeasurement,either:thosescientisttypesarewell-knownhoaxers,theyevenpretendtheybelieveinevolutionandglobalwarming,bothofwhichareleftieplotstostopclean-living,righteous-thinkingpeoplemakingtheobscenesumsofmoneythataretheirGod-givenright.
No,whatIhaveinmindiscommercialevidence:airlineflighttimes.YoucanlooktheseupontheInternet:makesureyouuseactualflightsthatexist,notflight-timecalculatorsthatassumearoundEarth.
Forcommercialreasonsalllargepassengerjetsflyataboutthesamespeed.Iftheydidn’t,othercompanieswouldgetallthebusinessfromtheslowones.Theyflytheshortestroute,subjecttolocalregulations,forsimilarreasons.Sowecanusetraveltimesasreasonablyaccurateestimatesofdistances.(Toreduce
theeffectsofwind,takeasuitableaverageofflighttimesinbothdirections—inpracticetheusualarithmeticmeanisgoodenough,butSeepage281.)Thesurveyor’stechniqueoftriangulation,whichconstructsanetworkoftriangles,canthenbeusedtomapoutthelocationsoftheairportsconcerned.ForthepurposeofshowingthataflatEarthdoesn’twork,wecanassumeit’sflatandseewhatthatimplies.Surveyorsusuallyworkwithoneinitialdistance,thebaseline,andcalculateeverythingelsefromtheanglesofthetriangle,butwehavetheluxuryofusingactualdistances(inunitsofaircraft-hours).
Flat-Earthairlinemap
Thefigureshowsatriangulationbasedonsixmajorairports.Giveortakealittlejiggling,thisistheonlyplanararrangementthatisareasonablefittothetraveltimes.StartwithLondonandaddCapeTowndistance12away.AfterthatplaceRiodeJaneiroandSydney.Theirlocationsareunique,exceptthattheentiremapcouldbereflectedleft-rightwithoutchanginganydistances.Thatambiguitydoesn’tmatter,butyoudohavetoconfirmthatRiodeJaneiroandSydneyareonoppositesidesofthelinefromLondontoCapeTown.Iftheywereonthesameside,theflighttimebetweenthemwouldbeabout11hours,butactuallyit’s18.YoucanaddLosAngelesnext,andfinallylocateTahiti,againusingextratimingstoremoveambiguities.
NowwecanusethehypothesisofaflatEarthtomakeaprediction.ThedistancefromTahititoSydney,measuredfromthemap,isabout35hours.(Asithappens,therouteviaRioandCapeTownisveryclosetoastraightline,andthesumofthedistancesis35.)Sothat’stheminimumtimeitshouldtaketotravelbyplane,notcountingstops.
Theactualfigureis8hours.Evenallowingforminorerrors,thedifferenceiswaytoolarge,andthehypothesisofaflatplanetmustberejected.Withanetworkwithmanymoreairports,andmoreprecisefigures,youcouldmapoutthebasicshapeofmuchoftheplanetveryaccurately—stillinunitsofaircraft-hours.Tosetthescaleyou’dneedtoknowhowfasttheplanesfly,ormakeatleastonedistancemeasurementsomeotherway.
UnitedNationslogo:azimuthalequiangularprojectionfromroundEarthtoflatdisc
Now,everywell-informedflat-Eartherisawareofformsofwordsandnon-standardphysicsthatcan‘explain’theseresults.Perhapssomekindofdistortionfieldaltersthegeometry,sothattheliteralimageofaplanewithitsusualmeasureofdistanceiswrong.Thisreallydoeswork:anazimuthalequiangularprojectionoftheEarthfromthenorthpoledoesjustthat,andyoucantransfereverythingfromaroundEarthtoaflatone,lawsofphysicsincluded,usingprojectionontoaflatdisc.Providedyoumissouttheregionaroundthesouthpole.TheUnitedNationslogodoesjustthat,andhasbeenusedbytheFlatEarthSocietyto‘prove’thatitsviewsarecorrect.However,thiskindofchangeistrivialandmeaningless.It’salogicallyequivalentmodeltoaroundEarthwithconventionalgeometry.Mathematicallyitisjustalessthancandidwaytoadmit‘it’snotflat’,withintheorthodoxmeaningofthatphrase.Soalteredmetricsandsimilarexcusesreallydon’thackit.
Effectsofwind?Maybethere’sareallyhighwindblowingfromTahititoSydney?Itwouldhavetoblowat750mph,butworsethanthat:thestraightlinepathbetweenTahitiandSydneyisveryclosetotheroutesTahiti-Rio-CapeTown-Sydneythatwe’vealreadytakenintoaccount.IfyoucouldgetfromTahititoSydneyreallyquicklybyexploitingthewind,atleastoneofthoseroutesistakingwaytoolong.
Thenextlineofdefencewouldbethestandarddenialistploy:it’sallahugeconspiracy.Yes,butbywhom?ThetimeslistedonInternetbookingsitescan’tbefarwrong,becausemillionsofpeopletravelbyaireverydayandmostofthemwouldnoticeifthescheduledtimeswerefrequentlywildlywrong.Buttheairlinesmightallbeconspiringtoflymoreslowlythannecessaryonsomeroutes,sothatmostofmymapshouldshrink,makingitpossibletogetfromTahititoSydneyinamere14hours.Youhavetodivideby4ormoretomakethispossible,soaconventionalpassengerjetcouldactuallygetfromLondontoSydneyin5hoursiftheairlinewasn’tdeliberatelydawdlingtoconvinceusthe
planetisround.Unlikeallegationsofscientistsconspiring,whichonlymakesensetopeople
whodon’tknowanyscientists,*thisonehasaflawthatisprettymuchfatal.Itrequiresmostairlinestolosehugeamountsofmoneyeverydayinwastedfuel,andtorefrainfromwipingoutthecompetitionbyflyingroutesinlessthanhalfthetimetheycurrentlytake.AconspiracytomaketheEarthappearround,usingtheairline-schedulemetric,wouldrequirehundredsofprivate-sectorcompaniestovoluntarilythrowawayvastsumsofmoney.Areyoumad?
Youcan,ofcourse,alwaysfallbackonthatoldstandby:whenallelsefails,ignoretheevidence.
Footnote
*I’mnotreferringtothepropositionthatscientistsaremostlyhonest.I’mreferringtothedelighttheyalltakeinprovingeachotherwrong,whichamongotherthingsishowtheygetpromoted.Massiveconspiracieswouldstillmakenosenseevenifallscientistswerecrooks.
123456789TimesXContinuedThere’snoneedtostopat9(Seepage115).Trymultiplying123456789by10,11,12,andsoon.Whatdoyounoticenow?
Seepage282fortheanswer.
ThePriceofFame
WładysławOrlicz
WładysławRomanOrlicz,aPolishtopologist,introducedwhatarenowknownasOrliczspaces.Thesearehighlytechnicalconceptsinfunctionalanalysis.Onedayhisfameprovedcounterproductive.Likemostofhiscompatriots,helivedinaverysmallapartment,andonedayheappliedtothecityofficialsaskingforabiggerone.Thereplywas:“Weagreethatyourapartmentisverysmall,butwemustdenyyourclaimsinceyouhaveyourownspaces.”
TheRiddleoftheGoldenRhombusFromtheMemoirsofDrWatsup
Thespectacularsuccessofourjointendeavoursencouragedmeoncemoretotakeupmedicalpractice,andIarrangedforasmallsurgerytobeconstructedinmyhouse.ButItookcaretoallowsufficientflexibilityforoccasionswhenSoamesmightrequiremyservices,withorwithoutadequatenotice.Sowhenthetelegramarrived,IhandedmypatientovertomylocumDrJekyll,andsummonedacabtotakemeto222BBakerStreet.
WhenIarrivedatSoames’slodgingsIfoundhimsurroundedbypiecesofpaper.Inhishandswasapairofscissors.
“Aprettypuzzle,”heremarked.“Merelyarectangularstripofpaper,tiedintoasimpleoverhandknot.Itishardtoimaginethataman’sfatemaydependuponit.”
Knottedstripofpaper
“GreatHeavens,Soames!Howcouldthatbe?”“Anastycaseofextortion,Watsup.Theevidencehingesontheshapethatis
formedwhentheknotispulledastightlyaspossible,andflattenedout.Isuspectthatitwillturnouttobethesymbolofasecretsociety,andifIcansoprove,mycasewillbecomplete.”Heheldtheknotbeforemyeyes.“So,Watsup,whatshapewillwesee,eh?”
Iquicklysketchedanoverhandknotinmynotebook.
Overhandknottiedinaclosedloopofstring
“Itiswellknownthatwhenanoverhandknotistiedinaclosedloop,ithasthreefoldsymmetry,”saidI,feelingunusuallyastute.“Iwouldthereforeexpecteitheratriangleorahexagontoform.”
“Letustrytheexperiment,then,”saidSoames.“Andthenweshalltacklethemoredifficulttaskofprovingthatoureyesdonotdeceiveus.”
Whatshapeistheflattenedknot?Tryit.Seepage282fortheanswerandaproof.
APowerfulArithmeticSequenceAnarithmeticsequence(asequenceofnumberswithconstantdifferences)ispowerfulifthesecondtermisasquare,thethirdtermacube,andsoon.Thatis,thekthtermisakthpower.(Thisimposesnoconditiononthefirsttermsinceallnumbersarefirstpowersofthemselves.)Forexample5,16,27haslength3andcommondifference11,and
5=5116=4227=33
Atrivialwaytogetapowerfulsequenceoflengthnistorepeatntimesthenumber2n!.Thisisafirstpower,asquare,acube,andsoon,uptoannthpower.Thecommondifferenceiszero.
In2000JohnRobertsonprovedthatexcludingsequenceslikethis,wherethesamenumberrepeats—commondifferencezero—thelongestpossiblepowerfulsequencehasfiveterms(length5).SeeJohnP.Robertson,Themaximumlengthofapowerfularithmeticprogression,AmericanMathematicalMonthly107(2000)951.Toobtainthissequence,startwiththenumbers1,9,17,25,33,whichformanarithmeticsequencewithcommondifference8,andmultiplyeachofthemby32453011241720.Theresultingnumbersalsoformanarithmeticsequence,withcommondifference8timesthisnumber.Theyare:
(1)10529630094750052867957659797284314695762718513641400204044879414141178131103515625
(2)94766670852750475811618938175558832261864466622772601836403914727270603179931640625
(3)179003711610750898755280216553833349827966214731903803468762950040400028228759765625
(4)263240752368751321698941494932107867394067962841035005101121985353529453277587890625
(5)347477793126751744642602773310382384960169710950166206733481020666658878326416015625
Thecommondifferenceis
84237040758000422943661278378274517566101748109131201632359035313129425048828125000
Ifthefivetermsarea1,a2,a3,a4,a5,then
a1isthefirstpowerofitself(obviously)a2=3078419575898491388288844129170837402343752isasquarea3=56357797471169485761035156253isacubea4=7162889984611066406254isafourthpowera5=510722993555156255isafifthpower
Wow!(It’seasiertocheckthatthetermsarethestatedpowersifyouworkwiththe
primefactors.)
WhyDoGuinnessBubblesGoDownwards?Anyonewhodrinksdarkstout,suchasGuinness,willhaveseensomethingthatappearstoflyinthefaceofconventionalphysics.Thebubblesinthebeermovedownwards.Atleast,theyseemto.Butbubblesarelighterthanthesurroundingfluid,sotheyexperienceabuoyancyforcethatpushesthemup.
It’sagenuinemystery,oratleastitwasuntil2012whenateamofmathematicianssolvedit.Appropriately,theywereIrish(orbasedinIreland):WilliamLee,EugeneBenilov,andCathalCumminsoftheUniversityofLimerick.
Thesameeffectoccursinotherliquids,butit’seasiertoseeinstoutbecauselight-colouredbubblesshowupmoreclearlyagainstadarkbeer.Itisenhancedbecausestoutbubblescontainnitrogenaswellasthecarbondioxidethatisfoundinallbeers,andnitrogenbubblesaresmallerandlastlonger.
Partoftheansweriseasy:we’reonlyseeingthebubblesthatareneartheglass.Thoseoutinthemiddlearehiddenfromviewbythedarkbeer.Somaybesomebubblesaregoingup,butothersdown.Whatthatfailstoexplainiswhyanybubblesgodown.Theyshouldn’t.
Untilafewyearsagowedidn’tevenknowwhetherthewholethingwasanopticalillusion.Onealternativeexplanationisthattheeffectiscausedbydensitywaves—regionswherebubblesbunchup.Thebubblesgoupwardsbutthedensitywavesgotheotherway.Thiskindofbehaviouriscommoninwaves.Forexample,thewaterinoceanwavesdoesnottravelalongwiththewave;itmostlygoesroundandroundinroughlythesameplace.Whatmovesisthelocationofthehighestpartsofthewater.Admittedly,wavesbreakingonabeachdogoupthebeach;however,someofthatistheeffectofshallowwater,andthewaterrunsbackdowntothesea.Ifthewaterweretravellingwiththewaves,itwouldhavetopileupeverhigheronthebeach,whichdoesn’tmakemuchsense.Althoughthewaterdoesn’tgobackwardstoanysignificantextent,thisfamiliarexampleshowsthedifferencebetweenwherethewatergoesandwherethewavesgo.Nowdoitwithbubbles.
It’safairlyplausibletheory,butin2004agroupofScottishscientistsledbyAndrewAlexander,workingwithcolleaguesinCalifornia,producedvideofootageprovingthatthebubblesreallydomovedownwards.TheteamreleaseditsresultsonStPatrick’sDay.Theyusedahigh-speedcameratoslowdownthemovementandtrackindividualbubbles.Theyfoundthatthebubblestouching
thewallsoftheglasstendedtostick,sotheycouldn’tmoveupwards.However,bubblesnearthemiddlewerefreetorise,whichcausedthebeertoflowupinthemiddleanddownatthesides,draggingthebubbleswithit.
FlowofGuinnessinaglass:downattheedges
TheIrishteamrefinedthisexplanation,showingthatit’snotcausedbybubblesstickingtothewalls.Whatmakesithappenistheshapeoftheglass.Stoutisusuallydrunkfromaglasswithcurvedsides,whichiswideratthetopthanatthebottom.Usingfluiddynamicscalculationsandexperiment,theteamfoundthatwhenthebubblesnearthewallrise,theygostraightup,asyou’dexpect.Butthewallslopesawayfromthevertical,soineffectthebubblesmoveawayfromthewall.Thebeernearthewallisthereforedenserthanthatinthemiddle,soittendstoslidedownthesideoftheglass,draggingthenearbyfluidwithit.Sothebeercirculates:upwardsinthemiddle,downwardsnearthesides.
Thebubblesarealwaysgoingupwardsrelativetothebeer,butattheedgesthebeergoesdownfasterthanthebubblesgoup,sothebubblesgodowntoo.Weseethebubbles,butwecan’teasilyseethemotionofthebeer.
Seepage286forfurtherinformation.
RandomHarmonicSeriesTheinfiniteseries
iscalledtheharmonicseriesbymathematicians.Thenameislooselyrelatedtomusic,wheretheovertonesofavibratingstringhavewavelengths1/2,1/3,1/4,andsoon,ofthestring’sfundamentalwavelength.However,theserieshasnomusicalsignificance.Itisknowntobedivergent,meaningthatthesumuptontermsbecomesaslargeaswewishifnislargeenough.Itdivergesveryslowly,butitdoesdiverge.Infact,thefirst2ntermsadduptomorethan1+n/2.Ontheotherhand,ifwechangethesignofeveryotherterm,wegetthealternatingharmonicseries
whichconverges.Itssumislog2,whichisabout0.693.ByronSchmulandwonderedwhathappensifsuccessivesignsarechosenat
random,bytossingafaircoinandassigningaplussignto‘heads’andaminussignto‘tails’.Heprovedthat,withprobabilityone,thisseriesconverges(theharmonicserieswouldcorrespondtotossingHHHHHH…forever,whichhasprobabilityzero).However,thesumdependsonthesequenceoftosses.
Thequestionnowarises:whatistheprobabilityofgettingaparticularsum?Thesumcanbeanyrealnumber,positiveornegative,sotheprobabilityofgettinganyspecificnumberiszero(thisisgenerallythecasefor‘continuousrandomvariables’).Thewaytodealwiththisistointroduceaprobabilitydistribution(ordensity)function.Thisdeterminestheprobabilityofgettingasuminanygivenrangeofvalues,saybetweentwonumbersaandb.Thisprobabilityistheareaunderthedistributionfunctionbetweenx=aandx=b.
Forharmonicseriesmodifiedusingrandomcointosses,theprobabilitydistributionlookslikethefigurebelow.It’sabitlikethefamiliarbellcurve,ornormaldistribution,butthetoplooksflat.Ithasleft–rightsymmetry,correspondingtoswapping‘heads’and‘tails’onthesymmetriccoin.
Probabilitydistributionforrandomharmonicseries
Thisproblemisanobjectlessonin‘experimentalmathematics’,inwhichcomputercalculationsareemployedtosuggestinterestingconjectures.Itlooksasthoughthecentralpeakisatheight0.25,thatis,1/4.Italsolooksasthoughthevaluesofthefunctionat–2and+2are0.125,thatis,1/8.In1995KentMorrisonconjecturedthatbothstatementsaretrue,butin1998hechangedhismind,havinginvestigatedtheconjecturesinmoredetail.Totendecimalplaces,thedensityhasvalue0.2499150393atx=0,slightlylessthan1/4.However,totendecimalplacesthevalueatx=2is0.1250000000,whichstilllookslike1/8.To45decimalplaces,however,thevalueturnsouttobe:
0.124999999999999999999999999999999999999999764
whichdiffersfrom1/8bylessthan10–42.Schmuland’spaper[Randomharmonicseries,AmericanMathematical
Monthly110(2003)407–416]explainswhythisprobabilityissocloseto,butnotexactly,1/8.Sohereaveryplausibleconjecturefromexperimentalevidenceturnsouttobefalse.Thisiswhymathematiciansinsistonproofs,justasHemlockSoamesinsistsonevidence.
TheDogsThatFightintheParkFromtheMemoirsofDrWatsup
TakingmyusualmorningconstitutionalinEquilateralPark,justoffMaryleboneRoadneartheDogandTrianglepublichouse,Iobservedacuriousincident,andonarrivingto222BBakerStreetIcouldnotrestrainmyselffromsharingitwithmycolleague.
“Soames,Ihavejustwitnessedacurious—”“Incident.Yousawthreedogsinthepark,”saidhe,notbattinganeyelid.“Buthow—ofcourse!Thereismudonmytrousers,andthespatterpatterns
indicate—”Soameschuckled.“No,Watsup,mydeductionhasanotherbasis.Ittellsme
notonlythatyousawthreedogsinthepark,butthattheywerefighting.”“Sotheywere!Butthatwasnotthecuriousincident.Itwouldhavebeen
curioushadthedogsnotfought.”“True.Imustrememberthatremark,Watsup.Mostapposite.”“Whatwascuriouswaswhatprecededthefight.Thedogsappeared
simultaneouslyatthethreecornersofthepark—”“Whichisanequilateraltrianglewhosesidesareall60yards,”Soames
interjected.“Uh,yes.Atthemomenttheyappeared,eachdogfacedthenextina
clockwisedirection,andimmediatelybegantoruntowardsit.”“Eachatthesamespeedof4yardspersecond.”“Ibowtoyourjudgement.Asaresult,allthreedogsfollowedcurvedpaths,
andcollidedsimultaneouslyatthecentreofthepark.Inaflash,theywerefighting,andIhadtoseparatethem.”
“Whencethetearsinyourcoatandtrousers,andtheteethmarksonyourleg,whichIseewereinflictedbyoneredsetter,oneretriever,andonecrossbetweenabulldogandanIrishwolfhound.Withalamefrontleftleg.”
“Ah.”“Andwearingaredleathercollar.Withabellonit.Whichhasrustedandno
longerrings.Wereyouobservantenoughtonoticehowlongittookthedogstocollide?”
“Ineglectedtolookatmypocket-watch,Soames.”“Oh,comenow,Watsup!Youlook,butyoudonotsee.However,inthis
casethetimecanbededucedfromthefactsalreadyestablished.”
Assumethedogsarepoints.Seepage286fortheanswer.
Thethreedogs
HowTallisThatTree?There’sanoldforester’strick(thetrickisold,nottheforester)forestimatingtheheightofatreewithoutclimbingitorusingsurveyingequipment.Itcanbeagreaticebreakeratgardenpartiesifthere’sasuitabletreeinthevicinity.IlearneditfromTobyBuckland,Diggingdeeper,AmateurGardening(20October2012)page59.Trousersaretherecommendedattire.
Standareasonabledistancefromthetree,withyourbacktowardsit.Bendoverandlookbackatitthroughyourlegs.Ifyoucan’tseethetop,moveawayuntilyoucan.Ifyoucanseeiteasily,movecloseruntilit’sjustvisible.Atthatpoint,yourdistancefromthebaseofthetreewillberoughlyequaltoitsheight.
Estimatingtheheightofatree
Thetechnique,ifImaycallitthat,isasimpleapplicationofEuclideangeometry.Itworksbecausetheangleatwhichmostofuscanlookupwardsthroughourlegsisroughly45˚.Sothelineofsighttothetopofthetreeisthehypotenuseofanisoscelesright-angledtriangle,andtheothertwosidesareequal.
Obviouslytheaccuracyofthemethoddependsonhowflexibleyourbodyis,butit’snottoofarwrongformanyofus.Bucklandremarks:“Haveago,it’scheaperthanyogaandoffersaviewontheworldthatmosthaven’tenjoyedsincechildhood!”
WhyDoMyFriendsHaveMoreFriendsThanIDo?OMG!EveryoneseemstohavemorefriendsthanIdo!
IthappensonFacebook,ithappensonTwitter.Ithappensonanysocialmediawebsite,andithappensinreallife.Ithappensifyoucountbusinesspartnersorsexualpartners.It’sahumblingexperiencewhenyoustartcheckingyourfriendstoseehowmanyfriendstheyhave.Notonlydomostofthemhavemorethanyoudo:onaverage,theyallhavemorethanyoudo.
Whyareyousounpopularcomparedtoeveryoneelse?It’sveryworrying.Butthere’snoneedtobeupset.Mostpeople’sfriendshavemorefriendsthantheydo.
Samplefriendshipnetwork
Thatprobablysoundsweird.Everyoneinagivensocialnetworkhasonaveragethesamenumberoffriends;namely,theaverage–thereisonlyone.Somehavemore,somehavefewer,buttheaverageis…theaverage.Itthenseemsintuitivelyplausiblethattheirfriendswill,onaverage,alsohavethatverysamenumberoffriends.Butisittrue?
Let’stryanexample.It’snotconcoctedtobeunusual;it’sthefirstoneIdrew.Mostnetworkswillbehavesimilarly.Thenetwork(above)shows12people,withlinesconnectingthefriends.(Weassumeallfriendshipsworkbothways.That’snotalwaysthecaseforsocialnetworks,buttheeffectstilloccursevenso.)Tabulateafewkeyfigures:
I’veusedboldfacetomarkthenumbersinthefinalcolumnthatarebiggerthanthenumbersinthesecond.ThesearethecaseswherepersonX’sfriendshave,onaverage,morefriendsthanpersonXhas.Eightoutoftwelvenumbersareboldface,andthere’sanotherwherethetwofiguresareequal.
Theaverageofthenumbersincolumn2is3.Thatis,theaveragenumberoffriends,acrosstheentiresocialnetwork,is3.Butthemajorityofentriesincolumn4arebiggerthanthat.Sowhat’swrongwithintuitionhere?
TheanswerispeoplelikeEthelandGwenwhohaveanunusuallylargenumbersoffriends,here5and6respectively.Becauseofthis,theygetcountedalotmoreoftenwhenwe’relookingathowmanyfriendspeople’sfriendshave.Andtheythencontributemoretothetotalincolumn3,hencetotheaverage.Ontheotherhand,peoplewithfewfriendsshowupmuchlessoften,andcontributeless.
Yourfriendsarenotatypicalsample.Peoplewithalotoffriendswillbeover-representedamongthem,becausethere’sagreaterchancethatyouareoneoftheirfriends.Peoplewithfewfriendswillbeunder-represented.Thiseffectskewstheaveragetowardsahighervalue.
Youcanseeithappeningincolumn3ofthetable.Thenumber5occursfivetimesincolumn3,oneforeachofEthel’sfriends;similarly6occurssixtimesincolumn3,oneforeachofGwen’sfriends.Ontheotherhand,Alice’scontributiontocolumn3(notinherrow,butwhensheoccursasafriendinotherrows)isjusttwo2’s:onefromBobandonefromCleo.SoEthelcontributes25,andGwenamassive36,whereaspooroldAlicecontributesjust4.
Tothemthathath,shallbegiven.Thisdoesnothappenforcolumn2:everyonecontributestheirfairshareto
theaverage,whichis3.Infacttheaverageofallthenumbersincolumn4is3.78,alotbiggerthan3.
Iprobablyoughttouseaweightedaverage:addallthenumbersincolumn3anddividebyhowmanyofthemthereare.Thisis3.33,againbiggerthan3.
Ihopeyoufeelhappiernow.
Seepage287foraproof.
Isn’tStatisticsWonderful?Statistically,42millionalligatoreggsarelaideveryyear.Ofthose,onlyhalfhatch.Ofthosethathatch,threequartersareeatenbypredatorsinthefirstmonth.Oftherest,only5%arealiveafterayear,foronereasonoranother.
Ifitwasn’tforstatistics,we’dallbeeatenbyalligators!
TheAdventureoftheSixGuestsFromtheMemoirsofDrWatsup
IthaslongdistressedmethatSoameshasaheartydislikeofdinnerparties.Hedespisessmalltalkandbecomesuneasyinthecompanyofwomen,especiallyattractivewomenlikemyfriendBeatrix.Butfromtimetotimeheisobligedtobitethebullet,graspthenettle,plaittheplatitude,andattendsocialeventsthatincludethefairersex.Atthese,hecanbefromonemomenttothenexttaciturn,obnoxious,charming,garrulous,orsomecombination.
Thisparticularoccasionwasamodesttête-à-têtewithAubreyandBeatrixSheepshear(brotherandsister)andCrispinandDorindaLambshank(husbandandwife).Iknewallfour,ofcourse;Beatrixisadelightfullady,unmarried,andwithoutacurrentsuitor,Ifirmlybelieve.Soamesknewonlyme,whichIfearedmightcausetheworsesideofhischaractertodominate,butIwashopingtowidenhissocialcircle.TheSheepshearsandLambshankshadnotmet,exceptforthemen,whobelongedtothesameclub.
AllthisbecamecleartoSoameswhentheguestsarrived,andsoonweweresittingtogether.Soames’spresencemadetheconversationspasmodic,soIventuredtopoursomeglassesofamodestbutacceptablesherry,givinghimadoubleshare.
“Howsingular!Iseebothatripleofmutualacquaintancesandatripleofmutualstrangers,”saidI,attemptingclumsilytobreaktheice.
“Atriplecannotbesingular,”Soamesmuttered,butatagesturefrommeheheldhispeace.Itoppeduphisglass.
Beatrixaskedmeforanexplanation,andIhastenedtocomply.“You,Aubrey,andIeachknowtheothertwo:atripleofmutualacquaintances.”
“Ithinkwearemorethanmereacquaintances,John,”shereplied.“Iamdelightedtohearit,dearlady,”saidI,“butIwasseekingawordthat
mightapplytoanypairofpeople.Incontrast,Soames,you,andDorindaaremutualstrangers,inthesensethatyouhavenotmetsociallyuntilnow.Ofcourse,Soames’sfamehasgonebeforehim.”
“Ithasindeed,”saidCrispin,givingmeasourlook.“Now,Ifindthisfactsomewhatremarkable—”“Youshouldnot,Watsup,”Soamesinterrupted.“Atleast,youshouldnot
consideritremarkablethatatleastonesuchtriple,acquaintancesorstrangers,
shouldoccur.”“Whynot?”askedAubrey.“Becauseatleastonesuchtriplemustoccurwheneversixpeopleare
gatheredtogether,”Soamesreplied.“Itmattersnotwhoknowswhom.”“Well,I’llbedashed,”saidAubrey.“Remarkable,what?”“Howcanyoubesosure,MrSoames?”Beatrixenquired,hereyesshining—
andnotjust,Isuspectedfromthesherry.“Because,mydearmadam,itcanbeproved.”“Oh.Dogoon,MrSoames.Ifindsuchmattersfascinating.”Soames
inclinedhishead,butIdetectedafaint,fleetingsmile.Hepretendstobeimmunetofemininecharms,butIknowthistobeasham.Hemerelylacksconfidence.Ihopedthiswouldcontinue,forBeatrixisprettyandmodestandwouldmakeagoodcatchforanycompatibleman.Myself,forexample.
“Theproofcanbeunderstoodmostsimplybywayofadiagram,”saidSoames.Herose,walkedacrosstothediningtable,andpickedupsomesideplatesandsomecutlery,brushingasidemyprotestationsalongwithseveralnapkins,themustard,andapottedaspidistra.
Soames’sdiagram“Theplatesrepresentthesixofus,”Soamessaid,writingourinitialsonthem
withastickofgreasepaint,whichhehadpresumablykeptasasouvenirfromthetimehehadcontemplatedacareeronthestage.“Aforkconnectingtwopeopledenotesthattheyareacquaintances;aknifedenotestheyarestrangers.”
“Lookingdaggersateachother,then,”saidBeatrix.Ihastenedtoapplaudherwitandrefillherglass.
“Forinstance,WatsupandIarejoinedbyaforkatthecentreofthetable,butIamjoinedtoeveryoneelsebyaknife.
“Now,asWatsuphassoastutelypointedout,WABisatriangleofforksand
SBDisoneofknives.Mycontention,however,isthathoweverwearrangeknivesandforks,therewillbeatleastonetriangleformedbyidenticaltypesofcutlery.”
“Coulditbeboth,MrSoames?”askedBeatrix.Hereyesfollowedhiseverymovement.
“Sometimes,madam,butnotalways.Togotoextremes,ifallimplementsareforksthenthereisnotriangleofknives,andiftheyareallknivesthenthereisnotriangleofforks.”
Beatrixnodded,aseriouslookonherface.“Itseems,then,”shemused,“thatasforksarereplacedbyknives,andtheopportunitytoformatriangleofforksdiminishes,thatofformingatriangleofknivesincreases.”
Soamesnodded.“Verywellput,madam.Theproofismerelyamatterofshowingthatthelattercomestopassbeforetheformerceases.Tobeprecise,letuschooseoneplate.Anyplate.Fiveimplementspointtowardsit.Atleastthreeofthemmustbeofthesamekind.Why?”
“Becauseifthereweretwoorfewerofeach,therewouldbeatmostfourimplements,”Beatrixsaidatonce.
“Verygood!”saidI,beforeSoamescouldofferasimilarcompliment.“Now,”saidhe,“letusconsiderasetofthreeidenticalimplements—letus
assumetheyareforks,theknifecaseissimilar—andlookattheplatestowhichtheypoint.Otherthantheoriginalchoice,ofcourse.Now,eitheroneofthoseplatesisjoinedtoanotherbyafork,or—”
“Allthreearejoinedbyknives!”shecried.“Inthefirstcase,wehavefoundatriangleofforks;inthesecond,oneofknives.Why,MrSoames,nowthatyouexplainitsoclearlyitseems—”
“Absurdlyobvious,”sighedSoames,takingalargesipofsherry.Thisremarksomewhatdeflatedher,andIwavedmyhandinapologyformy
friend’srudeness.Herinstantsmilewasmostgratifying.
ThisareaofmathematicsiscalledRamseyTheory.Seepage288forfurtherinformation.
HowtoWriteVeryBigNumbersHowmanygrainsofsandarethereintheuniverse?ThegreatestoftheancientGreekmathematicians,Archimedes,decidedtocombattheprevailingbeliefthattheanswerwasinfinite,byfindingawaytoexpressverylargenumbers.HisbookTheSandReckonerassumedthattheuniversewasthesizethatGreekphilosophersthoughtitwas,andthatitwasentirelyfilledwithsand.Hecalculatedthattherewouldbe(inourdecimalnotation)atmost1,000,…,000grainsofsand,with63successivezeros.
That’sbig,butnotinfinite.Aretherebiggernumbers?Mathematiciansknowthatthereisnolargest(whole)number.Theycanbe
asbigasyouwish.Thereasonissimple:iftherewerealargestnumber,youcouldmakeitlargerbyadding1.Mostchildrenwhohavemastereddecimalnotationquicklyrealisethatyoucanalwaysmakeanumberbigger(indeed,tentimesbigger)byputtinganextra0ontheend.
However,althoughthereisnolimitinprincipletohowlargeanumbercanbe,thereareoftenpracticallimitationsinthewaywechoosetowritenumbers.Forinstance,theRomanswrotenumbersusingthelettersI(1),V(5),X(10),L(50),C(100),D(500),andM(1,000),combiningthemingroupstogetintermediatenumbers.So1-4werewrittenI,II,III,IIII,exceptthatIIIIwasoftenreplacedbyIV(5minus1).Inthissystemthelargestnumberyoucanwriteis:
MMMMCMXCIX=4,999
andyoucanknockathousandoffthatifyoustopwiththreeM’s.However,theRomanssometimesneededbiggernumbers.Tosymbolisea
million,theyputabar(theirnamewasvinculum)overthetopofanM,toget .Ingeneral,abaroveranumbermultiplieditsvaluebyathousand,buttheyseldomusedthisnotation.Whentheydid,theyusedonlyonebar,soafewmillionwasasfarastheycouldget.Thelimitationsoftheirsymbolsystemshowthatthesizeofnumbersyoucanwritedowndependsonthenotationyouuse.
Nowadayswecangoalotfurther.Amillionis1,000,000—pathetic.Wecangetmuchbiggernumbersjustbyputtingmorezerosontheend,andmovingthecommasifnecessarytokeepthestandardgroupingintosetsofthreedigits.
(Mathematiciansgenerallyomitthecommasanyway,buttheyoftenreplacethembythinspaces:1000000.)IntheWesternworld,thestandarddictionarynamesforlargenumbersreflectthispractice:theystartwithmillion,billion,trillion,…andstopatcentillion.Sincehumanbeingsnevermanagetokeepthingssimple,especiallywhenitcomestomathematics,thesewordshave(oratleastusedtohave)differentmeaningsondifferentsidesoftheAtlantic.IntheUSAabillionis1,000,000,000,butintheUKitwas1,000,000,000,000—whichAmericanswouldcallatrillion.Butintoday’sinterconnectedworld,theAmericanusagehasprevailed,perhapsbecause‘milliard’—theUKtermforathousandmillion—is(a)obsolescentand(b)tooeasilyconfusedwith‘million’.Andabillionisaniceroundnumberforinternationalfinance,atleastuntiltheworld’sbanksthrewawaysomuchinthefinancialcrisisthatwehadtogetusedtothinkingintrillions.
Asimplerwaytowritethesenumbersistousepowersof10.So106denotes1followedbysix0’s,whichisamillion.The6iscalledanexponent.Abillionis109(or1012inold-fashionedUK-speak).Acentillionturnsouttobe10303(10600UK).Recognisedextensionstothestandarddictionarynamesgouptoamillinillion,103003.Thereareseveralsystemsandlifeistooshorttogointothemall,orindeeddistinguishproperlybetweenthem.
Twoothernamesforlargenumbers,alsofoundinmostdictionaries,aregoogolandgoogolplex.Agoogolis10100(1followedbyahundred0s);thenamewasinventedbyJamesNewman’snine-year-oldnephewMiltonSirotta.Sirottaalsosuggestedalargernumber,googolplex,whichhedefinedas‘1followedbywritingzerosuntilyougettired’.Acertainlackofprecisionledtoarefinement:‘1followedbyagoogolzeros’.
That’smoreinteresting,becauseitrunsintothesamesortofproblemthattheRomanshit,excepttheygottherealotsooner.Ifyouactuallytrytowriteagoogolplexdownindecimalnotation,1,000,000,000,…youwon’tgettotheendduringyourlifetime.Youwouldn’tgettoitduringthelifetimeoftheentireuniversetodate.Assumingthatconventionalcosmologicalcalculationsarecorrect,youprobablywouldn’tgettoitbeforetheuniverseended.Inanycase,you’drunoutofspacetowriteallthosezeros,evenifeachwerethesizeofaquark.
However,thereisacompactwaytowriteagoogolplex:iteratedexponents.Namely,
1010100
Andonceyoustartthinkingalongthoselines,youcangettosomeverybig
numbersindeed.In1976computerscientistDonaldKnuthinventedanotationforverylargenumbers,whichamongotherthingsturnupinsomeareasoftheoreticalcomputerscience.WhenIsay‘verylarge’Imeanverylarge—solargethatthere’snowayeventobegintowritethemdowninconventionalnotation.Thegoogolplex,whichis1followedby10100zeros,isdwarfedbymostofthenumbersyoucanexpressusingKnuth’sup-arrownotation.
Knuthstartsbywriting
a b=ab
So,forexample,10 2=100,10 3=1000,10 100isagoogol,and10 (10 100)isagoogolplex.Theusualconventionabouttheorderinwhichexponentsaretaken(startattherightandworkleftwards)letsuswritethismoresimplyas1010 100.Youdon’tneedalotofimaginationtocomeupwith10 10 10 10 10 1010,say.
Butthatisjustthestart.Let
a b=a a … a
whereaoccursbtimes.Theexponentialsareevaluatedstartingfromtheright,sothat(forexample)
a 4=a (a (a a))
Forexample,
2 4=2 (2 (2 2))=2 (2 4)=2 16=65,536
and
3 3=3 3 3=3 27=7,625,597,484,987
Thenumbersrapidlybecomeimpossibletowritedowndigitbydigit.Forinstance,4 4has155decimaldigits.Butthat’sthepoint up-arrownotationprovidesacompactwaytospecifygiganticnumbers.However,we’vebarelystarted.Nowlet
a b=a a … a
whereaoccursbtimesontheright-handside.Againthe sareevaluated
startingfromtheright.Yougetthepicture wecancontinuewith
a b=a a … aa b=a a … a
andsoon,whereasalwaysaoccursbtimesandweevaluatestartingfromtheright.
R.L.GoodsteindevelopedKnuth’snotationandsimplifiedit,leadingtoexpressionshecalledhyperoperators.JohnConwaydevelopedasimilar‘chainedarrow’notationwithhorizontalarrowsandbrackets.
Instringtheory,anareaoftheoreticalphysicsthataimstounifyrelativitywithquantummechanics,thenumber10 10 500turnsup itisthenumberofpotentiallydifferentstructuresforspace-time.AccordingtoDonPage,thelongestfinitetimeexplicitlycalculatedbyaphysicistisapaltry
10 10 10 10 10 1.1years
ThisisthePoincarérecurrencetimeforthequantumstateofablackholewiththemassoftheentireuniverse;thatis,thetimeitwouldtakebeforethissystemreturnedtoitsoriginalstateand,ineffect,historystartedtorepeatitself.
Graham’sNumberMathematiciansoccasionallyneedbiggernumbersthanphysicistsdo.Notjustforthefunofit:becausethesenumbersactuallyturnupinsensibleproblems.Graham’snumber,namedfortheAmericanRonGraham,arisesincombinatorics—themathematicsofcountingdifferentwaystoarrangeobjectsorfulfilconditions.
In1978GrahamandBruceRothschildwereworkingonaproblemabouthypercubes,multidimensionalanaloguesofthecube.Asquarehas4corners,acubehas8,afour-dimensionalhypercubehas16,andann-dimensionalhypercubehas2ncorners.Theycorrespondtoallpossiblesequencesofn0’sand1’sinasystemofncoordinates.
Takeann-dimensionalhypercube,anddrawlinesconnectingallpairsofcorners.Coloureachedgeeitherredorblue.Whatisthesmallestvalueofnforwhicheverysuchcolouringcontainsatleastonesetoffourcorners,lyinginaplane,suchthatalltheedgesjoiningthemhavethesamecolour?
Thetwomathematiciansprovedthatsuchanumbernexists,whichisfarfromobvious.Grahamhadearlierfoundasimplerproof,buthehadtousealargernumber:inKnuth’sup-arrownotationitisatmost:
Herethenumbersbelowthehorizontalbracesshowhowmanyarrowsoccurabovethebrace.Workbackwardsfromthebottomlayer:thereare3 3up-arrowsintheprevious(63rd)layer.Thenusethatnumberofarrowsinthe62ndlayertogetanewnumber.Thenusethatnumberofarrowsinthe61stlayer…!It’snotpossibletowritedownanyofthesenumbersinstandarddecimalnotation,sorry.Theyarefarworsethanagoogolplexinthatrespect.Butthat’stheircharm…
ThisisGraham’snumber,anditistrulygigantic.Andthensome.ThevaluefoundbyGrahamandRothschildissmaller,butstillridiculouslylarge,and
hardertoexplain,soIwon’t.Ironically,workersintheareaconjecturethatthenumbercanbemademuch
smaller.Infact,thatn=13willdo.Butthishasnotyetbeenproved.GrahamandRothschildprovedthatnmustbeatleast6;GeoffExooraisedthatto11in2003;thebestresultnowisthatnmustbeatleast13,provedin2008byJeromeBarkley.
Seepage289forfurtherinformation.
Can’tWrapMyHeadAroundItWhenscientistsmentionlargenumbers,suchastheageoftheuniverse(13.798billionyearsorabout4.35sextillionseconds)orthedistancetotheneareststar(0.237lightyearsorabout2.24trillionkilometres)wealltendtosaythingslike“Ican’twrapmyheadaroundthat”.Thesamegoesforthecostoftheglobalfinancialcrisis,oneofthehigherestimatesbeing£1.162trilliontotheUKeconomy.*Let’ssayaroundtrillion,£1012.
Millions,billions,trillions—inmanypeople’sminds,theseareallprettymuchthesame:toobigtowrapyourheadaround.
Thisinabilitytointernalisebignumbersaffectsourviewsonmanythings,especiallypolitics.Therewasmuchprotest,especiallyfromtheairlineindustry,whenEyjafjallajökullinIcelandspewedoutvolcanicashandgroundedmostoftheUK’saeroplanes.(Iwasn’ttoohappymyself:IwasduetoflytoEdinburghandhadtochangeplansrapidlyanddriveinstead.)Thecostwasestimatedat£100millionperday:£108.
Tobefair,thatwasthelosstoarelativelysmallnumberofcompanies.Butthescaleoftheoutcrywasprobablygreaterthanthatcausedbythefinancialcrisis.
Thebigsecretaboutcomparinglargenumbersisthatyoudon’thavetowrapyourheadroundthem.Indeed,itisprobablybestnotto.Themathematics—indeed,basicarithmetic—doesitforyou.Forexample,wecanaskhowlongtheflyingbanwouldhavetocontinueinordertocosttheeconomythesameamountthatthebankingcrisisdid.Thecalculationgoes:
Costofbankingcrisis:£1012Costperdayofvolcano:£1081012/108=104days=27years
Ifindthisperiodoftimeeminentlygraspable,andIhavenotroublerecognisingthatitisagreatdeallongerthanoneday.SoIcanworkoutthattheflyingbanwouldhaveneededtogoonfor27yearsbeforeitdidasmucheconomicdamageasthebankcrisis,withoutwrappingmyheadroundthelargerfiguresinvolvedinthecalculation.
Thisiswhatmathematicsisfor.Don’twrapyourheadroundthings:dothe
maths.
Footnote
*Thisfigurewasmorethanthefinalcostbecausethebankspaidmoneybackandsomeofitwastemporarysupport.ByMarch2011itwas£450billion,roughlyhalfasbig.
TheAffairoftheAbove-AverageDriverFromtheMemoirsofDrWatsup
Ithrewthenewspaperontothetableindisgust.“Isay,Soames—lookatthisridiculousstatistic!”
HemlockSoamesgrunted,andconcentratedonlightinghispipe.“Seventy-fivepercentofhansomcabdriversthinktheirabilityisabove
average!”Soameslookedup.“What’sridiculousaboutthat,Watsup?”“Well,I—Soames,it’sjustnotpossible!Theymusthaveanexaggerated
opinionofthemselves!”“Why?”“Becausetheaveragehastobeinthemiddle.”
Hansomcab from JohnThompson andAdolpheSmith,Street Life in London,1877
Thedetectivesighed.“Acommonmisconception,Watsup.”“Miscon—what’swrongwithit?”“Justabouteverything,Watsup.Suppose100peopleareassessedonascore
rangingfrom0to10.If99ofthemscore10andtheotherscores0,whatistheaverage?”
“Uh…990/100…whichis9.9,Soames.”
“Andhowmanyareaboveaverage?”“Uh…99ofthem.”“AsIsaid,amisconception.”Iwasnotsoreadilydiverted.“Buttheexcessissmall,Soames,andthedata
arenottypical.”“Iexaggeratedtheeffecttodemonstrateitsexistence,Watsup.Anydatathat
areskewed—asymmetric—arelikelytobehaveinasimilarmanner.Forexample,supposethatmostdriversarereasonablycompetent,asignificantminorityisappallinglybad,andaverytinynumberisexcellent.Whichdriversareaboveaverageinsuchcircumstances?”
“Well…thebadonesbringtheaveragedown,andtheexcellentonesdon’tcompensate…Myword!Thecompetentandexcellentdriversareallaboveaverage!”
“Indeed,”Soamesreplied.Herapidlysketchedagraphonasheetofscrappaper.“Withthesedata,whicharemorerealistic,theaverageis6.25,and60%ofthedriverslieabovethat.”
Soames’shypotheticaldrivingabilityscores,with60%ofdriversaboveaverage“SothearticleintheManchesterMirrographiswrong?”Ienquired.“Areyousurprised,Watsup?Veryfewofitsarticlesarecorrect,tobefrank.
Butthisonefallsintoacommontrap.Itconfusestheaveragewiththemedian—whichisdefinedtobethevalueforwhichhalfareaboveandhalfbelow.Thetwooftendiffer.”
“Soitisnotpossiblefor75%ofdriverstobeabovethemedian?”“Onlyifthenumberofdriversiszero.”“But75%ofdriverscouldbeaboveaverage?”“Yes.”“Andtheywouldn’thaveanundulyhighopinionoftheirownabilities?”Soamessighedagain.“That,mydearWatsup,isakettleofhorsesofa
differentcolouroffish.Thereisacommonformofcognitivebiascalledillusorysuperiority.Peopleimaginethemselvestobesuperiortoothers,evenwhentheyarenot.Almostallofussufferfromthisbias,withthenotableexceptionofmyself.AnarticleinQuantitativePhrenologyandCognitionlastmonthreportedthat69%ofSwedishcab-driversratedthemselvesabovethemedian.Andthatisdefinitelyillusory.”
Seepage289forgenuinemoderndata.
TheMousetrapCubeJeremiahFarrellinventedamagicwordcubeobeyingsimilarprinciplestothosegoverninghismagicwordsquares,Seepage20.HerethewordinvolvedisMOUSETRAPandthemagicnumberingofthelettersgoesM=0,O=0,U=2,S=6,E=9,T=18,R=3,A=1,P=0.Someofthewordsarepersonalnames,andothersareveryobscure.Forinstance,OSEisthenameofademon—andalsoofplacesinJapan,Nigeria,Poland,Norway,andSkye.Still,theamazingthingisthatitcanbedoneatall.
Successivelayersofthemousetrapmagicwordcube
SierpińskiNumbersNumbertheoristsseekinglargeprimesoftenconsidernumbersoftheformk2n+1,forspecificchoicesofk,asnvaries.Experimentsuggeststhatformostchoicesofk,thesenumbersincludealeastoneprime,oftenmore.Forinstance,ifk=1then1×2n+1isprimewhenn=2,4,8.Ifk=3then3×2n+1isprimewhenn+1,2,5,6,8,12.Ifk=5then5×2n+1isprimewhenn=1,3,7.(Ingeneral,wecandividekbyanyfactorsof2tomakeitoddandincludethesefactorsinthe2n.Sowemayassumekisoddwithoutlosinggenerality.Forexample,24×2n=3×23×2n=3×2n+3.)
Itistemptingtoconjecturethatforanyk≥2thereexistsatleastoneprimeoftheformk2n+1.However,in1960WacławSierpińskiprovedthatthereexistinfinitelymanyoddnumberskforwhichallnumbersoftheformk2n+1arecomposite.ThesearecalledSierpińskinumbers.
In1992JohnSelfridgeprovedthat78,557isaSierpińskinumber,byshowingthatallnumbersoftheform78,557×2n+1aredivisiblebyatleastoneofthenumbers3,5,7,13,19,37,73.Thesearesaidtoformacoveringset.ThefirsttenknownSierpińskinumbersare:
78,557271,129271,577322,523327,739482,719575,041603,713903,983934,909
Itiswidelybelievedthat78,557isthesmallestSierpińskinumber,butthathasnotyetbeenprovedordisproved.Since2002thewebsitewww.seventeenorbust.comhasbeenorganisingasearchforprimesoftheformk2n+1,whoseexistencewouldprovethatkisnotaSierpińskinumber.Whenthesearchstarted,therewere17possibleSierpińskinumberssmallerthan78,557,butthesehavebeeneliminatedonebyoneuntiljustsixremain:10,223,21,181,22,699,24,737,55,459,and67,607.Alongthewaytheprojecthasdiscoveredseveralverylargeprimes.
k eliminatedbyprimeoftheformk2n+1
4847 4847×23321063+15359 5359×25054502+1
(atthetimethefourthlargestknownprime)
1022319249 19249×213018586+121181226992473727653 27653×29167433+128433 28433×27830457+133661 33661×27031232+144131 44131×2995972+146157 46157×2698207+154767 54767×21337287+15545965567 65567×21013803+16760769109 69109×21157446+1
JamesJosephWho?
JamesJosephSylvester
JamesJosephSylvesterwasanEnglishmathematician,whoworkedwithArthurCayleyinmatrixtheoryandinvarianttheory,amongotherareas.Hehadalifelonginterestinpoetry,andoftenputextractsfrompoemsinhismathematicalresearcharticles.HemovedtotheUSAin1841,butreturnedshortlyafter.In1877hecrossedtheAtlanticagain,andtookupapositionasthefirstProfessorofMathematicsatJohnsHopkinsUniversity,andfoundedtheAmericanJournalofMathematics,stillgoingstrong.HereturnedtoEnglandin1883.
HisnamewasoriginallyJamesJoseph.WhenhiselderbrotheremigratedtotheUSA,theimmigrationofficialstoldhimhehadtohavethreenames:first,middle,andsurname.Forsomereasonthebrotheradded‘Sylvester’asanewsurname.SoJamesJosephdidthesame.
TheBafflehamBurglaryFromtheMemoirsofDrWatsup
LordBaffleham’sstatelyhomehadbeenburgledandsomeemeraldsandrubiesstolenfromthesafe.Soames,calledintoinvestigate,quicklybecamesuspiciousoftwovisitors,LadyEsmeraldaNickettandBaronessRubyRobham.Bothhadfallenonhardtimes,andhadnodoubtsuccumbedtotemptation.Butwherewastheproof?
Bothladiesadmittedpossessingsomejewels,butclaimedthemtobetheirown.SoameshadnotyetpersuadedInspectorRouladetoobtainasearchwarrant,whichmightsettlethematter,andhadnotbeenabletoinspecttheladies’jewelboxes.
“Thecase,”saidSoames,“hingesonjusthowmanyjewelsthetwoladiespossess.Ifthenumbersmatchwhatwasstolen,wehavethefinalpieceofevidenceweneed.Rouladeiswillingtorequestasearchwarrant,butonlyifwecantellhimthosetwonumbers.”
“Esmeraldastatedthatshehasonlyemeralds,”Imuttered,halftomyself.“AndRubysaysshehasonlyrubies.”
“Indeed.Iamsurethosestatementsaretrue.Now,thetestimonyofthemaidservantplacesthenumberofeachjewelsomewherebetween2and101,notexcludingthosetwonumbers.”
“Thecookisreluctanttotelltales,”Isaid.“ButIhavepersuadedhertotellmetheproductofthetwonumbers.”
“Andthebutler,equallyreticentbutopentopersuasionintheformoftengoldsovereigns,hastoldmetheirsum,”Soamesreplied.
“Thenwecanworkoutthetwonumbersbysolvingaquadraticequation!”Icriedinexcitement.
“Ofcourse,thoughwewon’tknowwhichnumberappliedtotheemeraldsandwhichtotherubies,”musedSoames.“Thedataaresymmetric.ButamatcheitherwaywouldbeenoughforInspectorRouladetoobtainasearchwarrant,whichwillnodoubtsuffice.”
“Ifyoutellmetheproduct,”Isaid,“Icansolvetheequation.”“Ah,mydearWatsup,youaresounsubtle,”Soamescomplained.“Letme
seeifIcandeducethenumberswithoutyourtellingme…Now,doyouknow
whatthetwonumbersare?”“No.”“Iknewthat,”saidSoames,tomyannoyance.Ifso,whyask?Butsuddenly
lightdawned.“NowIknowthenumbers,”Itoldhim.“Inthatcase,sodoI,Watsup.”
Whatwerethetwonumbers?Seepage290fortheanswer.
TheQuadrillionthDigitofπWecurrentlyknowthedecimalexpansionofπto12,100,000,000,050digits,acalculationperformedbyShigeruKondoin2013overaperiodof94days.Noonereallycareswhattheansweris,butthiskindofrecord-breakinghasledtosomeremarkablenewinsights,aswellasbeingagoodwaytotestnewsupercomputers.Oneofthemorecuriousdiscoveriesisthatitispossibletocomputespecificdigitsofπwithoutfindingthepreviousones.Butfornowwecandothisonlyinbase-16,orhexadecimal,notationfromwhichdigitsinbases8,4,and2(binary)canimmediatelybededuced.Thisideageneralisestoconstantsotherthanπ,andtobase3,butthereisnosystematictheoryyet.Nothinglikeitisknownfordecimalnotation,base10.
Theinitialdiscovery,theBBPformula,isstatedbelow:seealsoCabinetpage210.Itisaninfiniteseriesthatmakesitpossibletocalculateaspecifichexadecimaldigitofπwithoutcalculatinganyofthepreviousones.Sowecanbeconfidentthatthequadrillionthbinarydigitofπis0,thankstoProjectPiHex,andgoingevenfurtherthetwo-quadrillionthbinarydigitofπisalso0,thankstoa23-daylongcomputationbyoneofYahoo!’semployees.Despitethat,itwouldtakeanotherequallymassivecomputationtofindthepreviousdigit.
In2011DavidBailey,JonathanBorwein,AndrewMattingly,andGlennWightwickwroteasurveyofthisarea[Thecomputationofpreviouslyinaccessibledigitsofπ2andCatalan’sconstant,NoticesoftheAmericanMathematicalSociety60(2013)844–854].Theydescribedhowtocomputebase-64digitsofπ2,base-729digitsofπ2,andbase-4096digitsofanumbercalledCatalan’sconstant,startingatthe10trillionthplace.
ThestorystartswithaseriesknowntoEuler:
inΣ-notationforsums.Becauseofthepowersof2thatoccur,thisseriescanbeconvertedintoamethodforcomputingspecificbinarydigitsoflog2.Thecomputationsremainfeasible,buttakemuchlonger,asthepositionofthedigitconcernedgetslarger.
TheBBP(Bailey-Borwein-Plouffe)formulais
andthepowersof16makeitpossibletocomputespecifichexadecimaldigitsofπ.Since16=24,theseriescanalsobeusedtofindbinarydigits.
Isthisapproachlimitedtothesetwoconstants?From1997onwards,mathematicianssoughtsimilarinfiniteseriesforotherconstants,andsucceededforalargenumberofthem,including
π2log22πlog2ζ(3)π3log32π2log2π4ζ(5)
where
istheRiemannzetafunction.TheyalsosucceededforCatalan’sconstant
Someoftheseseriesyielddigitstobase3orsomepowerof3.Forexample,theamazingformulaofDavidBroadhurst
canbeusedtocomputedigitsofπ2tobase729=36.
IsπNormal?Thedigitsofπlookrandom,buttheycan’tbetrulyrandomsinceyoualwaysgetthesamenumberseverytimeyoucalculateπ,barringerrors.Itisgenerallythoughtthat,likealmostallrandomsequencesofdigits,everyfinitesequenceoccurssomewhereinthedecimalexpansionofπ.Indeed,infinitelyoften,though
withlotsofjunkinbetweensuccessiveoccurrences,andinthesameproportionyou’dexpectforarandomsequence.
Itcanbeprovedthatthisproperty,callednormality,holdsfor‘almostall’numbers:inanysufficientlylargerangeofnumbers,theproportionthatarenormalgetsascloseaswewishto100%.Butthisleavesaloophole,becauseanygivennumber,inparticularπ,mightbeanexception.Isit?Wedon’tknow.Untilrecentlythequestionlookedhopeless,butformulasliketheonesabovehaveopenedupanewlineofattack,whichmightjustsolvethequestionforbinary(orhexadecimal)digits.
Thelinkarisesthroughanothermathematicalprocedure:iteration.Herewestartwithanumber,applysomeruletogetanotherone,andrepeatedlyapplythatruletogetasequenceofnumbers.Forexample,ifwestartwith2andtheruleis‘formthesquare’thesequencegoes
241625665,6364,294,967,296…
Thebinarydigitsofanumberlikelog2canbegeneratedbytheiterativeformula
startingfromx0=0.Thesymbols(mod1)mean‘subtracttheintegerpart’,soπ(mod1)=0.14159….Thisformulawouldleadtoaproofthatlog2isnormaltobase2ifitcouldbeshownthattheresultingnumbersareuniformlyspreadovertherangefrom0to1.Such‘equidistribution’isquitecommon.Unfortunatelynooneknowshowtoproveitholdsfortheiterativeformulaabove,butit’sapromisingideaandmightgetthereeventually.
Thereisasimilarbutmorecomplicatediterativeformulaforπ:
Ifthisisequidistributed,πisnormalinbinary.Thisleadstoafinal,verystrange,discovery.Supposewestretchoutthe
rangefrom0to1byafactorof16,sothatyn=16xnrunsfrom0to16.Thentheintegerpartsofsuccessiveynrangefrom0to15.Experimentally,thesenumbersarepreciselythesuccessivehexadecimaldigitsofπ–3.Thishasbeencheckedonacomputerforthefirst10millionplaces.Ineffect,thisappearstoprovideaformulaforthenthhexadecimaldigitofπ.Thecomputationgetsharderandharderthefurtheryougo,andtook120hours.
Therearesolidreasonstoexpectthisstatementtobetrue,buttheydon’tamounttoarigorousproof.Itisknownthatveryfewerrors,ifany,occur.Sincenoneoccursforthefirst10millioniterations,thereisaboutaoneinabillionchancethatalatererrorwilloccur.However,that’snotaproof—justanexcellentreasontohopethatonecanbefound.
Afinalconjecture,alsobasedonsoundevidence,showshowstrangethisareais.Namely:nothinglikethiscanbedonefortheotherwell-knownconstante,thebaseofnaturallogarithms,roughlyequalto2.71828.Thereseemstobesomethingspecialaboutπcomparedtoe.
AMathematician,aStatistician,andanEngineer……wenttotheraces.Afterwards,theymetinthebar.Theengineerwasdrowninghissorrows.“Ican’tunderstandhowIlostallmymoney.Imeasuredthehorses,calculatedwhichwasmechanicallymostefficientandrobust,andfiguredouthowfasttheycouldrun—”
“That’sallverywell,”saidthestatistician,“butyouforgotindividualvariability.Ididastatisticalanalysisoftheirpreviousraces,andusedBayesianmethodsandmaximumlikelihoodestimatorstofindwhichhorsehadthegreatestprobabilityofwinning.”
“Anddidit?”“No.”“Letmebuyyouguysadrink,”saidthemathematician,pullingoutabulging
wallet.“Ididprettywelltoday.”Obviouslyherewasamanwhoknowssomethingabouthorses.Theothers
insistedonbeingtoldhissecret.Reluctantlyheagreed.“Consideraninfinitenumberofidenticalspherical
horses…“
LakesofWadaTopologyisoftencounterintuitive.Thismakesitdifficult,butalsointeresting.Here’sastrangetopologicalfactwithapplicationstonumericalanalysis.
Tworegionsoftheplanecanshareacommonboundarycurve;thinkoftheEnglish–ScottishborderortheAmerican–Canadianone.Threeormoreregionscanshareacommonboundarypoint:atFourCornersinAmerica,thestatesArizona,Colorado,NewMexico,andUtahallmeet.
FourCorners
Withsomeingenuityanynumberofregionscanbearrangedtosharetwocommonboundarypoints.Butitdoesn’tseempossibleforthreeormoreregionstohavemorethantwoboundarypointsincommon.Letaloneforthemalltohaveexactlythesameboundary.
However,itcanbedone.First,wehavetobepreciseaboutwhataboundarypointis.Supposewehave
someregionintheplane.Itneednotbeapolygon:itcanhaveaverycomplicatedshape—anycollectionofpointswhatsoever.Saythatapointliesintheclosureoftheregionifeverycirculardiscwithcentreatthatpointandnonzeroradius(howeversmall)containssomepointintheregion.Saythatapointliesintheinterioroftheregionifsomecirculardiscwithcentreatthatpointandnonzeroradiusiscontainedintheregion.Nowtheboundaryoftheregionconsistsofallpointsinitsclosurethatdonotlieinitsinterior.
Gotthat?Stuffontheedgebutnotinside,basically.Forapolygonalregion,boundedbyaseriesofstraightlinesegments,the
boundaryconsistsofthoselinesegments,sowhatwehavedefinedagreeswiththeusualconceptinthiscase.Itcanbeprovedthatthreeormorepolygonal
regionscannotallhavethesameboundary.Butthisisnottrueformorecomplicatedregions.In1917theJapanesemathematicianKunizōYoneyamapublishedanexampleofthreeregionsthathaveexactlythesameboundary.HesaidthathisteacherTakeoWadahadcomeupwiththeidea.Accordingly,theregions(oranythingofthesamekind)areknownastheLakesofWada.
Weconstructthethreeregionsstepbystepusinganinfiniteprocess.Beginwiththreesquareregions.
Startwiththreesquares…
Thenextendthefirstregionbyaddingatrenchthatwrapsroundallthreeregions.Dothissothateverypointontheboundaryofanysquareliesclosetothetrench.Also,makesurethatthetrenchdoesnotcloseuponitselftoleaveaholeintheresultingregion.
Digatrench…
Thenextendthesecondregionbyaddingayetnarrowertrenchthatwrapsroundallthreeregionsconstructedsofar.
Digathinnertrench…
Continuelikethat,withanevennarrowertrenchfromthethirdregion.Thengobacktothefirstregionaddingastillnarrowertrench,andsoon.
Repeatthisconstructioninfinitelyoften.Theresultingregionshaveinfinitelycomplicated,infinitelythintrenches.Butbecauseeachsuccessiveregiongetscloserandclosertoeverythingpreviouslyconstructed,allthreeregionshavethe
same(infinitelycomplicated)boundary.Thesameideaworksifwestartwithfourregionsormore:alloftheregions
constructedhavethesameboundary.TheLakesofWadawereoriginallyinventedtoshowthatthetopologyofthe
planeisnotasstraightforwardaswemightimagine.Manyyearslater,itwasdiscoveredthatsuchregionsarisenaturallyinnumericalmethodsforsolvingalgebraicequations.Thecubicequationx3=1,forexample,hasonlyonerealsolutionx=1,butitalsohastwocomplexsolutions ,wherei= .Thecomplexnumberscanbevisualisedaspointsinaplane,withx+iycorrespondingtothepointwithcoordinates(x,y).
Astandardmethodforfindingnumericalapproximationstoasolutionbeginswitharandomlychosencomplexnumber,thencalculatesasecondnumberinaspecificmanner,andrepeatsuntilthenumbersgetveryclosetogether.Theresultisthenclosetoasolution.Whichofthethreesolutionsitapproachesdependsonwhereyoustart,anditdoessoinaveryintricateway.Supposewecolourpointsinthecomplexplaneaccordingtowhichsolutiontheyleadto:saymediumgreyifthesolutionisx=1,lightgreyifthesolutionis ,darkgreyifthesolutionis .Thenthepointsthatarecolouredagivenshadeofgreydefinearegion,anditcanbeprovedthatallthreeregionshavethesameboundary.
UnlikeWada’sconstruction,theregionsherearenotconnected:theybreakupintoinfinitelymanyseparatepieces.However,itisstrikingthatregionsofsuchcomplexityarisenaturallyinsuchabasicproblemofnumericalanalysis.
Thethreeregionscorrespondingtosolutionsofthecubicequation
Fermat’sLastLimerickAchallengeformanylongagesHadbaffledthesavantsandsages.Yetatlastcamethelight:SeemsthatFermatwasright—Tothemarginaddtwohundredpages.
Malfatti’sMistakeFromtheMemoirsofDrWatsup
“Extraordinary!”Iexclaimed.Soamesglancedinmydirection,obviouslyannoyedatbeinginterruptedin
hisperusalofhisextensivecollectionofplastercastsofsquirrelfootprints.“Theanswerseemsobvious—yet,apparently,itiswrong!”Icried.“Theobvioususuallyis,”saidSoames.“Wrong,”headdedbywayof
clarification.“EverheardofGianFrancescoMalfatti?”Iasked.“Themultipleaxe-murderer?”“No,Soames,thatwas‘Hacker’FrankMacavity.”“Ah.Myapologies,Watsup,youareofcoursecorrect.Iamdistracted.My
specimenofRatufamacrouratracksisdisintegrating.Thegrizzledgiantsquirrel.”
“MalfattiwasanItaliangeometer,Soames.In1803heaskedhowtocutthreecylindricalcolumnsfromawedgeofmarbleinsuchamannerastomaximisetheirtotalvolume.Heassumedthattheproblemisequivalenttodrawingthreecirclesinsidethetriangularcrosssectionofthewedgeinsuchamannerastomaximisetheirtotalarea.”
“Anaivebutpossiblycorrectassumption,”Soamesreplied.“Thoughthecolumnsmightbecutonaslant.”
“Oh,Ihadnot…Butletussupposehisassumptiontobecorrect,sincethequestioncanalwaysbesuitablyrephrased.ItthenseemedobvioustoMalfattithatthesolutionmustcomprisethreecircles,eachtangenttotheothertwoandtoasideofthetriangle.”Idrewaquicksketch.
Malfatticircles
“Iseethefallacy,”saidSoames,inthatannoyinglyoffhandwaythatheoftenadoptswhenhehasinstantlygraspedcomplexitiesbeyondmostothermortals.
“IconfessIdonot,”Isaid.“Forifacircleliesinsidethetriangle,doesnotoverlaptheothers,andisnottangentinthatmanner,itcanbeenlarged.”
“Correct,”saidSoames.“Butthatmerelyprovesthesufficiency,notthenecessity,ofthetangencyconditions.”
“Iamawareofthat,Soames.But—howelsemightthecirclesbearranged?”“Theremightbeotherwaysforthetangencytooccur,ofcourse.For
example,Watsup:haveyouconsideredthesimplestcase,thatofanequilateraltriangle?”
Twopossiblearrangementsforanequilateraltriangle
“First,”saidSoames,“thereisMalfatti’sarrangement,theleft-handfigure.Butwhatabouttheright-handfigure?Again,nocirclecanbeenlarged,butthepatternoftangenciesisdifferent.Thesmallcirclesaretangenttothelargeone,butnottoeachother.Instead,eachistangenttotwosidesofthetriangle.”
Istaredatthefigures.“Totheeye,Soames,thefirstarrangementhaslargerarea.”
Helaughed.“Whichonlygoestoshow,Watsup,howeasilydeceivedtheeyeis.Supposethatthetrianglehassidesofunitlength.ThenMalfatti’sarrangementhasarea0.31567,buttheotheronehasarea0.31997,whichisveryslightlylarger.”
TherearetimeswhenSoames’seruditionleavesonebreathless.“Thedifferencemaybesmall,Soames,buttheimplicationisdecisive.Malfattiwaswrong.”
“Indeed.Moreover,Watsup,thedifferencebetweenMalfatti’sarrangementandthecorrectonecansometimesbemuchgreater.Forexample,ifthetriangleislong,thin,andisosceles,thenthecorrectsolutionstacksthethreecirclesontopofeachother,andtheareaisalmostdoublethatofMalfatti’sarrangement.”
Athinisoscelestriangle.Left:Malfatti’sarrangement.Right:Largestarea.
HepausedtohurlthecrumbledcastofthetracksofRatufamacrouraacrosstheroomintothefireplace.“Theironyis,”headded,“thatMalfatti’sarrangementisneverthebest.Thegreedyalgorithm—fitthelargestcirclepossibleinsidethetriangle,thenfindthelargestthatfitsintoaremaininggap,andfinallydothesameforthethirdcircle—isalwayssuperior,andindeedfindsthecorrectanswer.”
Seepage291forfurtherinformation.
SquareLeftoversPerfectsquaresendinoneofthedigits0,1,4,5,6,or9.Theydon’tendin2,3,7,or8.Infact,thefinaldigitofthesquareofanumberdependsonlyonthefinaldigitofthenumber:
Ifanumberendsin0thenitssquareendsin0.Ifanumberendsin1or9thenitssquareendsin1.Ifanumberendsin2or8thenitssquareendsin4.Ifanumberendsin5thenitssquareendsin5.Ifanumberendsin4or6thenitssquareendsin6.Ifanumberendsin3or7thenitssquareendsin9.
Numbertheoristsprefertophrasethiskindofeffectintermsofintegerstosomemodulus(page188).Ifthemodulusis10,thentheonlynumbersthatneedtobeconsideredare0–9:thepossibleremaindersondividinganynumberby10.Theirsquares(mod10)are
0149656941
andthelistofrulesforfinaldigitsofsquaresisadifferentwaytosaythesamething.
Asidefromtheinitial0,thelistofsquares(mod10)issymmetric:thenumbers1496appearafter5inreverse,6941.Thesymmetryarisesbecausenand10–nhavethesamesquaresmodulo10.Indeed,10–nisthesameas–n(mod10),andn2=(–n)2.Thesefournumbersthereforeoccurtwiceinthelist;0and5occuronce;but2,3,7,8don’toccuratall.It’snotverydemocratic,butthereyougo.
Whathappensifweuseadifferentmodulus?Thevaluesofthesquares,tothatmodulus,arecalledquadraticresidues.(Here‘residue’referstotheremainderondividingbythemodulus.)Therestarequadraticnon-residues.
Forinstance,supposethatthemodulusis11.Thenthepossibleperfectsquares(ofthenumberslessthan11)are
0149162536496481100
which,whenreduced(mod11),yield
01495335941
Sothequadraticresidues(mod11)are
013459
andthenon-residuesare
2678
Here’sashorttable:
modulusm squares(modm) quadraticresidues
2 01 013 011 014 0101 015 01441 0146 014341 01347 0142241 01248 01410141 0149 014077041 014710 0149656941 01456911 01495335941 01345912 014941014941 0149
Atfirstsight,therearefewclearpatternsasidefromthosealreadymentioned.Thisispartofthecharmofthearea,infact:whiletherearepatterns,ittakessomediggingtofindthem.Severalofthegreatestmathematiciansdevotedalotofattentiontothisarea,amongthemEulerandCarlFriedrichGauss.
Whenwesquareanumberwemultiplyitbyitself,andwhenitcomestomultiplication,whatmattermostinnumbertheoryaretheprimes.Soitpaystostartwithprimemoduli:2,3,5,7,11intheabovelist.Themodulus2isexceptional:theonlypossibleresiduesmodulo2are0and1,andbotharesquares.Forallotherprimes,abouthalfoftheresiduesaresquaresandtherestarenot.Moreprecisely,ifpisprimethenthereare(p+1)/2distinctquadratic
residuesand(p–1)/2non-residues.Thequadraticresiduesareusuallysquaresoftwodistinctnumbers,n2and(–n)2forsuitablen.However,0occursonlyoncebecause–0=0.
Compositemodulicomplicatethestory.Nowthesamequadraticresiduecansometimesbethesquareofmorethantwonumbers.Forexample1occursfourtimesformodulus8,asthesquareof1,3,5,and7.Thebestwaytomakesenseofallthisistousemodernabstractalgebra,butit’sworthtakingalookatmodulus15.Thishastwoprimefactors:15=3×5.Nowthelistofsquaresis:
n 01234567891011121314n2 01491106446101941
Sothequadraticresiduesmodulo15are
0=021=12;42;112;1424=22;72;82;1326=62;929=32;12210=52;102
Someresiduesoccuronce,sometwice,somefourtimes.Thosethatoccurfewerthanfourtimesaresquaresofnumbersthataredivisiblebyeither3or5,theprimefactorsof15.Alltheothernumbersoccurinsetsoffour,allhavingthesamesquare.
Thisisageneralpatternforanymodulusoftheformpqwherepandqaredistinctoddprimes.Thenumbersbetween0andpq–1thatarenotdivisiblebyporqsplitintosetsoffour,eachsethavingthesamesquare.(Thisfailsifoneoftheprimesis2:forexample10=2×5andwe’vealreadyseenthatinthiscasesquareseitheroccurinpairsorontheirown.)
Inalgebrawegetusedtotheideathateverypositivenumberhastwosquareroots:onepositive,theothernegative.Butinarithmeticmodulopq,mostnumbers(thosenotdivisiblebyporq)havefourdistinctsquareroots.
Thiscuriousfacthasaremarkableapplication,towhichwenowturn.
CoinTossingoverthePhoneSupposethatAliceandBobwanttoplayacoin-tossinggamewitha50–50outcome.Aswe’vealreadyseen(page131)AliceisinAliceSpringsandBobisinBobbington.Cantheytossthecoinoverthephone?Thebigsnagisthesameasforpoker.IfAlicetossesacoin,or,equivalently,carriesoutanyactivitywithtwoequallyprobableresults,andtellstheresulttoBob,hehasnoideawhethersheistellingthetruth.NowadaystheycoulddoitoverSkypeandwatchthecoinbeingtossed,buteventhen,thetosscouldberiggedbyfilmingseveraltossesaheadoftimeandsendingoneofthoseinstead.
Tossingacoinislikeplayingpokerwithapackofjusttwocards,sotheycouldadaptthemethodonpage132.However,there’sanotherelegantwaytoachievethesameresult,usingquadraticresidues.Here’show.
Alicechoosestwolargeoddprimenumberspandq.Shekeepsthemprivate,butsendstheirproductn=pqtoBob.YoumightthinkthatBobcouldthenfindpandqbyfactorisingn,butasfarasiscurrentlyknown,thereisnopracticalmethodtodothatwhenthenumbersaresufficientlylarge—say100digitsforeachofpandq.Thefastestcomputerusingthefastestknownalgorithmwouldtakelongerthanthelifetimeoftheuniverse.SoBobmustremainignorantoftheactualprimesinvolved.However,thereareveryquickwaystotesta100-digitnumbertoseewhetheritisprime.SoAlicecanfindpandqbytrial-and-error.
Bobchoosesarandomintegerx(modn),whichhekeepsprivate.Ifheisextremelypedantic,hecanquicklycheckwhetherxisamultipleofp
andq:notbydividingbythosenumbers,sincehedoesn’tknowthem,butbyfindingthehcfofxandnusingEuclid’salgorithm(page110).Iftheresultisnot1hethenknowseitherporq,sotheprocesshastoberepeatedwithanewx.Butinpracticeheneednotbother,sincewhenpandqhave100digitstheprobabilitythatporqdividesarandomlychosenxis2×10–100.
Bobnowcalculatesx2(modn),whichcanalsobedonequickly,andsendsittoAlice.TheyhaveagreedthatifAlicecancorrectlydeduceeitherxor–xshewins(‘heads’).Otherwise,sheloses(‘tails’).
BythepreviousitemAliceknowsthatintegersmodulopqthatarenotdivisiblebyporqhaveexactlyfoursquareroots.Sincexand–xhavethesamesquare,theseareoftheforma,–a,b,–bforsuitableaandb.Aliceknowsp,q,andx,whichimpliesthatshecancomputethesefoursquarerootsquickly.Two
ofthemmustbeBob’sxand–x;theothertwomustbedifferent.SoAlicehasa50%chanceofguessing xcorrectly—equivalenttotossingafaircoin.Shechoosesoneofthesefour,sayb,andsendsittoBob.
BobtellsAlicewhetherb= xornot;thatis,whethersheisrightorwrong.Ah—buthowdowestopBobcheating?AndhowdoesBobknowthatAlice
hasdonewhatsheissupposedtodo?Whetherornotb= x,BobcanbehappythatAlicehasplayedfairlyby
computingb2(modn).Thisshouldbethesameasx2.IfAliceloses,shecanconvinceherselfthatBobhasnotliedbyaskinghim
tosendhertheprimefactorspandqofn.Normallythiswouldbeimpossible,butifAlicehaslost,thenBobknowsallfoursquarerootsofx2,andthereisanumber-theoretictricktocalculatepandqquicklyfromthisinformation.Infact,thehighestcommonfactorofaandbisoneofthetwoprimes,andagainitcanbefoundusingEuclid’salgorithm.Theothercanthenbefoundbydivision.
HowtoStopUnwantedEchoesQuadraticresiduesmayseemtypicaloftheabstruseexplorationsofpuremathematicians:anintellectualgamewithnopracticaluses.Butit’samistaketothinkthatamathematicalideaisuselessjustbecauseitdoesn’tobviouslyderivefromapracticalproblemineverydaylife.It’salsoamistaketothinkthateverydaylifeisasstraightforwardasitappearstobeonthesurface.Evensomethingassimpleasapotofjaminasupermarketinvolvesmakingtheglass,growingthesugarcaneorbeet,refiningthesugar…andprettysoonyou’reintostatisticalhypothesistestingfordisease-resistantfruitandthedesignoftheshipusedtotransportvariouscomponentsorthefinishedproductroundtheglobe.Inaworldof7billionpeople,massfoodproductionisnotjustamatterofpickingsomeblackberriesandboilingthemup.
It’struethatthemathematicianswhofirstcameupwiththeseideashadnoparticularapplicationsinmind;theyjustthoughtquadraticresidueswereinteresting.Buttheywerealsoconvincedthatunderstandingthemwouldbeapowerfuladditiontothemathematicaltoolkit.Practicalpeoplecan’tuseatoolunlessitexists.Andwhileitmightseemtomakesensetowaitforanapplicationbeforeinventingasuitabletool,we’dstillbesittingincavesifwe’ddoneitthatway.“Whyareyouwastingtimebashingthoserockstogether,Ug?Youshouldbebashingmammothsovertheheadwithastickliketheotherboys.”
Quadraticresidueshavemanydifferentuses.Oneofmyfavouritesisthedesignofconcerthalls.Whenmusicreflectsoffaflatceiling,theresultisadistinctecho,whichdistortsthesoundandisgenerallyunpleasant.Ontheotherhand,aceilingthatabsorbssoundmakestheperformancesounddeadandfuzzy.Togetgoodacoustics,thesoundhastobeallowedtobounceback,butasadiffusespreadofsoundsratherthanasharpecho.Soarchitectsfitdiffusersontheceiling.Thequestionis:whatshapeshouldthediffusersbe?
Quadraticresiduediffuser(mod11)
In1975ManfredSchroederinventedadiffuserconsistingofaseriesofparallelgrooves,whosedepthsarederivedfromthesequenceofquadraticresiduesforsomeprimemodulus.Forexample,supposethattheprimeis11.We’vejustseenthatthesquaresof0–10,reducedmodulo11,are:
01495335941
andthesequencerepeatsthesevaluesperiodicallyforlargernumbers.It’ssymmetricaboutthemiddle,betweenthetwo3’s,becausex2=(–x)2moduloanyprime.Comparethepicturebelow,showingthesenumbersasrectangles,withtheshapeofthediffuserabove.Noticethatinthiscasethedepthsofthegroovesareobtainedbysubtractingtheresiduesfromsomeconstantdepth.Thishasnoseriouseffectonthemainmathematicalpoint.
Graphofquadraticresidues(mod11)
What’ssospecialaboutquadraticresidues?Onefeatureofasoundwaveisitsfrequency:howmanywaveshittheeareverysecond.Highfrequenciesgivehighnotes,lowfrequenciesgivelownotes.Arelatedfeatureisthewavelength:thedistancebetweensuccessivepeaks.High-frequencywaveshaveshortwavelength,andlow-frequencywaveshavelongerwavelength.Wavesofagivenwavelengthtendtoresonatewithcavitiesinthesurfacewhosesizeissimilartothatwavelength.Sowaveswithdifferentfrequenciesreactdifferentlywhentheyhitasurface.
Thequadraticresiduediffuserhasadelightfulmathematicalproperty:waves
withmanydifferentfrequenciesreacttoitinthesamemanner.Technically,itsFouriertransformisconstantacrossarangeoffrequencies.Schroederpointedoutanimportantconsequence:thisshapediffusessoundwavesofmanydifferentfrequenciesinthesamemanner.Inpracticethewidthsofthegroovesarechosentoavoidtherangeofwavelengthsthathumanscanhear,andtheirdepthsareaspecificmultipleofthesequenceofquadraticresidues,relatedtothewidth.
Whenthegroovesareparallel,asinthepicture,thesoundisdiffusedsideways,atrightanglestothedirectionofthegrooves.Thereisatwo-dimensionalanalogue,asquarearrayofrodsalsobasedonquadraticresidues,andthisdiffusesthesoundequallyinalldirections.Diffuserslikethatareoftenfoundinrecordingstudios,toimprovethesoundbalanceandgetridofextraneousnoises.
SoalthoughEulerandGausshadnoideawhattheirinventionwouldbeusedfor,orindeedwhetheritwouldeverbeusedforanything,itoftenplaysacrucialrole,behindthescenes,whenyoulistentorecordedmusic—beitclassical,jazz,country,rock,hiphop,crossoverthrashmetal,orwhateverelsetakesyourfancy.
Seepage292forfurtherinformation.
TheEnigmaoftheVersatileTileFromtheMemoirsofDrWatsup
“Solvingacrimeisoftenlikenedtofittingtogetherthepiecesofajigsawpuzzle,”Soamesremarkedoutoftheblue.Thebluebeingacloudofsmokethatenvelopedhishead,emanatingfromhispipe.
“Anaptsimile!”saidI,raisingmyheadfrommynewspaper.Hesmiledslyly.“Notso,Watsup.Onthecontrary,averypoorone.When
investigatingacrime,wedonotknowwhatthepiecesmightbe,norwhetherweareinpossessionofthemall.Notknowingthepuzzle,howcanwebecertainoftheanswer?”
“Surely,Soames,thatbecomesevidentwhenenoughoftheknownpiecesfittogetherintoanelegantpattern.”
Hesighed.“Ah,buttherecanbesomanypieces,Watsup.Andsomanypatterns.Decidingwhichiscorrecttakesacertain…jenesaisquoi.ButIdon’tknowwhat.”
Atthatmomenttherewasaknockonthedoorandawomanrushedin.“Beatrix!”Icried.“Oh,John!Ithasbeenstolen!”Andsherushedintomyarms,sobbing.Idid
mybesttocomforther,thoughIconfessmyownheartwasracing.Afteratime,shebecamecalm.“Pleasehelpme,MrSoames!Itisaruby
pendantinheritedfrommylatemother.Ilookedforitthismorning,andithadgone!”
“Donotdistressyourself,mydear,”saidI,pattinghershoulder.“SoamesandIwillapprehendthethiefandrecoverthejewel.”
“Didyoucomebycab?”Soamesasked.“Yes.Itiswaitingoutside.”“Thenweshalllosenotimeininvestigatingthesceneofthecrime.”Afterhalfanhourcrawlingonthefloor,samplingdustfromthecornersof
severalrooms,andinspectingthedoorstepandflowerbeds,Soamesshookhishead.“Nosignofabreak-in,MissSheepshear.Buttherearesmallscratchmarksonyourjewelcase.Veryfresh,andnotyours,forwhoevermadethemisleft-handed.”Heputthecasedown.“Haveanystrangersvisitedthehouserecently?Tradesmen,perhaps?”
“No…Oh!Thetilers!”
Twomenclaimingtobetilershadcometothebackdoorofferingtorenovatethebathroom.“Itisthenewfashion,MrSoames.Plainwhitesquaretiles,intowhichhasbeencutabluemotifmadefromtilesofamoreelaborateshape.TheDimworthyshadtheirsdonelastmonth,andfather—”Hervoicefailed,overcome,closetotears.Itookherhand.
“Areyouinthehabitofengagingunknowntradesmenatthedoorstep?”Soamesenquired.
“Why,no,MrSoames.Ordinarilywewoulddealonlywithareputablefirm.Buttheyareallbookedsolidformonths.Andtheseseemedhonest,decentmen.”
“Theyalwaysdo.Didyouleaveeitherofthemunattended?”Shethoughtforamoment.“Yes.Theassistantwaslefttomake
measurementsinthebathroomwhilehismastershowedmesamplemotifs.”“Ampletimetostealasmallbutvaluableitem.Theyareclever:bynotbeing
greedytheyensuredthatthetheftwasnotimmediatelynoticed.Didtheyleaveanydocumentation?”
“No.”“Havetheyreturnedsince?”“No,Iamawaitingawrittenestimateforthework.”“Iventuretopredictthatitwillnotarrive,madam.Itisamodusoperandi
thatinthetradewecalla‘distractionburglary’.”Overthefollowingweek,asteadystreamofladiesengagedSoames’s
serviceswithsimilartales.Thetradesmenvariedinappearance,butSoameswasunsurprised.“Disguised,”hesaid.
Thebreakthroughcameonthethirteenthcase,atthehomeofMrsAmeliaFotherwell.Soamesnoticedalumpofmudadheringtothebathroomdoor,inwhichwasembeddedasmallpieceofbone.ThecompositionofthemudandthenatureoftheboneledtoadirtybackyardnexttoasardinecanneryinoneofamazeoftinystreetsbehindtheAlbertDock.
“Sonowwebreakinandsearchforevidence?”saidI,reachingformyrevolver.
“No:thatmightalertthethief.WereturntoBakerStreetandassembleourcase.”
“Tellme,Watsup,”hesaid,aswesharedabottleofport.“Whatfeaturesdoallthesetheftshaveincommon?”Ipointedoutthosethatsprangtomind.“Verygood.Butyouhaveomittedthemostsignificantfeature.Themotifs.Youhavealistofthem,nodoubt?”
Itookoutmynotebook.Itread:
MrsWotton:threetilesforminganequilateraltriangle.Beatrix:fourtilesformingasquare.MissMakepiece:fourtilesformingasquarewithasquarehole.TheCranfordtwins:fourtilesformingarectanglewitharectangularhole.MrsBroadside:fourtilesformingaconvexhexagon.MrsProbert:fourtilesformingaconvexpentagon.LadyCunningham:fourtilesforminganisoscelestrapezium.MissWilberforce:fourtilesformingaparallelogram.MrsMcAndrew:fourtilesformingwindmillsails.MrsTushingham:sixtilesformingahexagonwithahexagonalhole.MissBrown:sixtilesforminganequilateraltrianglewithtrianglescutoffthecorners.DameJenkin-Glazeworthy:twelvetilesformingadodecagon(regular12-sidedpolygon)witharegular12-sidedstarhole.MrsFotherwell:twelvetilesformingadodecagonwith12-sidedstarholeshapedlikethebladeofacircularsaw.
“Aremarkablecollection,”saidSoames.“IthinkitistimetosendaBakerStreetIrreducibletoInspectorRoulade,askinghimtoraidthepremisesnearAlbertDock.”
“Whatareyouexpectingthepolicetofind?”“Recall,Watsup:eachladytoldusthathermotifwasmadefromanumber
ofidenticaltiles.”“Yes.”“Butthemotifsareveryvaried,suggestingthatalthougheachmotifuseda
singleshape,differentmotifsrequireddifferentshapesoftile.Theladiescannotdescribetheshapeexceptas‘irregular’,sowehavenoevidencethatthesameshapewasemployedforeachmotif.Ithereforeexpectthepolicetofindthirteenboxesofstrangelyshapedtiles:oneforeachmotif.”
AfteracoupleofhoursMrsSoapsudsappeared.“InspectorRoulade,MrSoames.”
TheInspectorentered,accompaniedbyaconstablebearingabox.“Ihaveplacedtwosuspectsunderarrest,”saidtheInspector.
“Roland‘therat’Ratzenbergand‘Bugface’McGinty.”“Yes,buthowonEarth—oh,nevermind.Icanholdthemfortwenty-four
hours.Buttheevidenceisweak.”Soameslooksshocked.“Surelyyoufoundallthoseboxesoftiles?Isnotthat
onemerelyasample?”TheInspectorshookhishead.“No:itistheirentirety.”
Soameswalkedovertotheboxandopenedit.Itcontainedtwelveidenticaltiles.“Oh,”saidhe.
“Itseemsthecasehascollapsed,”Iventured.“Icannotbelievethatsuchavariedlistofmotifscanallbemadefromasingleshapeoftile.”
ButSoamessuddenlybecamemoreanimated.“Youmaywellberight,”hesaid.“Unless…”Heproducedrulerandprotractorandbeganmeasuringoneofthetiles.
Afterafewmoments,asmilestoleacrosshisfeatures.“Clever!”saidhe.“Veryclever.”Heturnedtowardsme.“Ihavebeenextremelyfoolish,Watsup,andassumedwhenIshouldhavemaintainedanopenmind.DoyourememberourtopicofconversationjustbeforeBeatrixarrivedindistress?”
“Um—jigsawpuzzles.”“Indeed.Andthiscasehingesupononeofthemostremarkablejigsaw
puzzlesIhaveeverencountered.Lookatthistile.”“Itseemsaveryordinaryquadrilateral,”Isaid.“No,Watsup:itisaveryextraordinaryquadrilateral.Allowmeto
demonstrate.”Andhedrewadiagram.
Theversa-tile(thedashedlineisincludedtoexplainthegeometry)
“ThesidesABandBCareequal,andABCisarightangle,soanglesBACandBCAare45°,”Soamesexplained.“AngleACDis15°,soBCDis60°.AngleADCisagainarightangle,andthatmakesangleCADequalto75°.”
TheInspectorandIremainedunenlightened.Soameshandedmefourtiles.“Watsup,tryfittingthesetogethertomakeanelegantshape.Muchasadetectivemightfitcluestogethertocreateanelegantdeduction,toquoteyourearlieranalogy.”
“MayIturnthemover?”
“Anexcellentquestion!Yes,youmayturnanypieceoverifyouwish.”Iexperimentedforawhile.Suddenly,theanswerappearedbeforeme.
“Soames!Theymakeasquare—Beatrix’smotif!Howbeautiful!”
Watsup’sarrangement
Soamespeeredatmylittlejigsaw.“Indeeditis.Doyoustillcontendthatanelegantexplanationofhowseveralcluesfittogetherconstitutesdefinitiveevidencethatyouhavefoundtheguiltyparty?”
“Howelsecouldtheevidencefitsotightly,Soames?”“Howelseindeed?”Irealisedthathisquestionwasrhetorical.“Thereisa
holeinyourargument,Watsup,”hewenton,whenIdeclinedtoanswer.“Letuseliminateit.”Hereacheddownandrearrangedthepiecestoformacompletesquare.
Soames’salternative
“Oh,”saidI,shamefaced.“ThatisBeatrix’smotif,then.”“SoIconjecture.Butbenotdowncast:yourarrangementisMiss
Makepiece’s.”Lightdawned.“Youthinkthatcopiesofthisonetilecanformallthirteen
motifs?”
“Iamcertainofit.See:hereishowthreetilesformMrsWotton’smotif,anequilateraltrianglewithatriangularhole.”
Thethirdarrangement
“Goodheavens,Soames!”“Itisaremarkablyversatile—er—tile,”hereplied.“Thankstoitscunning
geometry.”“Soallwehavetodo—”Ibegan.“—istofindarrangementsthatfittheothertenmotifs!”Rouladefinishedfor
me.Soamesbegantoreamouthispipe.“IamsureIcansafelyleavethattoyou
gentlemen.”Thatevening,ItookacabtoBeatrix’sfather’shouse,stoppingonlytopick
upsomethingfromthejeweller’s.Shereceivedmeinthedrawing-room.Iplacedalongboxonthetable.“Openit,mydear.”Shereachedforithesitantly,hopedawningonherlovelyface.“Oh!John,youhaverecoveredmypendant!”Shetookmyhand.“HowcanI
everthankyou?”Suddenlyshefellsilent.“But—thisisnotmine.”Shereachedintotheboxandpickedoutasparklingjewel.“Itisanengagementring.”
“Soitis.Anditcanbeyours,”saidI,goingdownononeknee.
Canyoufindtheothertenarrangements?Seepage292fortheanswers.
TheThrackleConjectureAgraphisacollectionofdots(nodes)joinedbylines(edges).Whenagraphisdrawnintheplane,theedgesoftencross.In1972JohnConwaydefinedathrackletobeagraphdrawnintheplaneforwhichanytwoedgesmeetatanodeandotherwisedonotcross,ordonotmeetatanodebutcrossexactlyonce.ThenameissaidtohavebeeninspiredbyaScottishfishermancomplainingthathislinewas‘thrackled’—tangled.
Twothrackles
Thefigureshowstwothrackles.Theleft-handonehas5nodesand5edges,whiletherighthandonehas6nodesand6edges.Conwayconjecturedthat,foranythrackle,thenumberofedgesislessthanorequaltothenumberofnodes.Heofferedaprizeofabottleofbeerforaproofordisproof,butastheyearspassedandnoonesolvedtheriddle,theprizeroseto$1,000.
Boththracklesshownareclosedloops(nodesonacircle),drawnwithoverlaps.Itisknownthatanyclosedloopwithn≥5nodescanbedrawntoformathrackle.Ifso,thenumberEofedgescanbeequaltothenumbernofnodeswhenevern≥5.PaulErdoősprovedthattheconjectureistrueforanygraphwithstraightedges.ThebestboundonthesizeofEwasprovedbyRadoslavFulekandJánosPachin2011:
ForfurtherinformationSeepage292.
BargainwiththeDevilAmathematician,whohasspenttenfruitlessyearstryingtoprovetheRiemannHypothesis,decidestosellhissoultotheDevilinexchangeforaproof.TheDevilpromisestoshowhimtheproofinaweek,butnothinghappens.
Ayearlater,theDevilturnsupagain,lookinggloomy.“Sorry,couldn’tproveiteither,”hesays,handingbackthemathematician’ssoul.Hepauses,andhisfacelightsup.“ButIthinkIfoundareallyinterestinglemma…“
Atriskofspoilingthejoke,I’dbetterexplainthatinmathematicsalemmaisaminorpropositionwhosemaininterestistobeapotentialsteppingstonetowardssomethinginterestingenoughtodeservebeingcalledatheorem.Thereisnologicaldifferencebetweenatheoremandalemma,butpsychologicallytheword‘lemma’indicatesthatwhathasbeenprovedgoesonlypartwaytowardswhatisreallyrequired—
I’llgetmecoat.
ATilingThatIsNotPeriodicManydifferentshapestiletheplanewithoutleavinggapsoroverlapping.Theonlyregularpolygonsthatcandothisaretheequilateraltriangle,thesquare,andthehexagon.
Thethreeregularpolygonsthatcantiletheplane
Ahugerangeoflessregularshapesalsotiletheplane,suchastheseven-sidedpolygoninthenextpicture.Itisobtainedfromaregularheptagonbyflippingthreeofitssidesoveracrossthelinejoiningtheirends.
Left:Howtomaketheseven-sidedpolygontilefromaregularheptagon.Right:Spiraltiling.
Theregularpolygontilingsareperiodic,thatis,theyrepeatindefinitelyintwodifferentdirections,likewallpaperpatterns.Thespiraltilingisnotperiodic.However,theseven-sidedpolygonthatoccurstherecanalsotiletheplaneperiodically.
Howcanthisbedone?Seepage293fortheanswer.
Aretheretilesthatcantiletheplane,butcannotdosoperiodically?Thisquestionhasdeepconnectionswith
mathematicallogic.In1931KurtGödelprovedthatthereexistundecidableproblemsinarithmetic:statementsforwhichnoalgorithmcandecidewhethertheyaretrueorfalse.(Analgorithmisasystematicprocessthatisguaranteedtostopwiththecorrectanswer.)Histheoremimpliesanother,moredramaticone:therearestatementsinarithmeticthatcanneitherbeprovednordisproved.
Hisexampleofsuchastatementwasrathercontrived,andlogicianswonderedwhethermorenaturalproblemsmightalsobeundecidable.In1961HaoWangwasthinkingaboutthedominoproblem:givenafinitenumberofshapesfortiles,isthereanalgorithmthatcandecidewhetherornottheycantiletheplane?Heshowedthatifthereexistsasetoftilesthatcantiletheplane,butcannottileitperiodically,thereisnosuchalgorithm.HisideawastoencoderulesoflogicintotheshapesofthetilesanduseresultslikeGödel’s.Itworked,too:in1966RobertBergerfoundasetof20,426suchtiles,provingthatthedominoproblemisindeedundecidable.
Twentythousanddifferentshapesisratheralot.Bergerreducedthenumberto104;thenHansLa¨uchligotitdownto40.RaphaelRobinsonreducedthenumberofshapestosix.RogerPenrose’sdiscoveryin1973ofwhatwenowcallPenrosetiles(seeCabinetpage116)reducedthenumberstillfurther,toameretwo.Thisleftanintriguingmathematicalmystery:isthereasingleshapeoftilethatcantiletheplane,butcannottileitperiodically?(Thetile’smirrorimagecanbeusedaswell.)Theanswerwasfoundin2010byJoshuaSocolarandJoanTaylor[Anaperiodichexagonaltile,JournalofCombinatorialTheorySeriesA118(2011)2207–2231],anditis‘yes’.
Theirtileisshowninthefigure.Itisa‘decoratedhexagon’,withextra‘matchingrules’,anditdiffersfromitsmirrorimage.Thedecorationshavetofittogetherasshown.
FourcopiesoftheSocolar–Taylortileillustratingmatchingrules
Thenextfigureshowsthecentralregionofatilingoftheplane.Youcanseethatitdoesn’tlookperiodic.Thepaperexplainswhythistilingcanbecontinuedtocovertheentireplane,andwhytheresultcannotbeperiodic.Seetheirpaperfordetails.
CentralregionofatilingoftheplanebySocolar–Taylortiles
TheTwoColourTheoremFromtheMemoirsofDrWatsup
“Well,Soames,hereisajollylittleconundrumtobrightenyourmood.”ItossedtheDailyReporteracrosstomyfriendandcompanion,thenearlyfamousdetective,whowascurrentlysufferingfromafitofdepressionbecausehiscompetitoracrossthestreetwasdecidedlymorefamous,andlikelytoremainthatway.
Hetosseditbackwithasneer.“Watsup,Ihaveinsufficientenergytoread.”“ThenIshallreadittoyou,”Ireplied.“Itseemsthatthecelebrated
mathematicianArthurCayleyhaspublishedanarticleintheProceedingsoftheRoyalGeographicalSociety,askingwhether—”
“Whetheramapmaybecolouredwithatmostfourcolourssothatadjacentregionsreceivedifferentcolours,”Soamesinterrupted.“Itisalong-standingproblem,Watsup,andIfearitwillnotbeansweredinourlifetimes.”Isaidnothing,hopingtodrawhimout,sincethiswasthelongestsentencehehadutteredforthebestpartofaweek.Myployworked,forafteranawkwardsilencehecontinued:“AyoungmannamedFrancisGuthrieposedtheproblemtwoyearsbeforeIwasborn.Unabletosolveit,heemployedthegoodofficesofhisbrotherFrederick,astudentofProfessorAugustusDeMorgan.”
“Ah,yes,Gussie,”Iinterjected,havinghadsomeacquaintancewiththefamilyofthisadmirableeccentric,authorofABudgetofParadoxesandscourgeofmathematicalcrackpotseverywhere.
“DeMorgan,”Soameswenton,“madenoprogress,soheaskedthegreatIrishmathematicianSirWilliamRowanHamilton,whogavehimshortshrift.AndtheretheproblemlanguisheduntilCayleyonceagaintookitup.Thoughwhyhechosetopublishinthatparticularjournal,Ihavenoidea.”
“Possibly,”Iessayed,“becausegeographersareinterestedinmaps?”ButSoameswashavingnoneofit.
“Notinthismanner,”hehuffed.“Ageographerwillcolourregionsofamapaccordingtopoliticalconsiderations.Adjacencywouldnotaffectthechoice.Why,Kenya,Uganda,andTanganyikaarealladjacent,butonanymapoftheBritishEmpireallthreewillbecolouredpink.”
Iadmittedthatthiswasso.OurdearQueenwouldnotbeamusedwereit
otherwise.“ButSoames,”Ipersisted,“itremainsanintriguingquestion.Allthemoresobecausenooneseemsabletoanswerit.”
Soamesgrunted.“Letusmaketheattempt,”saidI,andquicklydrewamap.“Curious,”saidSoames.“Whyhaveyoumadeeveryregioncircular?”“Becauseanyregionwithoutholesistopologicallyequivalenttoacircle.”
Watsup’smap,andhowtocolourit
Soamespursedhislips.“Evenso,itisapoorchoice,Watsup.”“Why?Itseemstome—”“Watsup,manythingsseemtoyou,butfewactuallyare.Althoughanysingle
regionistopologicallyacircle,twoormoreregionsmayoverlapinamannerthatisimpossiblefortwoormorecircles.Asevidence,yourmaprequiresonlytwocolours.”Heshadedinabouthalfoftheregions.
“Well,yes,butIamcertainthatamoreelaboratemapofthesamekind—”Soamesshookhishead.“No,no,Watsup.Anymapcomposedentirelyof
circularregions,eveniftheybeofdifferentsizes,andoverlapinacomplicatedmanner,canbecolouredwithtwocolours.Assuming,asisalwaysthecaseinsuchquestions,that‘adjacent’requiresregionstoshareacommonlengthofborder,notjustisolatedpoints.”
Myjawdropped.“Atwocolourtheorem!Astonishing!”Soameshadthegracetoshrug.“Buthowcansuchatheorembeproved?”
Soamesleanedbackinhischair.“Youknowmymethods.”
Seepage293fortheanswer.
TheFourColourTheoreminSpaceSoameswasreferringtothecelebratedFourColourTheorem,whichtellsusthatgivenanymapintheplane,itsregionscanbecolouredusingatmostfourdistinctcolours,sothatregionsthatshareacommonborderhavedifferentcolours.(Here‘shareaborder’requiresthecommonbordertobeofnonzerolength;thatis,meetingatasinglepointdoesn’tcount.)Thisresultwasconjecturedin1852byFrancisGuthrie,andprovedin1976byKennethAppelandWolfgangHakenusingmassivecomputerassistance.*Theirproofhassincebeensimplified,butacomputerisstillessentialtocarryoutalargenumberofroutinebutcomplicatedcalculations.
Mighttherebeananalogoustheoremfor‘maps’inspaceratherthanonaplane?Nowtheregionswouldbesolidblobs.Alittlethoughtshowsthatsuchamapcanrequireanynumberofcolours.Suppose,forinstance,thatyouwantamapneedingsixcolours.Startwithsixdistinctspheres.Letsphere1putoutfivethintentaclesthattouchspheres2,3,4,5,and6.Thenletsphere2putoutfourthintentaclesthattouchspheres3,4,5,and6.Nowmoveontosphere3,andsoon.Theneachtentacledregiontouchesallfiveothers,sotheymustallhavedifferentcolours.Ifyoudidthiswith100spheres,you’dneed100colours;withamillionspheres,you’dneedamillioncolours.Inshort,thereisnolimittothenumberofcoloursrequired.
A‘map’inspaceneedingsixcolours
In2013BhaskarBagchiandBasudebDatta[Higher-dimensionalanalogues
ofthemapcoloringproblem,AmericanMathematicalMonthly120(October2013)733–736]realisedthatthisisnottheendofthestory.Thinkof‘maps’formedbyafinitenumberofcirculardiscsintheplanethateitherdonotoverlaportouchatacommonpoint.Supposeyouwanttocolourthediscssothattouchingdiscsgetdifferentcolours.Howmanydoyouneed?Itturnsoutthatagaintheansweris‘atmostfour’.
Infact,thisproblemisessentiallyequivalenttotheFourColourTheorem.Thistheoremcanbereformulatedastheproblemofcolouringthenodesofanetwork(orgraph,Seepage201)intheplane,withnoedgescrossingeachother,sothatiftwonodesarejoinedbyanedgethenthenodesreceivedifferentcolours.Justcreateanodeforeachregionofthemapandanedgebetweentwonodesifthecorrespondingregionsshareacommonboundary.Itcanbeprovedthatanynetworkintheplanecanbeproducedfromasuitablesetofcirclesbyjoiningthecentresofthosecirclesthattouch.Forinstance,hereisasetofcirclesneedingfourcolours,theassociatednetwork,andamapwithatopologicallyequivalentdistortionofthatnetworkthatalsoneedsfourcolours.
Left:Fourcirclesandtheassociatednetwork(greydotsandlines).Right:Amapwithatopologicallyequivalentnetwork,needingfourcolours.
Wecanextendthediscformulationnaturallytothreedimensions,byusingspheresinsteadofdiscs.Againeitherthespheresdonotoverlaportheytouchatacommonpoint.Supposeyouwanttocolourthespheressothattouchingspheresgetdifferentcolours.Howmanydoyouneed?BagchiandDattaexplainedwhythisnumbermustbeatleast5andnogreaterthan13.Theexactnumberiscurrentlyamathematicalmystery.Butyoumaybeabletoprovethatatleastfivecoloursareneeded.Theirresultimpliesthatsomethree-dimensionalmapsarenotequivalenttothoseobtainedfromspheres.
Seepage295fortheanswer.
Footnote
*SeeCabinet(pages10–16)foradiscussionoftheproblemanditseventualsolution.
ComicalCalculusForthisoneyouneedtoknowsomecalculus.If integrationsymbol,thentheexponentialfunctionexisitsownintegral:
ex= ex
Therefore
(1– )ex=0
So
Thecalculationseemstobenonsense;eventhefirstlinereallyoughttobeex= exdx.Andalatersteptakestheformula
1+y+y2+y3+y4+…=(1–y)–1
forsumminganinfinitegeometricseriesandreplacesyby .Thisformulaisvalidwhenyisanumber,lessthan1.But isn’tevenanumber,justasymbol.Howludicrous!
Despitewhich,thefinalresultisthecorrectpowerseriesforex.Thisisn’tacoincidence.Withtherightdefinitions(forexample, isan
operator,transformingafunctionintoitsintegral,andthe‘geometricseries’formulaworksforoperatorsundersuitabletechnicalconditions)everythingcanbemadeperfectlylogical.Butitdoeslookstrange.
TheErdoősDiscrepancyProblem
PaulErdoős
PaulErdoőswasaneccentricbutbrilliantHungarianmathematician.Heneverownedahouseorhadaregularacademicjob,preferringtotraveltheworldwithasuitcaseandsleepinthehomesofunderstandingcolleagues.Hepublished1,525mathematicalresearchpapers,collaboratingwith511othermathematicians—afigurenooneelsehascomenear.Hepreferredingenuitytodeepsystematictheories,andhisdelightwastosolveproblemsthatlookedsimple,butturnedoutnottobe.Hismainachievementswereincombinatorics,buthecouldturnhishandtomanyareasofmathematics.HefoundamuchsimplerproofofBertrand’spostulate(thereisalwaysaprimebetweennand2n)thantheoriginalanalyticoneofPafnutyChebyshev.Ahighpointofhiscareerwasaproofoftheprimenumbertheorem(thenumberofprimeslessthanxisapproximatelyx/logx)thatavoidedcomplexanalysis,previouslytheonlyknownroutetoaproof.
Hehadalifelonghabitofofferingmonetaryprizesforsolutionstoproblemsthathecameupwithbutwasunabletosolvehimself.Hewouldoffer$25forasolutionofsomethinghesuspectedtoberelativelyeasy,andthousandsofdollarsforsomethinghebelievedwasreallyhard.AtypicalexampleofhiskindofmathematicsistheErdoősDiscrepancyProblem,pricedat$500.Itwasposedin1932andsolvedearlyin2014.Itisaremarkableexampleofhowtoday’smathematiciansapproachlong-standingmysteries.
Theproblemstartswithaninfinitesequenceofnumbers,eacheither+1or–
1.Itmightbearegularsequence,suchas
+1–1+1–1+1–1+1–1+1–1…
oranirregular(‘random’)onesuchas
+1+1–1–1+1–1+1+1–1+1…
whichIgotbytossingacoin.Itneednotcontainthesameproportionof=and=signs.Anysequencewilldo.
Onewaytoseethatthefirstoftheseisregularistolookateverysecondterm:
–1–1–1–1–1…
Thesumsofthefirstntermsgo
–1–2–3–4–5…
anddecreasewithoutlimit.Ifwedothesameforthesecondsequenceweget
+1+1–1+1–1…
withsums
+10–10+1…
thatgoupanddown.Supposewetakeaspecificbutarbitrarysequenceof 1’s,andsetourselves
apositivetargetnumberC,whichcanbeaslargeaswewish—abillion,say.Erdoősaskedwhetherthereisalwayssomenumberdsuchthatthesumsoftermsdstepsapart,atpositionsd,2d,3d,andsoon,eitherbecomelargerthanCorsmallerthan-Catsomestage.
Havingreachedthetarget,theymaysubsequentlygivesumsbetweenCand-C:itisenoughtohitthetargetonce.However,therehastobeasuitablestepsizedforanytargetC.Ofcourse,ddependsonC.Thatis,ifthesequenceisx1,x2,x3,…,canwefinddandksothat
|xd+x2d+x3d+…+xkd|>C?
Theabsolutevalueofthesumontheleftisthediscrepancyofthesub-sequencedeterminedbythestep-sized,anditmeasurestheexcessof=signsover=signs(ortheotherwayround).
EarlyinFebruary2014AlexeiLisitsaandBorisKonevannouncedthattheanswertoErdoős’squestionis‘yes’ifC=2.Indeed,ifweselectad-stepsub-sequencefromthefirst1,161termsofany 1sequence,andchoosetheappropriatelengthk,theabsolutevalueofthesumexceedsC=2.Theirproofrequiresheavyuseofacomputer,andthedetailsrequirea13-gigabytedatafile.ThisismorethantheentirecontentsofWikipedia,at10gigabytes.Itiscertainlyoneofthelongestproofsever,andalreadytoolongforahumanbeingtocheck.
LisitsaisnowlookingforaproofforC=3,butthecomputerhasnotyetcompleteditscalculations.ItissoberingtothinkthatacompletesolutionrequiresunderstandingwhathappensforanychoiceofC.ThehopeisthatcomputersolutionsforsmallCmightrevealanewidea,whichahumanmathematiciancouldturnintoageneralproof.Ontheotherhand,theanswertoErdoős’squestionmightbe‘no’.Ifso,there’sareallyinterestingsequenceof1’soutthere,waitingtobedefined.
TheGreekIntegratorFromtheMemoirsofDrWatsup
Althoughmyfriend’sinvestigatorypowersaremainlydirectedtowardsthepursuitofcrime,fromtimetotimehisskillsareappliedintheserviceofscholarship.OnesuchinstancewasasingularquestthatwecarriedoutintheAutumnof1881atthebehestofawealthybutreclusivecollectorofancientmanuscripts.Withtheaidofatornpagefromanoldnotebook,alantern,abunchofskeletonkeys,andalargecrowbar,SoamesandIlocatedanenormousflagstoneandleveredituptouncoveraspiralstairwaythatleddowntoaconcealedchamberdeepbeneaththelibraryofafamousEuropeanUniversity.
Soamesconsultedthetornscrapofpaper,muchdamagedbyfireandwater.“TheLostIncunabulaoftheCartonari,”heexplained.
“Themagain!”HehadmentionedthisnameinpassingduringtheAdventureoftheCardboardBoxes(Seepage23),butdeclinedtosaymore.NowIpressedhimforfutherdetails.
“Thenamemeans‘cardboardmanufacturers’.ItisanItaliansecretsocietyorganisedalongthelinesoftheFreemasonsanddedicatedtothecauseofNationalism,beingimplicatedinthefailedrevolutionof1820.”
“Irecalltherevolutionwiththeutmostclarity,Soames.Butnottheorganisation.”
“Fewindeedareawareofitshiddenhand.”Heconsultedthescrapofpaper.“Thepageisalmostobliterated,butittakesnogreatexpertiseinhighermathematicstorecognisethatitissomeformofFibonaccicode,rewritteninDaVincimirrorscript,andtransformedintoasequenceofrationalpointsonanellipticcurve.”
“Evenachildcouldseethat,”saidI,lyingthroughmyteeth.“Quite.Now,ifIreadtheserunesaright,weshouldfindwhatweseek
somewhereontheseshelves.”Afteramoment,Iasked:“Soames,whatdoweseek?Youhaveplayedyour
cardsunusuallyclosetoyourchest.”“Itisknowledgethatholdsgreatdangers,Watsup.Isawnoneedtoexpose
youtoviolenceprematurely.Butnowthatwehavepenetratedtheinnersanctum—Ah!Hereitis!”HeextractedwhatIimmediatelyrecognisedtobeaparchmentcodex,blowingoffcenturiesofaccumulateddust.
“Whatthedevilisthat,Soames?”“Doyouhaveyourservicerevolver?”“Neverwithoutit.”“ThenitissafetotellyouthatinmyhandsIhold…theArchimedes
Palimpsest!”“Ah.”Now,Iwasawarethatapalimpsestisadocumentthathasbeenwrittenon
andthenscrubbedcleantoallowfurtherinscription,andthatscholarscanwithdifficultyreconstructthatwhichhasbeenobliterated,therebyretrievingahithertounknowngospelfromthelaundrylistofanobscureorderoffourteenth-centurymonks.Archimedeswasalsofamiliartome,asanancientGreekgeometerofprodigiousability.ItwasthereforeapparentthatSoameshadunearthedsomepreviouslyunknownmathematicaltext.Butheinsistedthatweshouldmakeourexitimmediately,beforetheInquisitorialVengeanceSquaddescendeduponus.
BackinthecomparativesafetyofBakerStreet,weinspectedthedocument.“Itisatenth-centuryByzantinecopyofahithertounknownworkof
Archimedes,”saidSoames.“ItstitlelooselytranslatesasTheMethod:itconcernsthatgeometer’scelebratedworkonthevolumeandsurfaceareaofasphere.Itshowsushowhecametodiscovertheseresults,offeringanunparalleledinsightintohisthoughtprocesses.”
Iwasstruckdumb,andnodoubtresembledagoldfishoutofwater.“Archimedesdiscoveredthatifasphereisinscribedinatightlyfitting
cylinder,thenthevolumeofthesphereisexactlytwothirdsthevolumeofthecylinder,anditsareaisthesameasthatofthecurvedsurfaceofthecylinder.Inmodernlanguage,iftheradiusisrthenthevolumeis andthearea4πr2.
“Now,Archimedeswassuchagreatmathematicianthathewasabletofindalogicallyrigorousgeometricproofofthesefacts,whichheincludedinhisbookOntheSphereandCylinder.Thereheusedacomplicatedmethodofproofnowknownasexhaustion.Oneofitstrickyfeaturesisthatonemustknowtheexactanswertotheproblembeforeprovingittobecorrect.Soithaslongbeenapuzzletoscholars:howdidArchimedesknowwhattherightanswermustbe?”
“Isee,”saidI.“Thislong-lostdocumentexplainshowhedidit.”“Exactly.Remarkably,itcomesclosetobeingananticipation,forthis
particularexample,oftheintegralcalculusofIsaacNewtonandGottfriedLeibniz,developedmorethantwothousandyearslater.But,asArchimedeswellknew,theideasusedinTheMethodlackrigour.Hencehisuseofexhaustion,averydifferentapproach.”
“Sohowdidhedoit?”Iasked.
Soamesstudiedthepalimpsestthroughhismagnifyingglass.“TheGreekisnotentirelyclassical,andoftenunclear,butthatposesnoseriousdifficultytoanexpertlinguistsuchasmyself.HaveIshownyoumypamphletonthedeciphermentofobscureancienttextsfromtheMediterraneanregion?Remindmetodoso.
“ItseemsthatArchimedesbeganwithasphere,acone,andacylinderofsuitabledimensions.Thenheimaginedtakingaverythinsliceofeach,andhangingthemonabalance:asliceofthesphereandasliceoftheconeononeside,asliceofthecylinderontheother.Ifthedistancesarechosencorrectly,themasseswillbalanceexactly.Sincemassisproportionaltovolume,thevolumesarerelatedbythelawofthelever.”
Slicingthesolidspriortohangingthemfromabalance:Seepage298fordetails
“Uh—prayremindmeofthislaw,”saidI.“Itwasunaccountablynotpartofthesyllabusatmedicalschool.”
“Itshouldhavebeen,”saidSoames.“Itwouldbeofgreatusewhendealingwithdislocatedjoints.Nomatter.Thelaw,whichwasdiscoveredandprovedbyArchimedes,statesthattheturningeffect,ormoment,ofagivenmassatagivendistanceisthemassmultipliedbythedistance.Forasetofmassestobalance,thetotalclockwisemomentmustequalthetotalanticlockwisemoment.Or,withappropriateassignmentofplusandminussigns,thetotalmomentmustbezero.”
“Er—”“Amassatagivendistancewillbalancehalfthatmassattwicethedistance,
provideditisontheotherarmofthebalance.”“Isee.”“Isuspectnot,butletmeproceed.Bysplittingthesesolidsintoinfinitely
manyinfinitelythinslices,andhangingthemappropriatelyonhisbalance,Archimedeswasabletoconcentratetheentiremassofthesphere,andthatofthecone,atasinglepoint.Theslicesofcylinder,whichareidenticalcircles,areplacedatdifferentdistances;togethertheyreconstructtheoriginalcylinder.Knowingthatthevolumeofthecone,henceitsmass,isonethirdthatofthecylinder,Archimedescouldthensolvetheresulting‘equation’forthevolumeofthesphere.”
“Amazing,”saidI.“Itseemsconvincingenoughtome.”
“ButnottoamathematicianoftheintellectualcalibreofArchimedes,”saidSoames.“Ifthesliceshavefinitethickness,theprocedureinvolvessmallbutunavoidableerrors.Butifthesliceshavezerothickness,theyhavezeromass.Thereisnouniquebalancepointwhenthemassesconcernedareallzero.”
Ibegantoseetheobjectiontotheprocedure.“Presumablytheerrorsbecomeeversmallerastheslicesbecomethinner?”Ihazarded.
“Theydo,Watsup,theydo.Andthemodernapproachtointegralcalculusconvertsthatstatementintoaproofthatthiskindofprocessgivessensibleanswers.However,theseideaswerenotavailabletoArchimedes.Soheusedanon-rigorousmethodtofindthecorrectanswer,andthatallowedhimtouseexhaustiontoprovetheanswerwascorrect.”
“Astonishing,”Isaid.“Wemustpublishthepalimpsest.”Soamesshookhishead.“AndriskthewrathoftheCartonari?Ivalueboth
ourlivestoohighlytoattracttheirattention.”“Sowhatshouldwedo?”“Wemustplacethemanuscriptsomewheresafe.Notbackinthelibrary,for
theymustbynowhavenoticeditsdisappearanceandsetsubtletraps.Iwillconcealitinsomeotherscholarlylibrary.No,donotaskwhich!Perhapsatsomelaterdate,whentimesarelesstroubledandtheinfluenceofsecretsocietieshaswaned,itwillberediscovered.Untilthen,wemustbecontentwithknowingthegreatgeometer’smethod,eventhoughwecannotrevealittotheworld.”
Hepaused.“Ihavealreadytoldyoutheformulasfortheareaandvolumeofasphere.Sohereisasimplelittleconundrumthatmightamuseyou.Whatshouldtheradiusofthespherebe,infeet,sothatitsareainsquarefeetisexactlyequaltoitsvolumeincubicfeet?”
“Ihavenoidea,”saidI.“Thenworkitout,man!”hecried.
Seepage296forthetruehistoryoftheArchimedespalimpsestandtheanswertoSoames’spuzzle.
SumsofFourCubesSumsoffoursquares,likemanymathematicalmysteries,havealonghistory.TheGreekmathematicianDiophantus,whoseArithmeticaofaboutAD250wasthefirsttextbooktouseaformofalgebraicsymbolism,askedwhethereverypositivewholenumberisasumoffourperfectsquares(0allowed).It’seasytoverifythisstatementexperimentallyforsmallnumbers;forexample,
5=22+12+02+026=22+12+12+027=22+12+12+12
Justwhenyouthinkthat8willneedanother12,hencefivesquares,4comestotherescue:
8=22+22+02+02
Experimentswithlargernumbersstronglysuggestthattheansweris‘yes’,buttheproblemremainedunsolvedforover1,500years.ItbecameknownasBachet’sproblem,afterClaudeBachetdeMéziriacpublishedaFrenchtranslationofArithmeticain1621.Joseph-LouisLagrangefoundaproofin1770.Simplerproofshavebeenfoundmorerecently,basedonabstractalgebra.
Whataboutsumsoffourcubes?Alsoin1770,EdwardWaringstatedwithoutproofthateverypositivewhole
numberisthesumofatmost9cubesand19fourthpowers,andaskedwhethersimilarstatementsaretrueforhigherpowers.Thatis,givenanumberk,istheresomefinitelimittothenumberofkthpowersneededtoexpressanypositivewholenumberbyaddingthemup?In1909DavidHilbertprovedthattheansweris‘yes’.(Oddpowersofnegativenumbersarenegative,andthatchangesthegameconsiderably,soforthemomentwerestrictattentiontopowersofpositivenumbers.)
Thenumber23definitelyrequires9cubes.Theonlypossibilitiesare8,1,and0,andthebestwecandois8+8+seven1’s:
23=23+23+13+13+13+13+13+13+13
Sowecan’tsucceedingeneralwithfewerthan9cubes.However,thatnumbercanbereducedifweignoreafinitenumberofexceptions.Forexampleonly23and239actuallyneed9cubes;allotherscanbeachievedusing8.YuriLinnikreducedthisto7byallowingafewmoreexceptions,anditiswidelybelievedthatthecorrectnumber,allowingfinitelymanyexceptions,is4.Thelargestknownnumberthatneedsmorethan4cubesis7,373,170,279,850,anditisconjecturedthatnolargernumberswiththatpropertyexist.Soitisverylikely—butremainsanopenquestion—thateverysufficientlylargepositivewholenumberisasumoffourpositivecubes.
But,asIsaidearlier,thecubeofanegativenumberisnegative.Thisallowsnewpossibilitiesthatdonotoccurforevenpowers.Forexample,
23=27–1–1–1–1=33+(–1)3+(–1)3+(–1)3+(–1)3
usingonly5cubes,whereaswithpositiveorzerocubesitrequires9,aswe’vejustseen.Butwecandobetter:23canbeexpressedusingjustfourcubes:
23=512+512–1–1000=83+83+(–1)3+(–10)3
Allowingnegativenumbersmeansthatthecubesinvolvedmightbemuchlarger(ignoringtheminussign)thanthenumberconcerned.Asanexample,wecanwrite30asasumofthreecubes,butwehavetoworkprettyhard:
30=2,220,422,9323+(–283,059,965)3+(–2,218,888,517)3
Sowecan’tworksystematicallythroughalimitednumberofpossibilities,aswecanwhenonlypositivenumbersareconsidered.
Experimentshaveledseveralmathematicianstoconjecturethateveryintegeristhesumoffour(positiveornegative)integercubes.Asyet,thisstatementremainsmysterious,buttheevidenceissubstantial.Computercalculationsverifythateverypositiveintegerupto10millionisasumoffourcubes.V.Demjanenkohasprovedthatanynumbernotoftheform9k 4isalwaysasumoffourcubes.
WhytheLeopardGotItsSpots
Leopardess,KananaCamp,Botswana
Leopardshavespots,tigershavestripes,andlionsareplain.Why?Itallseemsratherarbitrary,asiftheBigCatSalesCataloguelistscoatpatternsandevolutionpickswhicheverlooksprettiest.Butevidenceisaccumulatingthatit’snotlikethat.WilliamAllenandcolleagueshaveinvestigatedhowthemathematicalrulesthatdeterminethepatternsrelatetothecats’habitsandhabitats,andhowthisaffectsthewaythepatternsevolve.
Themostobviousreasonforevolvingpatternedcoatsiscamouflage.Ifacatlivesintheforest,spotsorstripeswillmakeithardtoseeamongthelightandshade.Catsthatoperateoutintheopen,ontheotherhand,willbeeasiertoseeiftheyhavestrongpatterns.However,theoriesofthiskindarelittlebetterthanjust-sostories,unlesstheycanbesupportedbyevidence.Experimentalverificationisdifficult:imaginepaintingoutatiger’sstripesforseveralgenerations,orfittingitanditsdescendantswithplainovercoats,toseewhathappens.Alternativetheoriesabound:markingsmightexisttoattractmates,ormerelybeanaturalconsequenceoftheanimal’ssize.
Themathematicalmodelofcatpatterningmakesitpossibletotestthecamouflagetheory.Somepatterns,suchastheleopard’sspots,areverycomplex,andthetypeofcomplexityiscloselyrelatedtothepattern’svalueascamouflage.Sotheresearchersclassifiedpatternsusingamathematicalscheme
inventedbyAlanTuring,inwhichthepatternislaiddownbychemicalsthatreacttogetheranddiffuseacrossthesurfaceofthedevelopingembryo.
Theseprocessescanbecharacterisedbyspecificnumbersthatdeterminetherateofdiffusionandthetypeofreaction.Thesenumbersactlikecoordinateson‘camouflagespace’,thesetofallpossiblepatterns,justaslatitudeandlongitudeprovidecoordinatesonthesurfaceoftheEarth.
Theresearchrelatesthesenumberstoobservationaldataon35differentcatspecies:whatkindofhabitatthecatsprefer,whattheyeat,whethertheyhuntbydayorbynight.Statisticalmethodsidentifysignificantrelationshipsbetweenthesevariablesandtheanimals’coatpatterns.Theresultsshowthatpatternsarecloselyassociatedwithclosedenvironments,suchasforests.Animalsinopenenvironments,suchassavannahs,aremorelikelytobeplain,likelions.Ifnot,theyusuallyhavesimplepatterns.Butcatsthatspendalotoftimeintrees,suchasleopards,aremorelikelytohavepatternedcoats.Moreover,thesepatternstendtobecomplex,notjustsimplespotsorstripes.Themethodalsoexplainswhyblackleopards(‘panthers’)arecommon,buttherearenoblackcheetahs.
Thedataargueagainstseveralalternativestocamouflage.Thesizeofthecatandthesizeofitspreyhavelittleeffectonpatterns.Sociablecatsarenomoreorlesslikelytobepatternedthansolitaryones,sothemarkingsareprobablynotimportantforsocialsignalling.Thisworkdoesnotprovethatcats’markingsevolvedforcamouflagealone,butitsuggeststhatcamouflageplayedakeyevolutionaryrole.
Lionsareplainbecausetheyprowlontheplains.Leopardsarespottybecausespotsarehardertospot.
Seepage299forfurtherinformation.
PolygonsForever
Keepgoingforever…Howbigdoesitget?
Here’satestofyourgeometricandanalyticintuition.Startwithacircleofunitradius.Drawthetightestfittingequilateraltrianglethatyoucanroundit;thendrawthetightestfittingcircleroundthat.Repeat,butatsuccessivestagesuseasquare,aregularpentagon,aregularhexagon,andsoon.
Ifthisprocessgoesonforever,doesthepicturebecomearbitrarilylarge,ordoesitremainwithinaboundedregionoftheplane?
Seepage299fortheanswer.
TopSecretInthe1930saRussianmathematicsprofessorwasrunningaseminarinfluiddynamics.Twooftheregularattendees,alwaysdressedinuniform,wereobviouslymilitaryengineers.Theyneverdiscussedtheprojecttheywereworkingon,whichwaspresumablytopsecret.Butonedaytheyaskedtheprofessorforhelpwithamathematicalproblem.Thesolutionofacertainequationoscillated,andtheywantedtoknowhowtochangethecoefficientstomakeitmonotonic.
Theprofessorlookedattheequationandsaid:“Makethewingslonger!”
TheAdventureoftheRowingMenFromtheMemoirsofDrWatsup
IhaveoftenbeenastonishedbySoames’sabilitytoperceivepatternsinthemostunpromisingofcircumstances.Nobetterexamplecouldbefoundthanthatwhichoccurredintheearlyspringof1877.
AsIwalkedacrossEquilateralParktowardshislodgings,afreshlymintedsuncastdappledlightandshadethroughascatteringoffluffyclouds,andthehedgerowsrangwithbirdsong.Onsogloriousadayitseemedveritablyindecenttoremainindoors,butmyeffortstodragmyfriendawayfromcataloguinghiscomprehensivecollectionofusedmatchsticksmetwithindifference.
“Manyacasehashingeduponthetimeittookamatchtoburn,Watsup,”hegrumbled,transferringameasurementfromhisdividersintoanotebook.
Disappointed,Iopenedthenewspaperatthesportspages,andmyeyewascaughtbyatimelyreminderofaneventthatevenSoameswouldnotwishtomiss.Ithadcompletelyescapedmymindamidthebuzzingofthebeesandtheblossomingtrees.WithinthehourthetwoofuswereseatedontheriverbankwithaluncheonbasketandseveralbottlesofapalatableBurgundy,awaitingthestartoftheannualrace.
“Whomdoyoufavour,Soames?”HestoppedmeasuringthelengthoftheburnmarkonanearlyScottish
lucifer,forhehadinsistedonbringinganumberofmatchstickswithhimtohelppassthetime.“Theblueteam.”
“Dark,orlight?”“Yes,”hesaidenigmatically.“Imean:OxfordorCambridge?”“Yes.”Heshookhishead.“Oneofthose.Thevariablesaretoocomplexto
makeaprediction,Watsup.”“Soames,myquerywasaboutsupport,notprediction.”Hegavemeascathinglook.“Watsup,whyshouldIsupportmenwithwhom
Iamnotacquainted?”WhenSoamesisinamood,thereisalwaysareason.Inoticedthathewas
layingoutmatchsticksinpatternsresemblingthebonesonakipper,andaskedhimwhy.
“Ihavebeenobservingthedistributionoftheoars,andIamwonderingwhysuchaninefficientarrangementhasbecometraditional.”
IlookedatthetwoboatsastheylinedupontheThamesfortheannualUniversityBoatRace.“Traditionisofteninefficient,Soames,”Ichided.“Foritconsistsofdoingthingsthesamewaytheyhavealwaysbeendone,insteadofaskinghowbesttodothem.ButIseenoinefficiencyhere.Thereareeightrowers,andtheoarspointalternatelytotheleftandtheright.Itisknownastandemrigging.Thatseemssymmetricandsensibletome.”
Tandemrigging(arrowshowsbowofboat)
Soamesgaveadissatisfiedgrunt.“Symmetric?Pah!Notatall.Theoarsononesideoftheboatareallinfrontofthecorrespondingoarsontheotherside.Sensible?Theasymmetrycreatesatwistingforcewhentherowerspullontheoars,causingtheboattoveertooneside.”
“That,Soames,isonereasonwhythereisacoxswain.Whohasarudder.”“Whichcreatesresistancetotheforwardmotionoftheboat.”“Oh.Buthowelsecouldtheoarsbearranged?Itisnotpossibletosittwo
rowerssidebyside.”“Thereare68alternatives,Watsup;34ifwecountleft–rightreflectionsas
beingthesame.OurGermanandItalianfriendsusedifferentarrangements,tobespecific.”Helaidouttwoskeletalarrangementsofmatchsticks.
Left:Germanrigging.Right:Italianrigging.
Istaredatthem.“Surelysuchstrangearrangementssufferfromevenworseproblems!”
“Perhaps.Letussee.”Hepursedhislips,deepinthought.“Thereareinnumerablepracticalissues,Watsup,whichwouldrequireamorecomplexanalysis.Nottomentionmorematchsticks.SoIshallcontentmyselfwiththesimplestmodelIcandevise,inthehopeofgainingusefulinsights.Iwarnyounow,theresultswillnotbedefinitive.”
“Fairenough,”saidI.
Resolvingtheforces.NotethatPpointsforwardsandRpointsawayfromthecentreoftheboatbecausetheoutermostendoftheoarisheldfixed(toagoodapproximation)bytheresistanceofthewater.Don’tforget
thattherowerfacesthesternandpullstheoartowardshim.
“Letusthenconsiderasingleoar,andcalculatetheforcesactingontherowlockwhereitpivots,duringthatpartofthestrokewhentheoarisinthewater.ForsimplicityIshallassumethatallrowershavethesamestrength,androwinperfectsynchrony,sotheyexertidenticalforcesFatanygiveninstant.IthenresolvethisforceintoacomponentPparalleltotheaxisoftheboat,andRatrightangles.”
“Alloftheseforcesvarywithtime,”saidI.Henodded.“Whatmattersiswhatmechanicianscallthemomentofeach
force—theextenttowhichitturnstheboataboutsomechosenpoint.This,youwillrecallfromourencounterwiththeArchimedesPalimpsest,isfoundbymultiplyingtheforcebyitsperpendiculardistancefromthatpoint.”
Itwasmyturntonod.IwassureIhadrememberedsomethingofthekind.“Imarkthepositionofthesternmostoarbyadot.Thiswillbeourchosen
point.Now,theforcePhasmomentPdaboutthepointatwhichtherowlockmeetsthecentralaxisoftheboat,iftheoarisontheleft-handside.Butifitisontheright-handside,themomentis–Pdsincetheforceactsintheoppositedirection.Noticethatthesemomentsarethesameforallfouroarsonthesamesideoftheboat.Inconsequence,thetotalmomentofalleightoarsis4Pd–4Pd,whichis0.”
“Thetwistingforcescancelout!”“FortheparallelforcesP,yes.However,themomentoftheforceRis
differentforeachoar,foritdependsonthedistancexbetweenthatoarandtheoneatthestern.Infact,itisRx.Ifsuccessiveoarsareseparatedbythesamedistancec,thenxtakesthevalues
0cR2cR3cR4cR5cR6cR7cR
astheoarsrunfromthesterntothebow.Thereforethetotalmomentis
0 cR 2cR 3cR 4cR 5cR 6Rc 7cR
wherethesignisplusforoarsontheleftoftheboat,butminusforthoseontheright.”
“Why?”“Forcesontheleftsideturntheboatclockwise,Watsup,whereasthoseon
therightturnitanticlockwise.Wecansimplifythisexpressionto
( 0 1 2 3 4 5 6 7)cR
wherethepatternofplusandminussignsmatchesthesequenceofsidesatwhichtheoarsareplaced.
“Nowconsidertandemrigging.Herethesequenceofsignsis
+–+–+–+–
sothecombinedturningmomentis
(0–1+2–3+4–5+6–7)cR=–4cR
Duringthefirstpartofthestroke,Rpointsinwards,butoncetheoarstartstotrailbackwardsthedirectionofRreversesanditpointsoutwards.Sotheboatfirstturnsinonedirection,thenintheother,creatingawigglingmotion.Thecoxhastousetheruddertocorrectthis,and—asIhavesaid—thiscreatesresistance.
“WhatoftheGermanrigging?Nowthecombinedturningmomentis
(0–1+2–3–4+5–6+7)cR=0
whatevercandRmightbe.Sothereisnotendencytowiggle.”“WhatoftheItalian?”Icried.“Oh,doletmetry!Thecombinedturning
momentis
(0–1–2+3+4–5–6+7)cR=0
aswell!Howremarkable.”“Quite,”repliedSoames.“Now,Watsup,hereisaquestionforyouragile
mind.AretheGermanandItalianrigs—ortheirleft–rightreflections,whichdifferonlytrivially—theonlywaystomaketheturningforcesequalzero?”Hemusthaveseenthelookonmyface,forheadded:“Thequestionboilsdowntoseparatingthenumbersfrom0to7intotwosetsoffour,eachhavingthesame
sum.Whichmustbe14sinceallsevennumbersaddupto28.”
Seepage300fortheanswer,andfortheresultofthe1877BoatRace.
TheFifteenPuzzleThisisanoldfavourite,butnonetheworseforthat.It’safascinatingcasewherealittlemathematicalinsightcouldhavesavedanawfullotofwastedeffort.Plus,Ineedittosetupthenextitem.
In1880aNewYorkpostmasternamedNoyesPalmerChapmancameupwithwhathecalledtheGemPuzzle,andthedentistCharlesPeveyofferedmoneyforasolution.Thistriggeredabriefcraze,butnoonewonthecash,soitquicklydieddown.TheAmericanpuzzlistSamLoyd*claimedthathehadstartedacrazeforthispuzzleinthe1870s,butallhereallydidwastowriteaboutitin1896,offeringaprizeof$1,000,whichrevivedinterestforatime.
Thepuzzle(alsocalledtheBossPuzzle,GameofFifteen,MysticSquare,andFifteenPuzzle)startswith15slidingblocksnumbered1–15arrangedinasquare,withanemptysquareatbottomright.Blocksareinnumericalorder,exceptfor14and15.Yourtaskistoswapthe14and15,leavingeverythingelseasitwas.Youdothisbymovinganyadjacentblockintotheemptysquare,andrepeatingthesemovesasoftenasyouwish.
Asyoumovemoreandmoreblocks,thenumbersgetjumbledup.Butifyou’recareful,youcanunjumblethemagain.It’seasytoassumethatanyarrangementcanbeobtainedifyou’recleverenough.
Fifteenpuzzle.Left:Start.Middle:Finish.Right:Colouringtheblocksforanimpossibilityproof.
Footnote
*That’snotatypo:hedidn’tspellitwithadoubleL.
Loydwashappytooffersuchagenerous(atthetime)prize,becausehewasconfidenthewouldneverhavetopayup.Thereare16!potentiallypossible
arrangements:allpossiblepermutationsoftheblocks(15numberedplusoneempty).Thequestionis:whichofthesearrangementscanbereachedbyaseriesoflegalmoves?In1879WilliamJohnsonandWilliamStoryprovedthattheanswerisexactlyhalfofthem;and—wouldn’tyoujustknowit—thearrangementthatgetsthecashisintheotherhalf.TheFifteenPuzzleisinsoluble.Butmostpeopledidn’tknowthat.
Theimpossibilityproofinvolvescolouringthesquareslikeachessboard,asintheright-handfigure.Slidingablockineffectswapsthatblockwiththeemptysquare,andeachswapchangesthecolourassociatedwiththeemptysquare.Sincetheemptysquaremustendupinitsoriginalposition,thenumberofswapsmustbeeven.Everypermutationcanbeobtainedfromsomeseriesofswaps;however,halfofthemuseanevennumberofswapsandtheotherhalfuseanoddnumber.
Therearemanywaystoachieveanygivenpermutation,buttheyareeitheralloddoralleven.Thedesiredresultcouldbeobtainedwithjustoneswap,interchanging14and15,butoneisodd,soyoucan’tachievethispermutationwithanevennumberofswaps.
Thisconditionturnsouttobetheonlyobstacle:legalmovesleadtoexactlyhalfofthe16!possiblerearrangements.Now16!/2=10,461,394,944,000;suchalargenumberthathowevermanytimesyoutry,mostpossibilitiesremainunexplored.Thiscouldencourageyoutothinkthatanyarrangementmustsurelybepossible.
TheTrickySixPuzzleIn1974RichardWilsongeneralisedtheFifteenPuzzleandprovedaremarkabletheorem.Hereplacedtheslidingblocksbyanetwork.Theblocksarerepresentedbynumbersthatcanbeslidalonganedge,provideditisconnectedtothenodecurrentlybearingtheblanksquare.Theblanksquarethenmovestoanewlocation.Thefigureshowsthestartingpositionoftheblocks.Nodesarelinkedifthecorrespondingsquaresareadjacent.
NetworkrepresentingtheFifteenpuzzle
Wilson’sideaistoreplacethisnetworkbyanyconnectednetwork.Supposeithasn+1nodes.Initially,onenode,markedbythebox,isempty(fromnowon,thinkofitasnode0)andtheresthaveanumber(1–n)sittingonthem.Thepuzzleamountstomovingthesenumbersaroundthenetwork,byswapping0withthenumberonanadjacentnode.Therulesspecifythat0mustendupwhereitstarted.Theothernnumberscanbepermutedinn!ways.Wilsonasked:whatfractionofthesepermutationscanbeachievedbylegalmoves?Theanswerclearlydependsonthenetwork,butnotasmuchasyoumightexpect.
Thereisoneobviousclassofnetworksforwhichtheanswerisunusuallysmall.Ifthenodesformaclosedring,theinitialarrangementistheonlyoneyoucanreachbylegalmoves,because0hastoreturntoitsstartingpoint.Alltheothernumbersstayinthesamecyclicorder;there’snowayforanumbertoworkitswayroundthesideofanotherone.RickWilson’sTheorem(sonamedtoavoidconfusionwithanothermathematicalWilson)statesthatasidefromclosedrings,eitherallpermutationscanbeachieved,orexactlyhalfofthem(theevenones).
Withexactlyonegloriousexception.
Thetheoremrevealsasurprise.Auniquesurprise:anetworkwithsevennodes.Sixformahexagon,andtheotheronesitsinthemiddleofadiameter.Thereare6!=720permutations;halfofthisis360.Buttheactualnumberthatcanbereachedisonly120.
Wilson’sexceptionalnetwork
Thereasoninginvolvesabstractalgebra,namelysomeelegantpropertiesofgroupsofpermutations.Detailsarein:AlexFinkandRichardGuy,Rick’strickysixpuzzle:S5sitsspeciallyinS6,MathematicsMagazine82(2009)83-102.
AsDifficultasABC
Fromtimetotimemathematicianshaveapparentlycrazyideasthatturnouttohavehugeimplications.TheABCConjectureisoneofthem.RememberFermat’sLastTheorem?In1637PierredeFermatconjecturedthat
ifn≥3therearenononzerointegersolutionstotheFermatequation
an+bn=cn
Ontheotherhand,thereareinfinitelymanysolutionswhenn=2,suchasthePythagoreantriple32+42=52.Ittook358yearstoproveFermatwasright,withtheworkofAndrewWilesandRichardTaylor(seeCabinetpage50).Jobdone,youmightthink.Butin1983RichardMasonrealisedthatnoone
hadlookedcloselyatFermat’sLastTheoremforfirstpowers:
a+b=c
Youdon’tneedtobeawhizzatalgebratofindsolutions:1+2=3,2+2=4.ButMasonwonderedwhetherthequestiongetsmoreinterestingifyouimposedeeperconditionsona,b,andc.Whateventuallyemergedwasashiningnewconjecture,theABC(orOesterlé–Masser)Conjecture,whichwillrevolutionisenumbertheoryifanyonecanproveit.It’ssupportedbyavastquantityofnumericalevidence,butaproofseemselusive,withthepossibleexceptionofworkofShinichiMochizuki.I’llgetbacktothatonceweknowwhatwe’retalkingabout.Morethantwothousandyearsago,EuclidknewhowtofindallPythagorean
triples,usingwhatwewouldnowwriteasanalgebraicformula.In1851JosephLiouvilleprovedthatnosuchformulaexistsfortheFermatequationwhenn≥3.Masonwonderedaboutthesimplerequation
a(x)+b(x)=c(x)
wherea(x),b(x),andc(x)arepolynomials.Apolynomialisanalgebraiccombinationofpowersofx,like5x4–17x3+33x–4.Again,it’seasytofindsolutions,buttheycan’tallbe‘interesting’.The
degreeofapolynomialisthehighestpowerofxthatoccurs.Masonprovedthatifthisequationholds,thedegreesofa,b,andcarealllessthanthenumberofdistinctcomplexsolutionsxoftheequationa(x)b(x)c(x)=0.Itturnedoutthat
W.WilsonStothershadprovedthesamethingin1981,butMasondevelopedtheideafurther.Numbertheoristsoftenlookforanalogiesbetweenpolynomialsandintegers.
ThenaturalanalogueoftheMason–Stotherstheoremwouldbethis.Supposethata+b=c,wherea,b,andcareintegerswithnocommonfactor.Thenthenumberofprimefactorsofeachofa,b,andcislessthanthenumberofdistinctprimefactorsofabc.Unfortunately,thisisspectacularlyfalse.Forexample,14+15=29,which
hasonlyone(necessarilydistinct)primefactor,whereas14=2×7and15=3×5bothhavetwoprimefactors.Oops.Undaunted,mathematicianstriedtomodifythestatementuntilitlookedlikeitmightbetrue.In1985DavidMasserandJosephOesterlédidjustthat.Theirversionstates:
Foreveryε>0,thereareonlyfinitelymanytriplesofpositiveintegers,withnocommonfactors,satisfyinga+b=c,suchthatc>d1+ε,whereddenotestheproductofthedistinctprimefactorsofabc.
ThisistheABCConjecture.Ifitweretobeproved,manydeepanddifficulttheorems,provedoverpastdecades,withenormousinsightandeffort,wouldbedirectconsequences,andthereforehavesimplerproofs.Moreover,allofthoseproofswouldbeverysimilar:dosomeminorandroutinesettingup,andthenapplytheABCTheorem,asitwouldthenbecome.AndrewGranvilleandThomasTucker[It’saseasyasabc,NoticesoftheAmericanMathematicalSociety49(2002)1224–1231]writethataresolutionoftheconjecturewouldhave“anextraordinaryimpactonourunderstandingofnumbertheory.Provingordisprovingitwouldbeamazing.”BacktoMochizuki,awell-respectednumbertheoristwithasolidtrackrecord
ofresearch.In2012heannouncedaproofoftheABCConjectureinaseriesoffourpreprints—papersnotyetsubmittedforofficialpublication.Contrarytohisintentions,thisattractedtheattentionofthemedia,thoughitwassurelyunrealistictoimaginethatthiscouldpossiblyhavebeenavoided.Expertsarecurrentlycheckingthe500orsopagesofradicallynewmathematicsinvolved.Thisistakingalotoftimeandeffortbecausetheideasaretechnical,complicated,andunorthodox;however,nooneisrejectingitbecauseofthat.Oneerrorhasbeenfound,butMochizukihasstatedthatitdoesnotaffecttheproof.Hecontinuestopostprogressreports,andtheexpertscontinuechecking.
RingsofRegularSolids
Eightidenticalcubesfittogether,facetoface,tomakeacubetwicethesize.Eightcubesalsofittogethertoforma‘ring’—asolidwithaholeinit,topologicallyatorus.
Aringofcubes
Withabitmoreeffort,youcandothesamewiththreeotherregularsolids:theoctahedron,dodecahedron,andicosahedron.Inallfourcasesthesolidsareexactlyregularandtheyfittogetherexactly:thisisobviousforcubesandisasimpleconsequenceofsymmetryfortheotherthreesolids.
Ringsofoctahedrons,dodecahedrons,andicosahedrons
However,therearefiveregularsolids,andthismethoddoesn’tworkfortheoneremainingtype,atetrahedron.Soin1957HugoSteinhausaskedwhetheranumberofidenticalregulartetrahedronscanbegluedface-to-facesothattheyformaclosedring.Hisquestionwasansweredayearlater,whenS.Świerczkowskiprovedthatsuchanarrangementisimpossible.Tetrahedronsarespecial.However,in2013MichaelElgersmaandStanWagondiscoveredabeautiful
eightfoldsymmetricringmadefrom48tetrahedrons.WasŚwierczkowskiwrong?
ElgersmaandWagon’sring.Left:Perspectiveview.Right:Viewfromabovetoshoweightfoldsymmetry.
Notatall,asElgersmaandWagonexplainedintheirarticleabouttheirdiscovery.Ifyoumaketheirarrangementusinggenuinelyregulartetrahedrons,theyleaveasmallgap.Youcanclosethegapbyelongatingtheedgesshownasthickerlines,from1unitto1.00274,adifferenceofonepartinfivehundred,whichthehumaneyecan’tdetect.
Thegap,exaggerated
Świerczkowskiasked:ifyouuseenoughtetrahedrons,andfitthemtogetherinaringtoleaveagap,howsmallcanthegapbe?Canyoumakeitassmallasyouplease,relativetothesizeofasingletetrahedron,byusingsufficientlymany?Theanswerisstillnotknownifthetetrahedronsarenotallowedtointersect,butElgersmaandWagonprovethattheansweris‘yes’iftheycaninterpenetrate.Forexample,438ofthemleaveagapthatisaboutonepartintenthousand.
ElgersmaandWagon’s438interpenetratingtetrahedrons
Theyconjecturethattheanswerremains‘yes’evenwhenthetetrahedronsarenotallowedtointersect,butthearrangementshavetobemorecomplicated.Asevidence,theyhavefoundaseriesofringswithever-narrowergaps.Thecurrentrecord,discoveredin2014,isanalmost-closedringof540non-intersectingtetrahedronswithgap5×10–18.Seepage302forfurtherinformation.
ElgersmaandWagon’sringof540non-intersectingtetrahedrons.
TheSquarePegProblem
Thismathematicalmysteryhasbeenopenformorethanacentury.Isittruethateverysimpleclosedcurveintheplane(onethatdoesnotcrossitself)containsfourpointsthatarethecornersofasquarewhosesideisnotzero?
Asimpleclosedcurveandasquarewhosecornerslieonit
‘Curve’hereimpliesthatthelineiscontinuous,nobreaks,butitneednotbesmooth.Itcanhavesharpcorners,andindeeditcanbeinfinitelywiggly.Weinsistthatthesideofthesquareisnonzerotoavoidthetrivialanswer:choosethesamepointforallfourcorners.ThefirstprintedreferencetotheSquarePegProblemappearedin1911ina
reportonaconferencetalkbyOttoToeplitzthatapparentlyclaimedaproof.However,noproofwaspublished.In1913ArnoldEmchprovedthestatementistrueforsmoothconvexcurves,sayingthathehadn’theardaboutitfromToeplitz,butfromAubreyKempner.Thestatementhasbeenprovedcorrectforconvexcurves,analyticcurves(definedbyconvergentpowerseries),sufficientlysmoothcurves,curveswithsymmetry,polygons,curveswithnocuspsandboundedcurvature,star-shapedtwice-differentiablecurvesthatmeeteverycircleinfourpoints…Yougetthepicture.Lotsoftechnicalhypotheses,nogeneralproof,no
counterexample.Perhapsit’strue,perhapsnot.Whoknows?Therearegeneralisations.TheRectangularPegProblemaskswhetherforany
realnumberr≥1,everysmoothsimpleclosedcurveintheplanecontainsthefourverticesofarectanglewithsidesintheratior:1.Onlythesquarepegcaser=1hasbeenproved.Thereareafewextensionstohigherdimensionsunderverystrongconditions.
TheImpossibleRoute
FromtheMemoirsofDrWatsup
Itiswithaheavyheart…Ithrewdownmypen,overcomewithgrief.ThatDevil’sspawn!Professor
Mogiarty’smachinationshadcausedtheprematuredemiseofoneofthegreatestdetectivesevertohavelimpedthestreetsofLondondisguisedasanelderlyRussianfishmonger.ThefinestmindIhaveeverencountered,snuffedoutbyacriminalwho—untilSoamesdispatchedhimatsuchcost!—hadafingerineveryfouldeedinthekingdom.Exceptfortheidiotwhooftenparkshiscarriagedirectlybeneathourwindow,wherehishorse—Pleasebearwithyourhumblescribeashewipesawayamanlyteartorecount
thetragicevents.Soameshadbeeninablackmoodforaweek.ItwaswhenIsawhimfitting
thesixthpadlocktothewindowandliningupthethirdGatlinggunthatIbegantosuspecthismindwastroubledinsomemanner.“Youmightsaythat,”saidhe.“Sowouldyoursbeifyouhadnarrowly
dodgedafallinggrandpianoonyourwaytothebarber’s—aChickering,bytheway,Icouldtellinstantlyfromthecast-ironframe.BeforeIhadgatheredmywits,Iwasforcedtoleapasidefromarunawaybrewer’sdraydrawnbyfourcarthorses,whichexplodedasplitsecondafterIhadhadtheforesighttotakecoverbehindaconvenientwall.Thepromptcollapseofthewallintoadeepcavitycameclosetodisturbingwhatlittleequilibriumremainedtome,butImanagedtoswingmyselftosafetyusingagrappling-ironthatIhabituallycarryinmypocketforsucheventualities.Itfoldsupforconvenience,andthecordislightbutstrong.Afterthat,thingsbecameatriflefraught.”IfIhadnotknownmyfriendbetter,Iwouldhavethoughthewasrattled.“Hasitoccurredtoyou,Soames,thatperhapssomeoneistryingtodoyou
harm?”Hesnortedinadmirationatmyastuteness,orsoIpresumed.“ItisMogiarty,”
hestatedflatly.“ButthistimeIhavethemeasureofhim.Evenaswespeakmycunningplaniscomingtofruition,andeverypolicemaninLondonisdescendingonthat…Wellingtonofcrime…andhisminions.Soontheywillbebehindbars,andthen…therope!”
Therewasaknockatthedoor.Anurchinappeared.“Telegramfer‘isnibs!”Soamestookthepaperandhandedtheurchinathruppenybit.“Thegoingrateissixpence,”saidtheurchin.“Whosaysso?”“’imacrosstheroad,guv.ThatMrSher—”“It’llbetuppenceandacliproundtheearifyoudon’tgoaway,”saidSoames.
Theurchinleft,mutteringunderhisbreath.Soamesopenedthefoldedpaper.“Nodoubtnewsoftheoperation’ssucce…”Hisvoicetrailedoff.“Whatisit?”Iaskedanxiously.Hisfacehadgonedeathlypale.“Mogiartyhasescaped!”“How?”“Disguisedasapoliceman.”“Thecunningfiend!”“ButIknowwherehehasgone,Watsup.Youhavetenminutestogohome
andpack.Thenwewillbetakingthecross-channelferry,assortedtrains,abrougham,adog-shay,anomnibus,andtwodonkeys.Oneeach.”“But—Soames!BeatrixandIhavebeenmarriedforlessthanamonth!I
cannotleave—”“Yournewbridewillhavetogetusedtothiskindofthingeventually,
Watsup,ifwearetocontinueourcollaboration.”“True,but—”“Notimelikethepresent.Absencemakestheheartgrowfonder.Adogisa
man’sbest—well,enoughclichés.Herbrotherwilltakecareofherwhileyouaregone.Anabsenceofsixweeksshouldbeample.”Irealisedthathewouldnotaskthisofmewithoutacompellingreason.He
neededme.Imustrisetotheoccasion,nomatterthepersonalcost.“Verywell,”Isaid,direforebodingsnotwithstanding.“Beatrixwillunderstand.Wherearewegoing?”“TotheSchtickelbachFalls,”hesaidquietly.Igaveaninvoluntaryshudder.Itwasanametostriketerrorintotheheartof
eventhemostaccomplishedmountaineer.“Soames!Thatissuicide!”Heshrugged.“ItiswhereweshallfindMogiarty.Butfirstwemustgetthere.”
Hepulledoutamap.
Soames’smap
“ThemapshowstheappropriateregionofSwitzerland.Observethenetworkofrivers.Theyriseinthenorthandflowacrossthecountry’sborders.TheSchtickelbachFallsareattheendofasmallriverbranchingoffalargerone.”“WheredoestherivergoaftertheFalls?”“Itplungesbeneaththeearth,intosomeundergroundpassage.Nooneknows
whereitreappears.”“Itisstrangegeology,Soames.”“TheSwisslandscapeisatorturedone,Watsup.Now,therearesixbridges,
whichIhavemarkedA,B,C,D,E,F.ThesearetheonlybridgeswithinSwissbordersthatjointheareasoflandshown.TheomnibusterminusisatthesmalltownofFroschma¨usekrieg.FromtherewewillhiredonkeysandproceedtotheFalls.WemuststayinSwitzerland:itwillbehardenoughtocrossanationalborderunnoticedonce,anditwouldbefoolhardyintheextremetorepeattheattempt.Ihavealreadyworkedoutaroute,butyoumayhaveabetteridea.”Istudiedthemap.“Why,itissimple!WecrossbridgeA.”“No,Watsup.Itistoodirect.Mogiartywillbeexpectingthat,itisabridgetoo
far.WemustleavebridgeAuntillastinthehopeofthrowinghimoffthescent.Andwemustcrosseachbridgeatmostoncetominimisethechanceofattractingattentionandbeingidentified.”“ThenwemustbeginwithbridgeB,”saidI.“TheonlycontinuationisviaC,
thenD.AtthatpointwehaveachoiceofEorF.BothleadtotheFalls,sowemayaswelluseE.Done!”“AsIsaid,wemustleaveAuntillast.NotE.”“Oh,yes.ThenweproceedacrossA—no,thatisadeadendwithnofurther
connectiontotheFalls.SoweleaveAforlaterandcrossF…Butno:thatalsoisadeadend.”
TwopathsthatdonotreachtheFalls
Soamesgruntednon-committally.Icheckedmyanalysis.“PerhapsbridgeF…no.ThesameproblemsariseifweuseFinsteadofE,aftercrossingD.Thereisnosuchpath,Soames!”Athoughtstruckme.“Unlessthereisatunnel,orsomeotherwaytocrossoneoftherivers.Aferry?Acanoe?”“Thereisnotunnel,orferry,orcanoe,andwedonotneedtocrossanyrivers.
Bridgesanddrylandsuffice.”“Thenthethingisimpossible,Soames!”Hesmiled.“ButWatsup:Ihavealreadytoldyouthatthereisaroute
satisfyingthestatedconditions.Indeed,therearenolessthaneightessentiallydifferentroutes—bywhichImeanthatthebridgesarecrossedinadifferentorder.”“Eight?IconfessIdonotseeevenone,”saidI,exasperated.
IsSoamesright?Seepage303fortheanswer.
TheFinalProblem
FromtheMemoirsofDrWatsup
Isleptbadlyandawokeatsunrise,tofindSoamesalreadydressed,bright-eyed,andbushy-tailed.“Timeforbreakfast,Watsup!”hedeclaredinaheartyvoice.Ifhewasapprehensiveabouttheforthcomingencounter,heconcealeditperfectly.Assoonaswehadfinishedourplatesofbread,meat,andSwisscheese,we
mountedourdonkeysandmadeourwayupanarrowtrack.Aftersomemileswetetheredourtrustysteeds,havingreachedthebaseoftheSchtickelbachFalls.Awildtorrentofwaterplungedbetweenthesheersidesofatoweringchasm,todisappearintoadeepholeintheground,creatingagloriousrainbowthatglitteredintheafternoonsun.AsteeprockypathledtothetopoftheFalls.Asweapproached,asilhouette
appearedontheskylineaboveus.“Mogiarty,”saidSoames.“Therecanbenomistakingthatevilprofile.”He
tookouthispistolandflickedoffthesafety-catch.“Thefiendistrapped,forthereisnowaydownsavethispathway.Notanypaththatamancantakeandstilllive,atanyrate.Waithere,Watsup.”“Nay,Soames!Iwillaccomp—”“Youwillnot.Thedutytoridtheworldofthisvilecreatureisminealone.I
willsignalwhenitissafeforyoutojoinme.Promisemeyouwillremainhereuntilyoureceiveit.”“Whatsignal?”“Youwillknowwhenthetimecomes.”Iassented,despiteprofoundmisgivings,andheascended,quickly
disappearingfromviewbehindarockycrag.ThelastIsawofhimwasapairofstoutclimbingboots.Iwaited.Allwassilence.Then,suddenly,Iheardshouting.ThewindcarriedawaythewordsandI
couldnotmakethemout.ThenIheardtheindubitablesoundsofaprolongedstruggle,andseveralgunshots.Therewasascream,andsomethingplungedpastmeamidthetorrent.ItwasshroudedbysprayandmovedsorapidlythatIcouldnotmakeitout,butitwasroughlythesizeofaman.Ortwomen.
Shockedtothecore,IneverthelessdidasSoameshadtoldme,andwaited.Nosignalcame.AtlastIdecidedthatsomethinghadgonewrong,whichrelievedmeofmy
promise.Iclamberedupthepath.Atthetop,therockclimbedfurtherskywardsinahugeoverhang,barringmyway.Amossyledgeledtowardstheprecipicefromwhichthewaterfallplunged.OfSoamesandMogiartytherewasnosign.But,dampenedbythespray,themossheldfainttracesoffootprints.Theprintstoldacleartaletoanyonewhohadabsorbedlessonsindetection
fromthemaster.IrecognisedtheindistinctimpressionswithachevronpatternasSoames’sboots;theotherprintswithazigzagwereevidentlyMogiarty’s.Thetwosetsofprintsledtothelipofthechasm,andherethegroundwaschurnedintothickmudbywhathadevidentlybeenthestruggleIhadheard.Isuckedinmybreathinhorror,fornofootprintsreturnedfromthatterrible
brink.IretainedenoughpresenceofmindtoattemptwhatSoameswouldhavedone,
facedwithsuchevidence.Beingcarefulnottosuperimposemyownfootprints—forthelocalconstabulary,ineptastheywouldundoubtedlybe,wouldnodoubtwishtoinspectthescene—Imadeathoroughstudy.SoameshadclearlywalkedbehindMogiarty,forhisprintssometimes
overlaidthoseofthecriminal,butnotviceversa.Mogiarty’sprintsseemeddeeperthanSoames’s,butthen,myfriendwasalwayslightoffoot.Thedismalconclusionwasclear.SoameshadpursuedMogiartytotheedgeoftheprecipice;therehadbeenastruggle;bothmen,nodoubtstillgrappling,hadplungedtotheirdoom.Theirbodieswerenowdeepundergroundinsomedankchamber,nevertoberecovered.Itrudgeddespondentlybacktothepath,wherethebarerockshowedno
footprints.Theclifftoweredaboveme,unscalable.IreasonedthathadSoamesprevailed,hewouldhavesignalledandbeawaitingmyarrival.HadMogiartyprevailed,hewouldhavebeenawaitingmeinstead,armedtotheteeth.Therewasnoconceivabledoubtthatbothmenhadmetthesameterribleend.Yet,evenasIbeganmydescent,myfriend’svoiceseemedtoechoinmy
mind,andthetonewasmocking.Wasmysubconscioustryingtotellmesomething?Griefovercamemyanalyticalabilities,andItrudgeddownhilltowardsthedonkeys.BywhichIrefertooursteeds;however,theSwisspolicewouldbenext.
TheReturn
FromtheMemoirsofDrWatsup
IthadbeenthreeyearssinceSoames’snoblesacrificehadridtheworldofMogiarty.Thedetective’spremisesat222BhadpassedtohisbrotherSpycraft,andIhadtakenupmedicalpracticeinearnest.Astoopedfigureintatteredclotheslimpedintomysurgery.“Isyouthat
doctorbloke?Whatwrotethemdee-tectivestoriesinthemmagazines?”Iacknowledgedmymedicalstatus.“Idowrite,butregrettablytheStrandhas
declinedmysubmissionstodate.”“Oh.Mustbethatothergeezer.Butyou’lldo.Igottaterriblepaininmeleg,
Doc.”“Thatwillbesciatica,”Itoldhim.“Itiscausedbyabackproblem.”“Inmeleg?”“Thenervesinyourlegaretrappedsomewhereinyourspine.”“Ohgawd!Igotnervesinmeleg?”“Lieonthecouchand—”Iobservedthedirtonhisclothes.“No,firstletme
getaclothforyoutolieon.”Iturnedmybackandopenedacupboard.“Noneedforthat,Watsup,”saidafamiliarvoice.Iturned,stared—andfainted.WhenIcameto,Soameswasbendingovermewavingsmellingsaltsunder
mynose.“Idoapologise,oldchap!Ihadpresumedyouhadlongagoworkedoutmy
cunningdeceptionandwhyithadbeennecessary.”“Notatall.Ithoughtyoudead.”“Ah.Well,yousee,whenIpushedMogiartyofftheprecipice,andsawhow
thefootprintswouldappeartoanyonelessastutethanmyself,Irealisedinaflashthatfatehadpresentedmewithagoldenopportunity.”“Yes!Isee!”Icried.“AlthoughMogiarty’shenchmenintheBritishIsleshad
beenapprehended,severalremainedatlargeonthecontinent.Iftheythoughtyoudead,youcouldweaveyourwebandtrapthem.Soyousuppliedsomemisleadingevidence,goodenoughtoconvincethebunglingSwisspolice.Sincethenyouhavespentallyourwakinghourspursuingthecriminalscumthatremained.Onebyoneyoueliminatedthem.Youtrackedthelastonedownin—
oh,Casablancaorsomeotherexoticlocale—andhewillnotbothertheworldagain.Sonowyoucanrevealthatyouarestillalive.”“Abrilliantsequenceofdeductions,Watsup.”Isilentlycongratulatedmyself.
“Thoughwronginalmosteveryparticular.”Soamesexplained:“Theonethingyougotrightwasthatthiswasaheaven-
sentopportunityformetodisappear.Butmyreasonswerenotwhatyouimagine.Ihadrunupasignificantnegativebalancebettingonthehorses,lackedthefundstoreimbursemybookmaker,andfacedthethreatofseriousbodilyharm.Havingfinallyaccumulatedthenecessaryfunds,Ipaidoffthedebtandrejoinedsociety.”Ifounditdifficulttotakein.“Iunderstandyourposition,Soames.Itcould
happentothebestofus.Buthow—?”
HowdidSoamesescapeanddisappear?Makeyourdeductionsbeforereadingon,fornarrativeimperativerequirestheimmediatepresentationoftheanswer.
TheFinalSolution
Soamessettledhimselfbesidethefire.“Ithappenedlikethis,Watsup.WhenIreachedthetopofthepath,Mogiartywaswaitingformebehindarock.Heknockedmeout,andwascarryingmetowardstheprecipicetohurlmeintotheabyss.FortunatelyIregainedconsciousnessandreachedformypistol.Intheensuingstrugglesomeshotswerefired,butnoonewashit.Mogiarty’sexertionscausedhimtoslipandfalltohisdeath.Iwasfortunatenottohavejoinedhim.”Hesaidthisinamatter-of-factvoice,asifithadbeenoflittleimport.“Intendingtosummonyou,Ilookedbackandsawasinglesetoffootprints
madebyMogiarty’sboots.Theyledfromthepathtothebrinkofthechasm,andnothingledtheotherway.IsawatoncethattheywereslightlydeeperthantheyshouldhavebeenforamanofMogiarty’sweight—acluethatIhopedyouwouldspot,Watsup,andknewthatthepolicewouldnot.Ithenwalkedbackwardstothesafetyofthepath,overlayingmyownprints,beingcarefultomakethemappearasthoughIhadcometheotherway.”“Thethoughtdidcrossmymind,Soames.ButIdismissedit,sinceIwas
unawareofyourgamblingdebts,soIcouldnotthinkofamotive.Buttheledgewasempty,thecliffimpossibletoclimb!Howdidyouconcealyourself?”Heignoredmyquestion.“IrealisedthatifIfailedtosignal,youwould
eventuallydecidethatyourpromisewasnolongervalid,andclimbthepath.Itwasbuttheworkofamomenttoclimbtotheledgeabove,wheretheoverhanghidmefromview.Therestyoucanguess.”“But—thecliffisunclimbable!”Icried.Heshookhisheadsadly.“Mydear,Watsup,Idistinctlyremembertellingyou
ofafoldinggrappling-ironthatIalwayscarryforsucheventualities.”(Seepage240.)“Youreallymustnotforgetsuchvitalinformation.Oftenonetinyfactisallittakestounlockagreatmystery.”Ihungmyhead,forIhadoverlookedthatitemofequipmentuntilthatvery
moment.Iattemptedawrygrin.“Why,Soames!Howabs—olutely,uh,ingenious!”Hegaveathinsmile,andchangedthesubject.“Acupoftea,Watsup?”“Thatwouldbedelightful,Soames.”“ThenIshallaskMrsSoapsuds—”
Thedoorswungopenandourlandlady’sheadpokedaroundit.“CanIbeofservice,MrSoames?”“—tomakeusapot,”sighedthedetective.
TheMysteriesDemystified
or,ifnot,castinanewlightbysundryextractsfromDrJohnWatsup’sextensivearchiveofcasenotes,presscuttings,andSoamesianmemorabilia;withoccasionalcontributionsfromothersources
TheScandaloftheStolenSovereign
Magnifyingglassinhand,SoamesinspectedeveryinchoftheGlitz’skitchensandaccounts.Hehadthecarpetsliftedtoseewhatwasbeneath—aremarkablecollectionbutnotofrelevancetothistale—andsearchedManuel’scrampedlodgingsintheattic.Hesampledthecontentsofseveralbottlesinthebar.Infact,hehadarrivedathisconclusionsbeforeHisLordshiphadfinisheddescribingthefactsofthecase,butitwouldneverdotomaketheprocesslookeasy,andtheopportunityofafreemaltwhiskeyshouldnotbedeclinedwithoutgoodreason.TheowneroftheGlitzHotelwaswaitinginasumptuouslyfurnishedprivate
room,pacingthefloorandglaring.“Haveyourecoveredmystolensovereign,Soames?”“No,myLord.”“Pah!IknewIshouldhavetriedMrSher—”“Ihavenotrecovereditbecausetherewasnostolensovereign.Itwasnever
missinginthefirstplace.”“But£27plus£2donotmake£30!”“Iagree.Butthereisnoreasonwhytheyshould.Thesumsaddupifyoudo
themcorrectly.”AndSoameswrote:
“Thesumof£30isnolongertheissue,”saidSoames.“Itwas,afterall,thewrongbill.Themenhavenowpaid£27,MyLord,andweshouldsubtract£2togetthe£25owedtothehotel.Notaddit.”“But—”“Youroriginalcalculationappearstomakesensebecausethenumbers29and
30aresoclosetogether.Butsuppose,forexample,thatthebillhadactuallybeen£5,sothewaiterwasgiven£25toreturn,keeping£1asatipandgivingthemeach£8.Nowthemenhadeachpaid£2,atotalof£6.Manuelhasretainedonly
£1.Thetotalofthesetwoamountsis£7.Youwouldnowask:wherehastheother£23gone?Buttheactualbillwas£5,andthehotelhasbeenpaidexactlythat.Sohowcan£23bemissingfromthehotel’sshare?Ithasbeensharedbythethreecustomers,whohavegivenasmallpartofittoManuel.”Humphshaw-Smatteringturnedpink.“Humph,”hesaid.“Pshaw.”Hepulled
himselftogether.“Yourfee,sir?”“Twenty-ninesovereigns,”saidSoames,notmissingabeat.
NumberCuriosity
1001100001100000011000000001100000000001100000000000000001
Ialsoaskedwhyitworks.That’saharderquestionbecauseyouhavetothinkandnotjustcalculate.Insteadofaformalproof,let’sjustlookatatypicalcase:11×909091.First,rewriteitintheoppositeorderas909091×11.Thisis909091×10+909091×1;thatis,9090910+909091.Addthemlikethis:
Whatnext?Startingfromtheright,0+1=1,soweget
Then1+9=0carry1:
Nowwehavetoaddthecarrytothe9and0,whichagaingives0carry1.Thisleadstoacascadeofcarries,eachconvertinga9to0carry1,untilwegetto
Finallyonlythecarrydigitisleft,leadingtotheanswer
TrackPosition
Mazesolution
ForfurtherinformationseeR.Penrose,Railwaymazes,inALifetimeofPuzzles,(eds.E.D.Demaine,M.L.Demaine,T.Rodgers),A.K.Peters,WellesleyMA2008,133–148.PicturesoftheLuppittMillenniumMonumentcanbefoundat:
http://puzzlemuseum.com/luppitt/lmb02.htm
SoamesMeetsWatsup
“Adecimalpoint?”Watsuphazarded.“No,youaskedforawholenumber.”Hepaused,struckbyasuddenrealisation.“Didyoutellmethatthesymbolmustgobetweenthetwodigits,MrSoames?”“No.”“Didyouinsistthatthedigitsbeseparatedbyaspace?”“Mydrawingwasperhapsambiguous,butIdidnotspecifyaspace.”“Ithoughtso.Wouldthismeetyourconditions?”AndWatsupwrote:
“Whichequals7.”
GeomagicSquares
Howtoassemblethejigsawsforrows,columns,anddiagonals
WhatShapeisanOrangePeel?
LaurentBartholdiandAndréHenriques.OrangepeelsandFresnelintegrals,MathematicalIntelligencer34No.4(2012)1–3.Youcandownloadasimilararticlefromarxiv.org/abs/1202.3033.
HowtoWintheLottery?
No.Thestatementsmadeareallcorrect,butthedeductionisfallacious.Toseewhy,considerthelotterythatrunseveryweekinthelittleknown
provinceofLilliputia.Herethereareonlythreeballs—1,2,3—andtwoofthemaredrawn.Youwinbygettingthosetworight.Therearethreepossibledraws:
121323
andtheseareallequallylikely.Thefirstnumberis1withprobability2/3,morelikelythan2withprobability
1/3or3withprobability0.Thesecondnumberis3withprobability2/3,morelikelythan2with
probability1/3or1withprobability0.So,bythesameargument,puntersshouldchoose13tomaximisetheir
chances.However,eachofthethreepossibilitiesisequallylikely,sothisisrubbish.Ingeneral,1ismorelikelytobethesmallestnumberinthedrawbecausein
thiscasetherearemorelargernumbersthanthereareforanyotherchoice.Notbecause1ismorelikelytobedrawn.Thesameeffectappliestotheotherpositions,butnotasobviously.
TheGreenSocksCaperIncident
“FrommydeepknowledgeofLondon’slowlife,itisimmediatelyobviouswhotheculpritis,”Soamesannounced.“Who?”“Thatisofnoconcernuntilwehaveformalproofofhisguilt,Watsup.
NothinglesswillconvinceInspectorRouladeoftheMetropolitanPolicewhenwepresenthimwithourconclusions.First,wemustlistthepossiblewaystodistributethecoloursamongthegarments.”“Icandothat,”saidWatsup.“Ihavesomesmallcommandofelementary
combinatorics.Ithasprovedusefulwhendecidingwhichlimbtoamputatefirst.”Andhewrote:
BGWBWGGBWGWBWBGWGB
“Thelettersdenotethecoloursofthegarments,intheorderjacket,trousers,socks,”Watsupexplained.“Nocolourrepeats,becauseofthewitnessreports,sotheonlypossibilitiesarethesixpermutationsofthethreeletters.”“Verygood,”saidSoames.“Andwhatshouldournextstepbe?”“Uh—totabulateallofthewaystodistributethegarmentsamongthethree
men.Thatwilltakesometime,Soames,becausethereare…uh,6×5×4…120combinations.”“Notso,Watsup.Withalittlethoughtwecaneliminatemostofthematthe
outset.Letusbeginbyfocusingonjustoneofthesuspects—GeorgeGreen,say.Suppose,forthesakeofargument,thatGreenwearsagreenjacket,browntrousers,andwhitesocks:caseGBW.”“Ah,butdoeshe?”“SoIhypothesise,forthesakeofargument.Ifthatiscorrect,itfollowsthat
theothertwosuspectscannotwearagreenjacket,orbrowntrousers,orwhitesocks,sinceonlyoneofeachtypeofgarmenthasagivencolour.SoforthosemenwecaneliminateGWB,BGW,andWBGfromthefivepossibilitiesremaining.ThatleavesonlyBWGandWGB.Which,youobserve,arecyclicpermutationsofGBW.WecanassignthesechoicestoBillBrownandWallyWhiteinonlytwoways.”Soamesbegantocompilehistable:
“ButSoames,”criedWatsup,“perhapsGeorgeGreendoesnotwearthegarmentsGBW!”“Quitepossibly,”saidSoames,unperturbed.“Thesearemerelythetoptwo
rowsofmytable.IcanmakesimilardeductionsfortheotherfivepossiblelistsforGeorgeGreen.And,ofcourse,againthepermutationsarecyclic.Therearethustwelvepossibilitiesaltogether.”Watsupcopiedouttheresultingtable:
Whenhehadfinished,Soamesnodded.“Andnow,mydearWatsup,allthatremainsistousetheevidencetoeliminatetheimpossiblecombinations—”“Becausethen,whateverremains,howeverimprobable,mustbetrue!”
Watsupcried.“Icouldnothaveputitbettermyself.Thoughinthiscase,themost
improbablefeatureisthatonlyoneofthesevillainswasinvolved.Iwouldhaveexpectedaconspiracy.“Anyway,ConstableWuggins—anadmirablefellow,Watsup,whomakesup
inperseveranceforwhathelacksinimagination—statedthatBrown’ssockswerethesamecolourasWhite’sjacket.ThatmeansthatBrown’stripleoflettersmustendwiththesameletterthatbeginsWhite’striple.Thateliminatesrows1,3,5,7,9,11,andreducesthetableto
“Next,IdeterminewhichcombinationssatisfythegoodConstable’ssecondcondition:thatthepersonwhosenamewasthecolourofWhite’strousersworesockswhosecolourwasthenameofthepersonwearingawhitejacket.Thisispurelyamatterofkeepingaclearhead.Forexample,inrow2White’strousersarewhite,sothepersonwhosenameisthecolourofWhite’strousersisWhitehimself.Hissocksaregreen.DoesGreenwearawhitejacket?No,hisjacketisgreen.Soweeliminaterow2.”“I’mnotsureIquite—”“Oh,verywell,letmedrawupanothertable!”AndSoameswrote:
“Onlyrows4,and12remain.Whichfurtherreducestheoriginaltableto
“Finally,ConstableWugginstellsusthatthecolourofGreen’ssockswasdifferentfromthenameofthepersonwearingthesamecolourtrousersasthejacketwornbythepersonwhosenamewasthecolourofBrown’ssocks.”
“Thateliminatesrow12,leavingonlyrow4.“Sonowitremainsonlytoseewhowaswearingthegreensocksinrow4.As
Isuspectedfromthestart,itwasWallyWhitewiththeentryWBG.”
ConsecutiveCubes
233+243+253=12,167+13,824+15,625=41,616=2042
Thiscanbefoundbytryingnumbersinturn.Amoresystematicmethodistoletthemiddlenumberbenandobservethat(n–1)3+n3+(n+1)3=3n3+6n=m2forsomem.Som2=3n(n2+2).Theterms3,n,n2=2havenocommonfactorsasidefromperhaps2and3.Thereforeanyprimefactorgreaterthan3mustoccurtoanevenpower(perhaps0)inbothnandn2+2.Thefirsttwonumberstosurvivethistestare4and24,and24providesasolutionbut4doesnot.
AdonisAsteroidMousterian
Thenumbersshouldbeassignedlikethis:
Order3:A=0,D=3,I=2,N=0,O=1,S=6.Order4:A=0,D=12,E=1,I=2,O=3,R=8,S=0,T=4.Order5:A=0,E=1,I=2,M=0,N=5,O=3,R=10,S=15,T=20,U=4.
Thesquaresbecome
Changingletterstonumbersandadding
Formoremagicwordsquaresandsimilarconstructions,see:JeremiahFarrell,Magicsquaremagic,WordWays33(2012)83–92.
Availableat:
http://digitalcommons.butler.edu/wordways/vol33/iss2/2
TwoSquareQuickies
1923187456,thesquareof30384.
Sincewewantthelargestnumberofitstype,it’safairbetthattheanswerstartswith9,sothisreallyhastobetriedfirst,evenifitturnsouttobefalse.Soitmustliebetween912345678and987654321,bearinginmindthatalldigitsaredifferentandthereisno0.Thesquarerootsoftheseare30205.06and31426.96.Soallwehavetodoissquarethenumbersbetween30206and31426andseewhichgiveallninenonzerodigits.Thereare1221suchnumbers.Workingbackwardsfrom31426weeventuallygetto30384.Nowthatwe’vefoundasolutionstartingwith9,wedon’tneedtoworryaboutstartingwith8orlower.2139854276,thesquareof11826.
Thewaytofindthisissimilar.
TheAdventureoftheCardboardBoxes
1Theboxeshavedimensions6×6×1and9×2×2.
Supposethedimensionsoftheboxesarex,y,zandX,Y,Z.TheirvolumesarexyzandXYZ.Thelengthofribbonis4(x+y+z)and4(X+Y+Z).Dividingoutthefactorof4,wemustsolve
xyz=XYZx+y+z=X+Y=Z
innonzerowholenumbers.Thatis,findtwotriplesofnumberswiththesameproductandthesamesum.Thesmallestsolutionis(x,y,z)=(6,6,1)and(X,Y,Z)=(9,2,2).Theproductis36andthesumis13.2Thesmallestsolutionforthreeboxesis(20,15,4),(24,10,5),and(25,8,6).Nowtheproductis1,200andthesumis39.
Inpassing,wecananswerathirdquestion,whichdidnotfeatureinSoames’sinvestigation:3Supposethattheribbonsaretiedinthemanneroftheleft-handpicture,withxbeingwidth,ydepth,andzheight.Thentheequationsbecome
xyz=XYZx+y+2z=X+Y+2Z
Ifwereplacex,y,zbyx,y,2z,andsimilarlyforX,Y,andZ,weareagainseekingtwo triples of numbers with the same product (now 2xyz = 2XYZ) and sum.However,zandZmustbeeven.Thisisthecaseinsolution(1)ifwearrangethesidesintherightorder,anditleadstothesmallestsolution(6,3,1)and(9,2,1).MyattentionwasdrawntothisproblembyMoloyDefromKolkata,India,
whohasalsofoundthesmallestsetsoffour,five,andsixwholenumberswiththesamesumandproduct:
Fourpackages
(54,50,14)(63,40,15)(70,30,18)(72,25,21)
sum=118,product=37,800.
Fivepackages
(90,84,11)(110,63,12)(126,44,15)(132,35,18)(135,28,22)sum=185,product=83,160.
Sixpackages
(196,180,24)(245,128,27)(252,120,28)(270,98,32)(280,84,36)(288,70,42)sum=400,product=846,720.
TheRATSSequence
Thenexttermis1345.Theruleis:‘Reverse,Add,ThenSort’.RATS.By‘sort’Imeanrearrangeinto
ascendingorder.Anyzerosareomitted.Forexample:
16+61=77alreadyinnumericalorder77+77=154,reorderas145145+541=686,rearrangeas668668+866=1534,rearrangeas1345
JohnHortonConwayhasconjecturedthatwhichevernumberyoustartwith,eventuallythesequenceeithergoesroundandroundsomerepeatingcycleorgetsintotheever-increasingsequence
wherethenindicatesnotapowerbutnidenticaldigitsrepeated.
MathematicalDates
Thenexttriplepalindromedaywillbeon21:1221/122112.Thenextpalindromewas20:0230/032002(Britishsystem).
TheHoundoftheBasketballs
“Indeed,madam,DrWatsupisquiteright,”Soamesconfirmed.“Therealisationthatonlyfourballsweremovedmakestherequiredarrangementobvious.”“Whatisit?”“That,madam,isinformationwhich,accordingtoyourownstatements,
shouldonlybedisclosedtotheseniorextantmemberofthemaleline.”“Namely,LordEdmundBasquet,”Ipointedout.“Whoisinacoma.Which
posesaconsiderablediff—”“Rubbish!”saidLadyHyacinth.“Youcantellme.”Itwasevidentfromher
facethatnothingwoulddeterherfromthiscourseofaction.“Verywell,”saidSoames,sketchingrapidly.“Thepood—er,giantslavering
hound—musthavemovedthefourbasketballsshowninwhitetothepositionsshowninblack.Oreitherofthetworotationsofthissolution.Butyousaidthattheorientationoftheconfigurationdoesnotmatter.”NowIunderstoodthepointofhisobscurequeryearlier.
Theoriginalconfiguration
“Wonderful!”saidLadyHyacinth.“IwillinstructWillikinstomakeitso.”“Butwillthatnotinfringetheconditionsoftheceremony?”Ienquired.“Ofcourse,DrWatsup.Butthereisnorationalreasontofearanyadverse
consequences.Thatancienttabooislittlemorethanaloadofold,er,superstition.”Amonthlater,SoameshandedmetheManchesterGarble.
“GreatScott!”Icried.“LordBasquethasdiedandBasquetHallhasburnedtotheground!Thefamily’sinsurancecompanyhasdeclinedpaymentbecauseMalevolentForcesofPureEvilareexcluded,andthefamilyisnowruined!LadyHyacinthhasbeenconfinedtoanasylumfortheincurablyinsane!”Soamesnodded.“Purecoincidence,I’msure,”hesaid.“Perhapswith
hindsightIshouldhavetoldHerLadyshipaboutthepoodle.”
DigitalCubes
370,371,and407.Despitethisproblemallegedlyhavingnomathematicalsignificance,youhave
tobequitegoodatmathstofindthefoursolutions,andevenbettertoprovetherearen’tanyothers.I’llsketchoneapproach.Sincenumberswithleadingzerosareexcluded,thereareonly900
combinationstotry.Butwecancutthisdown.Thecubesofthetendigitsare0,1,8,27,64,125,216,343,512,and729.Thesumofthethreedigitsis999orless,sowecanruleoutanynumbercontainingtwo9’s,two8’s,an8anda9,andsoon.Supposeonedigitis0.Thenthenumberisasumoftwocubesfromthelist.
Ofthe55suchpairsonly343+27=370and64+343=407havetherequiredproperty.Wemaynowassumenodigitis0.Supposeoneofthemis1.Asimilar
calculationleadsto125+27+1=153and343+27+1=371.Wemaynowassumenodigitis0or1.Nowwehaveasmallerlistofcubesto
workwith.Andsoon.Shortcuts,suchasconsideringoddandevennumbers,shortenthe
calculationsfurther.It’sabitlong,butasystematicapproach(asSoamesalwaysrecommends)getstherewithoutanyseriousproblemsalongtheway.
NarcissisticNumbers
Hereweallowleadingzeros.
Clueless!
Watsup’ssolutiontocluelesspseudoku
“Soames!”Icried.“I’vesolvedit!”“Yes,themurderesswasGra¨finLiselottevonFinkelstein,ridingher
thoroughbredPrinzIgorandtowingthreecarthorsesbehindtoobscurethetracksinthe—”“No,no,Soames,notyourcase!Thepuzzle!”Hegavemyscrawledsolutionacursoryglance.“Correct.Aluckyguess,no
doubt.”“No,Soames,Ireasoneditoutusingthelogicalprinciplesthatyouhave
impresseduponmyconsciousness.First,Irealisedthatthenumbersineachregionmustsumto20.”“Becausethetotalofthenumbersinallsquaresis(1+2+3+4)×4=40,to
bedividedequallybetweenthetworegions,”Soamessaiddismissively.“Exactly.Now,onceithadoccurredtometoconcentrateonthelargerregion,
thesolutionbegantofallintoplace.Thatregionhasfourcellsinarowalongthebottom,whichmustcontain1,2,3,4insomeorder,andthoseaddto10,whatevertheordermaybe.Sotheremainingthreerowsmustalsoaddto10.Theonlywaythiscanhappenisifthetoprowcontains1,2,3insomeorder;thesecondrowcontains1,2insomeorder,andthethirdrowcontains1.”“Why?”“Anyotherchoicewouldmakethetotaltoobig.”“Youareindeedlearning,Watsup.Verygood:continue.”Ismiledatthisfaintpraise,sincegettinganypraisefromSoameswaslike
makingmarmaladefromtheIsleofWight.“Well—it’snoweasytoverifythatthereisonlyonepossiblewaytocompletethearrangement.Thenumbersinthe
secondpieceareforced:forexample,thetoprowmustendin4,andthentheother4’smustgodownthediagonal;thenthetwo3’sareforced,andfinallythe2goesintheremainingposition.”ThispuzzlewasinventedbyGerardButters,FrederickHenle,JamesHenle,
andColleenMcGaughey,Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.Seealsothewebsite:
http://www.math.smith.edu/~jhenle/clueless/
ABriefHistoryofSudoku
ThetwobasicallydifferentsolutionstoOzanam’spuzzleare:
Remember:eachofthesegivesriseto576solutionsbypermutingthevaluesandsuits,sodon’tbesurprisedifyoursolutionslookdifferent.Ifyoustartwiththetoprow (orrearrangeyoursolutionintothisform)youneedonlythinkaboutpermutingtheotherthreerows.
Once,Twice,Thrice
TheCaseoftheFace-DownAces
“Itisalltrickery,Watsup.Withtherightpreparation,thetrickworksautomatically,nomatterwhichsequenceoffoldstheaudiencechooses.”“Dashedclever,what?”saidI.Soamesgrunted.“WhenWhodunnipreparedthepack,heplacedthefouraces
inpositions1,6,11,and16fromthetopdown.So,whenthepackwasdealtintoasquare,theaceslayalongthediagonalfromtoplefttobottomright.Buttheywerefacedown,soofcourseyouwouldnothavebeenawareofthedeception.“Imagineturningthecardsthatliealongthediagonalfaceup.Thenthesquare
wouldhaveapatternlikeachessboard,withtheacesalongthediagonal:
Whodunni’sinitialarrangementwiththediagonalcardsturnedover
“Now,thisarrangementhasawonderfulmathematicalproperty.Howeveryoufoldthesquare,atanystagethecardsthatendupinagivenpositionwillallfacethesameway:eitheralluporalldown.”“Really?”“Letustry.Forinstance,wemightbeginbyfoldingalongthecentralvertical
line.Thinkofthetoprowofcards.Thethirdcard(up)turnsover(down)andisplacedontopofthesecondcard—alsodown.Andthefourthcard(down)turnsover(up),andisplacedontopofthefirstcard—alsoup.”Ibegandimlytoseehowitworked.“Andthesamegoesfortheotherrows?”
“Indeed.Thisfirstfoldcreatesarectangle,madefromcardsorsmallpilesofcards.Thecardsineachpileallfacethesameway(upordown),andthesetofpileshasthesamechessboardpatternofupsanddownsastheoriginalsetofcards.Sothesamethinghappensforthenextfold,andthenext.Bythetimewereachasinglepile,allofthecardsinthepilefacethesameway.”“Yes,but—whenwestarted,thecardsonthediagonalwerethewrongwayup
comparedtothechessboardpattern,”saidI.Itwasintendedasanobjection,buthebeamedatmyinsight.“Exactly!So,
afterfolding,theywillagainbethewrongwayup.Soinsteadofapileofsixteencardsallfacingthesameway,youwillhaveapilewithtwelvecardsfacingoneway,andthefouracesfacingtheother.”Itwasdevilishlycunning.Thechessboardpatternhaswhatmathematicianscall‘coloursymmetry’.The
foldlinesactlikemirrors,andthemirrorimageofeachcardsitsontopofacardthatfacestheotherway.Thisideaisusedtostudyhowtheatomsincrystalsarearranged.Thecunningbitistoturnthemathematicsintoaneffectivecardtrick.Whodunnididn’tdothat.Followinghisusualmodusoperandi,hestolethetrickfromitsinventorArthurBenjamin,amathematicianandmagicianatHarveyMuddCollegeinCalifornia.
JigsawParadox
Neithershapeisatriangle.Thefirstbulgesslightlyupwardsalongtheslopingedge,thesecondbulgesslightlydownwards.That’swherethemissingsquarehasgone.
TheCatflapofFear
Soamesnoddedinsatisfaction.“Ihaveit,Watsup!Cirrhosisgoesout,Dysplasiagoesout,Aneurysmgoesout,Cirrhosiscomesbackin,Borborygmusgoesout,Cirrhosisgoesout.”Webeganthedelicateprocessofenticingcatsoutthroughtheflapand
stuffingthembackin.“Takecare,Soames!”Iwhispered.“Onemistakeandthisentireareawillbeasmokingcrater.IdonotwishtopresentmyselformycatsatthePearlyGatesquiteyet.Iamwearingunpressedtrousers,andthecatsneedbrushing.”“Donotconcernyourself,Watsup,”saidhe,grabbingCirrhosisbeforethe
wretchedanimalcouldhoofitoverthefence.“Youcanhavetotalconfidenceinmysolution.”“Idonotdoubtit,Soames,”Ireplied,castinghastilyaboutforsomething
solidtohidebehind.“Er—howdidyoumakeyourdeductions?”Heborrowedmynotebookandapencil.“Thereare16possibilitiesforwhichcatsareinthehouse:ABCD,ABC,
ABD,andsoonuntilnonearepresent(callthis*).Useanarrow todenoteapossiblemove:onecatthroughtheflap.“ThefirstconditionrulesoutACandABC.ThesecondrulesoutBDand
BCD.ThethirdrulesoutAD.ThefourthrulesoutCD.ThefifthrulesoutthechangeA *.ThesixthrulesoutB *.“Now,ABCD ACDorABD.However,ACD AC,AD,orCD,allof
whichareruledout.ThereforeABC ABD.SinceABD ADandABD BDareruledout,wemusthaveABD AB.ButAB AispointlesssinceAcan’tgooutifnoothersarepresent.SoAB B.However,Bcannotthengoout,sosomeothercatmustcomebackin.ButB ABinvolvesAcomingstraightbackin,andB BDisruledout,soB BC.NowBC C *.“Thereisalsoavisualwaytoseethis,whichinsomewaysissimpler,”he
added,andsketchedadiagram.“Thispictureshowsall16possiblecombinationsofcats,withthethinlinesrepresentingpossiblechangesasacatgoesinorout.Theblackdotsareruledout,thetwocrossesruleouttwoofthelines.TheboldlineistheonlypaththatrunsfromABCDto*usingonlypermitteddotsandlines,butneverbacktracks.”
Catflapconditions
Shortlyafterwards,Iwasreunitedwithmyfurryfriends.“Soames,howcanIeverthankyou?”Icried,claspingtheanimalsjoyouslytomychest.Helookeddownathisjacket.“Bybrushingyourcatsmoreoften,Watsup.”
PancakeNumbers
1No.2Somestacksoffourpancakesneedfourflips;forexample,theonebelow.Seethefigurebelowthatfortheothertwo.Nostackneedsmorethanfourflips.
Astackthatneedsfourflips
Here’sasystematicmethodforprovingthosestatements.Thediagramshowstherequiredfinalarrangement1234atthetop,wherethesizesarelistedinorderfromthetopdown.Weworkbackwardsfromthis.Thesecondlineshowsallarrangementsthatcanbeobtainedfrom1234byoneflip.Thesearealsothearrangementsthatcanreach1234inoneflip.(Thesameflipdonetwiceputseverythingbackwhereitwas.)Thethirdlineshowsallarrangementsthatcanbe
obtainedfromthefirstlinebyoneflip.Thesearealsothearrangementsthatcanreach1234intwoflips.Noticethatpreciselyoneentryinthethirdrowcanbereachedfromtwointhesecond,namely1324.Sothestructureofthediagramlooksslightlyasymmetricthere.Rows1,2,3contain21ofthe24possiblestackorders.Themissingonesare
2413,3142,and4231.Row4showshowthesecanbeobtainedfromrow3byonemoreflip—or,reversingtheseriesofflips,howtoconvertthemto1234infourflips.(Theotherlinkstorow4areomittedsincetheymakethediagrammorecomplicatedandwedon’tneedthem.)Thefigureaboveinanswer2isthearrangement2413convertedintostacks.
Stacksthatneed1,2,3,or4flipstoputtheminorder
3Eitherthebiggestpancakeisatthetop,orit’snot.Ifnot,insertthespatulaunderitandflip.Nowit’satthetop.Insertthespatulaatthebottomofthestackandfliptheentirething.Nowthebiggestisatthebottom.Soatmosttwoflipswillputthebiggestpancakeatthebottom.Leaveitthereandrepeatforthenextbiggestpancake:atmosttwoflipswillplaceitdirectlyontopofthebiggest.Repeatforthenextbiggest,andsoon.Ittakesatmosttwoflipstogeteachsuccessivepancakeinitscorrectposition,soatmost2nflipswillachievethisfortheentirestackofnpancakes.4P1=0,P2=1,P3=3,P4=4,P5=5.Thepancake-sortingproblemgoesbacktoJacobGoodmanin1975,who
publisheditunderthepseudonymHarryDweighter.(Sayitaloudandpronouncethesurnameas‘Dwayter’togetthejoke.)Thesolutionisknownforallnupto19,butnotfor20.Theresultsare:
Thepancakenumberstendtoruninsequence,goingupby1asnincreases.ForexamplePnis3,4,5,6whenn=3,4,5,6.Butthispatterngoeswrongwhenn=7becauseP7=8,not7.Afterthat,thereisajumpof2atn=11,andagainatn=19.Theupperestimateof2nflips,myanswertoquestion3,canbeimproved.In
1975WilliamGates(yes,theBillGates)andChristosPapadimitrioureplaceditby(5n+5)/3.GatesandPapadimitrioualsodiscussedtheburntpancakeproblem.Hereeach
pancakeisburntononeside,whichcouldbethetoporthebottom,andyouhavetogetalltheburntsidesonthebottomaswellasstackingthepancakesintherightorder.In1995DavidCohenprovedthattheburntpancakeproblemneedsatleast3n/2flips,andcanbesolvedwithatmost2n–2flips.Ifyou’rethinkingoftacklingn=20,bearinmindthatthereare
2,432,902,008,176,640,000
differentstackstostartfrom.
TheCaseoftheCrypticCartwheel
“Thediameterofthewheelis58inches,ofcourse,”saidSoames.“ItisanelementaryapplicationofPythagoras’stheorem.”Ithoughtaboutthis.Ihavesomesmallfacilityingeometryandalgebra.“Let
metry,Soames.Itaketheradiusofthewheeltober.Theshadedtriangleinyourdiagramisright-angled,withhypotenuserandtheothersidesbeingr–8andr–9.So,asyouhinted,IcanapplyPythagoras,getting
(r–8)2+(r–9)2=r2
Thatis,
r2+34r=145=0
Thinkofatriangle…
Istaredatthesymbols,temporarilyhalted.“Thequadraticfactorises,Watsup:
(r–29)(r–5)=0
“Soitdoes!Whichmeansthatthesolutionsarer=29andr=5.”
“Yes.Butyoumustrememberthatthediameteris2r,whichis58or10.However,thesolution10inchesisruledoutbecausethediameterismorethan20inches.Allthatremains—”“Is58inches,”Ifinishedforhim.
TheV-shapedGooseMystery
FlorianMuijresandMichaelDickinson,Birdflight:Flywithalittleflapfromyourfriends,Nature505(16January2014)295–296.StevenJ.Portugalandothers,Upwashexploitationanddownwashavoidance
byflapphasinginibisformationflight,Nature505(16January2014)399–402.
AmazingSquares
Themainideacanbeexpressedinfullgeneralityusingalgebra,butI’llforegoformalitiesandillustrateitbyexample.Lookattheprocessinreverse,startingwith
92+52+42=82+32+72
andexpandingitto
892+452+642=682+432+872
It’seasytocheckthefirstequation,whichishoweverythinggetsstarted,butwhydoesthesecondequationhold?Theactualvalueofatwo-digitnumber[ab]is10a+b.Sowecanwritethe
left-handsideas
(10×8+9)2+(10×4+5)2+(10×6+4)2
whichis
100(82+42+62)+20(8×9+4×5+6×4)+92+52+42
Similarly,theright-handsidebecomes
100(62+42+82)+20(6×8+4×3+8×7)+82+32+72
Comparingthese,thefirsttermsareequalbecause62+42+82isjust82+42+62inadifferentorder,andthethirdtermsareequalbecausewestartedfromthose.Sowejustneedtoseewhetherthemiddletermsareequal;thatis,whether
8×9+4×5+6×4=6×8+4×3+8×7
Infact,bothare116.Everythinguptothispointwouldhaveworkedifwe’dusedanythreesingle-
digitnumbersinplaceof8,4,6.Sowejusthavetochoosethesenumberstomakethefinalexpressionsequal.Therestoftheexplanationissimilar.
TheThirty-SevenMystery
WithsomeproddingfromSoamesalongtheway,Ieventuallyrealisedthatthekeytothemysteryistheequation111=3×37.Thethree-digitnumbersthatproducelonglistsofrepeateddigitswhensubjectedtomyprocedureturnouttobethosethataremultiplesof3.Thisisthecasefor123,234,345,456,and126,forexample.Forsuchnumberstheprocedureisequivalenttomultiplyingmanyrepetitionsofasmallernumber,onethirdthesize,by3×37,whichis111.Asanexample,considerSoames’s486.Thisis3×162.Therefore,
multiplying486486486486486486by37isthesameasmultiplying162162162162162162by111.Since111=100+10+1wecandothisbyaddingtogetherthenumbers
162162162162162162001621621621621621620162162162162162162
Fromrighttoleft,weobtain0+0+2=2,then0+2+6=8.Afterthat,weget2+6+1,6+1+2,1+2+6,overandoveragain,untilwegetneattheleft-handend.Butthesearethesamethreenumbersaddedinvariousorders,sotheresultisthesameineachcase—namely9.WhenSoamesfirstexplainedthistome,Iraisedanobjection.“Yes,butwhat
ifthethreenumbersaddtomorethan9?Thenthereisacarrydigit!”Hisreplywasbriefandtothepoint.“Yes,Watsup:thesamecarrydigitevery
time.”Ieventuallyrealisedthatthismeansthatagain,onedigitwillrepeatmanytimes.“Thereare,ofcourse,moreformalproofs,”Soamesremarked,“butIthink
thisonemakestheideaclear.”Afterwhichhereturnedtohischairwithapileofnewspapersandsaidnomorethatevening,whileIwentdownstairstobegaplateofGorgonzolasandwichesfromMrsSoapsuds.[ThisitemwasinspiredbysomeobservationsmadebyStephenGledhill.]
AverageSpeed
We’reusingthewrongmean.Weshouldbeusingtheharmonicmean(explainedbelow),notthearithmeticmean.Wenormallydefinethe‘averagespeed’forsomejourneytobethetotal
distancetravelleddividedbythetotaltimetaken.Ifthejourneyisbrokenupintopieces,thentheaveragespeedforthewholetripisnot,ingeneral,thearithmeticmeanoftheaveragespeedsforthepieces.Ifthepiecesaretravelledinequaltimes,thearithmeticmeanworks,butitdoesn’tiftheycoverequaldistances,whichisthecasehere.Equaltimesfirst.Supposeacardrivesatspeedafortimet,andthenatspeed
bforthesametimet.Thetotaldistanceisat+bt,travelledintime2t.Sotheaveragespeedis(at+bt)/2t,whichequals(a+b)/2,thearithmeticmean.Next,equaldistances.Nowthecardrivesdistancedatspeeda,takingtimer.
Thenitdrivesdistancedagain,atspeedb,takingtimes.Thetotaldistanceis2d,andthetotaltimeisr+s.Toexpressthisintermsofthespeedsaandb,observethatd+ar=bs.Sor=d/aands=d/b.Theaveragespeedistherefore:
Thissimplifiesto2ab/(a+b),whichistheharmonicmeanofaandb.Itisthereciprocalofthearithmeticmeanofthereciprocalsofaandb,wherethereciprocalofxis1/x.Thisoccursbecausethetimetakenisproportionaltothereciprocalofthespeed.
FourCluelessPseudoku
Cluelesspseudokuanswers
ThesepuzzlesalsocomefromGerardButters,FrederickHenle,JamesHenle,andColleenMcGaughey.Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.
ThePuzzleofthePurloinedPapers
“Charlesworthwasthethief,”saidSoames.“Areyoucertain,Hemlock?Muchhangsuponyourbeingright.”“Therecanbenodoubt,Spycraft.Theirstatementsare:Arbuthnot:Burlingtondidit.Burlington:Arbuthnotislying.Charlesworth:ItwasnotI.Dashingham:Arbuthnotdidit.
Weknowthatoneofthemenspeakstrulyandtheotherthreelie.Therearefourpossibilities.Letustrytheminturn.“IfonlyArbuthnotistellingthetruththenhisstatementinformsusthat
Burlingtonistheguiltyparty.However,Charlesworthislying,soitwasCharlesworth.Thisisalogicalcontradiction,soArbuthnotisnottellingthetruth.“IfonlyBurlingtonistellingthetruththen—”“Charlesworthislying!”Icried.“SoitwasCharlesworth!”
Soamesglaredatmeforstealinghisthunder.“Thatisso,Watsup,andtheotherstatementsareconsistentwithit.SowealreadyknowthatCharlesworthisthethief.However,itisworthcheckingtheothertwopossibilitiestoavoideventheremotepossibilityoferror.”“Absolutely,oldchap,”saidI.Hetookouthispipebutdidnotlightit.“IfonlyCharlesworthistellingthe
truththenBurlington’sstatementisfalse,soArbuthnotistellingthetruth,againacontradictionsinceheislying.“IfonlyDashinghamistellingthetruththenthesamecontradictionarises.“SotheonlypossibilityisthatBurlingtonisthesolepersontellingthetruth,
andthatconfirmsthatthethiefisCharlesworth.AsWatsupsoastutelydeduced.”“Thankyou,gentlemen,”saidSpycraft.“IknewIcouldrelyonyou.”He
gesturedandashadowyfigureenteredtheroom.Theyheldawhisperedconversation,andheleftagain.“TheCaptain’sresidencewillbesearchedforthwith,”saidSpycraft.“Iamconfidentthedocumentwillbefoundthere.”“ThenwehavesavedtheEmpire!”Ireplied.“Untilthenexttimesomeoneleavessecretdocumentsontheseatofacab,”
saidSoamesdrily.OnourwayoutIwhisperedtomycompanion:“Soames,ifSpycraftisan
expertinprimenumbers,whatonearthishedoingworkingincounter-espionage?Therecanbenopossibleconnection,canthere?”Hestaredatmeforamoment,andshookhishead.Whethertoconfirmthe
absenceofanyconnection,ortowarnmenottopursuethematter,Idonotknow.
AnotherNumberCuriosity
123456×8+6=9876541234567×8+7=987654312345678×8+8=98765432123456789×8+9=987654321
It’snottotallyclearwhat‘ought’tocomenext:perhaps
1234567890×8+10
whichis9876543130,ormaybeIshouldhavereplacedthat0by10anddone
somecarryinguntilitallsettlesdown,giving
1234567900×8+10
whichis98765432730.Ithinkwecanagreethat,eitherway,thepatternstopshere.
ProgressonPrimeGaps
TheElliott-HalberstamConjecture[PeterElliottandHeiniHalberstam,Aconjectureinprimenumbertheory,SymposiaMathematica4(1968)59-72]isverytechnical.Writeπ(x)forthenumberofprimeslessthanorequaltox.Foranypositiveintegerqandahavingnofactor(otherthan1)incommonwithq,letπ(x;q,a)bethenumberofprimeslessthanorequaltoxthatarecongruenttoa(modq).Thisisapproximatelyequaltoπ(x)/ (q)where isEuler’stotientfunction,thenumberofintegersbetween1andq–1thathavenofactorincommonwithq.Considerthelargesterror
TheElliott-HalberstamConjecturetellsushowbigthiserroris:itstatesthatforallθ<1andA>0thereexistsaconstantC>0suchthat
forallx>2.Itisknowntobetrueforθ< .
TheSignofOne:PartTwo
Here’sonesolution:
SeeTheSignofOne:PartThree,page115,foranexplanation.
Euclid’sDoodle
Youcoulddoitbyhandusingprimefactors,givenadayortwo.You’dhavetoworkoutthat
44,758,272,401=17×17,683×148,89113,164,197,765=5×17,683×148,891
Thenyou’dconcludethatthehcfis17,683×148,891,whichequals2,632,839,553.UsingEuclid’salgorithm,thewholecalculationgoeslikethis:
(13,164,197,765;44,758,272,401) (13,164,197,765;31,594,074,636)(13,164,197,765;18,429,876,871) (5,265,679,106,
13,164,197,765)(5,265,679,106;7,898,518,659) (2,632,839,553;5,265,679,106)(2,632,839,553;2,632,839,553) (0;2,632,839,553)
Sothehcfis2,632,839,553.
123456789TimesX
12345679×1=12345678912345679×2=246913578
12345679×3=37037036712345679×4=49382715612345679×5=61728394512345679×6=74074073412345679×7=86419752312345679×8=98765431212345679×9=1111111101
Thesemultipleshaveallninenonzerodigitsinsomeorder,exceptwhenwemultiplybysomethingdivisibleby3(thatis,3,6,and9).
TheSignofOne:PartThree
Since
wecanusetherepresentationof7withtwo1’sfrompage279toget62withfour1’s.ForalongtimeSoamesandWatsupdespairedofgetting138withfour1’s,
butbyusingWatsup’sinsightaboutsquarerootsandfactorials,andbeingsystematic,theyeventuallydiscoveredthatitispossibletoget138usingonlythree1’s.Againthestartingpointis7writtenwithtwo1’s,andthen
andfinally
whichisacleverwaytomultiplyby3usingjustoneextra1.
TossingaFairCoinIsn’tFair
PersiDiaconis,SusanHolmes,andRichardMontgomery,Dynamicalbiasinthecointoss,SIAMReview49(2007)211–223.Foranon-technicalsummaryseePersiDiaconis,SusanHolmes,andRichard
Montgomery,Thefifty-onepercentsolution,What’sHappeningintheMathematicalSciences7(2009)33–45.Similareffectsoccurfordice—infact,notjustfortheusualcubebutforany
regularpolyhedron.SeeJ.Strzalko,J.Grabski,A.Stefanski,andT.Kapitaniak,Canthedicebefairbydynamics?InternationalJournalofBifurcationand
Chaos20No.4(April2010)1175–1184.
EliminatingtheImpossible
“Youromission,”saidSoames,“wastofailtoobservethatthewinemaymove,aswellastheglasses.Imerelypickupthesecondandfourthglasses,andpourtheircontentsintotheseventhandninth.”
MusselPower
MoniquedeJager,FranzJ.Weissing,PeterM.J.Herman,BartA.Nolet,andJohanvandeKoppel.Lévywalksevolvethroughinteractionbetweenmovementandenvironmentalcomplexity,Science332(4June2011)1551–1553.
ProofThattheWorldisRound
Onpage74wesawthatwhencalculatingaveragespeedsoverafixeddistanceweshouldusetheharmonicmean,notthearithmeticmean.Theharmonicmeanalsoturnsupintheestimationofthedistancebetweentwoairportswhenthewindspeedistakenintoaccount,forasimilarbutslightlydifferentreason.Togetasimplemodel,assumethattheaircraft’sspeedrelativetotheairisc,itspathisastraightlineandthewindblowsalongthatlineinafixeddirectionwithspeedw.Assumebothcandwareconstant.Thena=c–wandb=c+w,andwewanttoestimatedfromthetimesrands.Togetridofw,wefirstsolveforaandb,gettinga=d/randb=d/s.Therefore
c–w=d/rc+w=d/s
Adding,2c=d(1/r+1/s).Soc=d(1/r+1/s)/2.Iftherehadbeennowind,asingletripwouldhavetakentimet,whered=ct.Therefore
t=d/c=d/[d(1/r+1/s)/2]=1/[(1/r+1/s)/2]
whichistheharmonicmeanofrands.Inshort:ifweareworkinginunitsofaircraft-hours,thenthissimplemodelof
theeffectofwindimpliesthatweshouldusetheharmonicmeanofthetraveltimesinthetwodirections.
123456789TimesXContinued
12345679×10=123456789012345679×11=135802467912345679×12=148148146812345679×13=160493825712345679×14=172839504612345679×15=185185183512345679×16=197530862412345679×17=209876541312345679×18=222222220212345679×19=2345678991
Thesemultipleshavealltendigits0–9insomeorder,exceptwhenwemultiplybysomethingdivisibleby3…Untilwegetto19,whenthepatternstops(19isnotamultipleof3:answerhastwo9’sandno0).Butitpicksupagain:
12345679×20=246913578012345679×21=2592592569
(21isamultipleof3,sorepetitionsOK)12345679×22=271604935812345679×23=2839506147
Thenextexceptionsoccurat28and29.Itworksfor30–36,andthenfailsfor37.Istoppedcalculatingatthatpoint.What’sgoingonhere?Ihavenoidea.
TheRiddleoftheGoldenRhombus
Soamesfinishedtighteningtheknot,flattenedit,andheldituptothelight.“Why,itisapentagon!”Icried.“Moreprecisely,Watsup,itappearstobearegularpentagonwithone
diagonalvisibleandthreemorehidden.Observetheabsenceofadiagonallinerunninghorizontally.Werethattobeadded,forinstancebyfoldingthestriponemoretime,wewouldobserve—”
Theflattenedknot(dottedlinesshowhiddenedges)
“Afive-pointedstar!Apentacle!Usedinblackmagictoconjuredemons!”Soamesnodded.“Butwithoutthatfinalfold,andthereforehavingoneedge
missing,thepentacleisincompleteandthedemonwillescape.Sothesymbolrepresentsathreattounleashdemonicforcesupontheworld.”Hegaveahumourlesssmile.“Ofcourse,therearenodemonsinasupernaturalsense,andtheycanneitherbeconjurednorunleashed.Buttherearehumansofademonicdisposition—”“SuchastheAl-Jebraterroristorganisation!”Icried.“Theyhavepursuedme
fromAl-Jebraistanwithweaponsofmathsinstruction!”“Calmyourself,Watsup.No,theorganisationIhaveinmindisthe
MathemagicalAssociationofNumerica.Itisanobscuregroup,whichIstronglysuspecttobeafrontforoneofMogiarty’sdevilishschemes.Ihaveencountereditbefore,andInowhavethefinallinkinthechaintostrikeablowagainstthefoulProfessoranddestroythispartofhisworldwidewebofcriminalityforever.Provided…““Providedwhat,Soames?”
“ProvidedthatwecanofferincontrovertibleproofwhenthecasecomestoCourt.Howdoweknowthatthepentagonisregular?”“Isthatnotabsurdlysimple?”“Onthecontrary,youwillshortlybeassuringmethatitisincrediblysubtle
andpossiblyfalse—thoughinpointoffactthetrueansweriswhatonewouldnaivelyguess.Idaretosurmisethatonceweestablishthatfact,allelsewillfollow,buttheknot’sappearancetotheeyeisnotenough.Ishall,however,assumethatthearrangementoflinesinthefigureiscorrect,sowehaveapentagonwithfourofitsdiagonals.Isittrulyregular?Thatremainstobeseen.Iftrue,itmustbeaconsequenceoftheconstantwidthofthestrip.“Letus,then,labelthecornersinthemannerofthegreatEuclidof
Alexandria,andpursueourgeometricaldeductions.”
Theflattenedknot,labelled.LineCDisomittedbecausewedonotyetknowwhetheritisparalleltoBE.
ImustwarnmyreadernowthattheremainderofthediscussionwillappealonlytothosewithsomefacilityinEuclideangeometry.“Ibegin,”saidSoames,“withafewsimpleobservations.Theycanbeproved
withoutgreatdifficultyusingbasicgeometry,soIomitthedetails.“First,noticethatiftwostripswithparalleledges,havingthesamewidth,
overlap,thentheirintersectionisarhombus—aparallelogramwithallfoursidesequal.Moreover,iftwosuchrhombuseshavethesamewidthandthesameside,theyarecongruent—theyhaveexactlythesamesizeandshape.Thediagramoftheflattenedknotthereforeincludesthreemutuallycongruentrhombuses.”“Whyonlythree?”Iasked,puzzled.“BecauseCDandBEdonotcoincidewithedgesofthestrip,sowecannotyet
saythesameofCDRBorDESC.ThisiswhyIhavenotdrawnlineCD.”Ihadnotnoticedthat.“Itisincrediblysubtle,then,Soames.Infact,itmight
evenbefalse!”
Threecongruentrhombusesintheflattenedknot
Hesighed,Iknownotwhy.“Nowwecometothecentralpointinmydeductions.Thediagonalsofarhombusbisectitsangles,andoppositeanglesareequal.”SoamesmarkedfouroftheangleswiththeGreekletterθ(theta),seetheleft-handfigure.
Left:Fourequalangles.Right:Fivemoreangles,allequaltothefirstfour(ingrey).
“Forsimilarreasons,angleCABisalsoequaltoθ.SincerhombusesDEATandPEABarecongruent,Icanmarkfourmoreangleswithθ.Thisleadstotheright-handfigure.“Now,Watsup:whatinstantlyspringstomind?”“Thereareanawfullotofθ’s,”Irepliedatonce.HegrimacedandIheardalowgrowlinhisthroat,Iknownotwhy.“Itisas
plainastheneckonaverytallgiraffe,Watsup!ConsidertriangleEAB.”Iconsideredit,initiallywithoutenlightenment.Well…thetrianglealsohada
lotofθs.Infact…allofitsangleswerecomposedofθ’s!NowIsawit.“Theanglesofatriangleaddto180°,Soames.Inthistriangle,theanglesareθ,θ,and3θ.Sotheirsum5θisequalto180°,whenceθ=36°.”“Wewillmakeageometerofyouyet,”saidhe.“Nowtherestoftheproofis
easy.ThelinesDE,EA,AB,andBCareofequallength,beingsidesofcongruentrhombuses.Theangles∠DEA,∠EAB,and∠ABCareallequal,sincetheyoccurincongruentrhombuses,andoneofthemis∠EAB,equalto3θ,whichis108°.Thereforeallthreeanglesare108°.Butthisistheinteriorangleofaregularpentagon.”“ThereforeDEABCarethecornersofaregularpentagon,andIcancomplete
thefigurebydrawingsideCD!”Icried.“Howabsur—”Icaughthiseye.“Er,elegant,Soames!”
Heshrugged.“Atrifle,Watsup.EnoughtowrecktheMathemagicalAssociationofNumericaandcauseMogiartysomemomentaryannoyance.Themanhimself…Ifearhewillproveafarhardernuttocrack.”
WhydoGuinnessBubblesgoDownwards?
E.S.Benilov,C.P.Cummins,andW.T.Lee.WhydobubblesinGuinnesssink?arXiv:1205.5233[physics.flu-dyn].
TheDogsThatFightinthePark
“Thedogstook10secondstocollide,”Soamesdeclared.“Ishalltakeyourwordforit,”saidI.“But,merelytosatisfymycuriosity,
howdidyouarriveatthatfigure?”“Theproblemissymmetric,Watsup,andsymmetryoftensimplifies
reasoning.Thethreedogsarealwaysatthecornersofanequilateraltriangle.Thisrotatesandshrinks,butkeepsitsshape.Therefore,fromtheviewpointofoneofthedogs—sayA—itisalwaysrunningstraighttowardsthenextdogB.”“Doesnotthetrianglerotate,Soames?”“Indeeditdoes,butthatisirrelevant,forwemayperformthecalculationina
rotatingframeofreference.Whatmattersishowfastthetriangleshrinks.DogBisalwaysrunningat60°tothelineAB,becausethedogsalwaysformanequilateraltriangle.SothecomponentofitsspeedinthedirectionofdogAis1/2×4=2yardspersecond.ThereforeAandBareapproachingeachotheratacombinedspeedof4+2=6yardspersecond,andcovertheinitialseparationof60yardsin60/6=10seconds.”
WhatdogBdoesintheframeofreferenceofdogA
WhyDoMyFriendsHaveMoreFriendsThanIDo?
Supposethesocialnetworkhasnpeople,andpersonihasxifriends.Thentheaveragenumberoffriends,overallmembers,is
Tothinkaboutcolumn3inthetable,theweightedaverageofhowmanyfriendseachfriendjofpersonihas,weuseastandardmathematicaltrickandworkonpersonjinstead.Theyturnupasafriendtoxjpeople—namely,theirownfriends—andtheycontributexjtothetotalforeachofthosefriends.Sothecasesforwhichpersonjoccursasafriendcontributex2jtothetotal.Thenumberofentriesincolumn3isx+1+…+xn.Sotheweightedaverageofhowmanyfriendseachfriendhasis
Iclaimthatforanychoicesofthexjwealwayshaveb>a,unlessallthexjareequal,inwhichcaseb=a.Thisfollowsfromastandardinequalityrelatingtheaveragetowhatengineerscallthe‘rootmeansquare’(squarerootoftheaverageofthesquares):
withequalityonlywhenallxjareequal.Squaringthisandrearranging,wegeta<bexceptwhenallxjareequal,asrequired.Forfurtherinformation,see:
https://www.artofproblemsolving.com/Wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality
TheAdventureoftheSixGuests
Soames’sremarkisanexampleofRamseyTheory,abranchofcombinatoricsnamedafterFrankRamsey,whoprovedamoregeneraltheoremofasimilarkindin1930.HisbrotherMichaelbecameArchbishopofCanterbury.Let’sworkuptoitgently.Supposethatanumberofpeopleareseatedroundatable,witheveryonebeingconnectedtoeveryoneelsebyeitheraforkoraknife.Pickanytwonumbersfandk.ThenthereissomenumberR,dependingonfandk,suchthatifatleastRpeoplearepresenttheneitherfofthemarejoinedbyforks,orkbyknives.ThesmallestsuchRisdenotedbyR(f,k)andcalledtheRamseynumber.
Soames’sproofshowsthatR(3,3)=6.Ramseynumbersareextraordinarilydifficulttocalculate,exceptinafewsimplecases.Forexample,itisknownthatR(5,5)liesbetween43and49,buttheexactvalueremainsamystery.Ramseyprovedamoregeneraltheoreminwhichthenumberoftypesof
connection(knife,fork,whatever—coloursareamorecommonimage,butSoamesworkswithwhateveristohand)canbeanyfinitenumber.Theonlyknownnon-trivialRamseynumberformorethantwotypesofconnectionisR(3,3,3),whichis17.Thereareinnumerablegeneralisationsofthisidea.Inveryfewcasesisthe
exactnumberconcernedknown.Thepaperthatstarteditalloffis:F.P.Ramsey,Onaproblemofformallogic,ProceedingsoftheLondonMathematicalSociety30(1930)264–286.Asthetitlesuggests,hewasthinkingaboutlogic,notcombinatorics.
Graham’sNumber
R.L.GrahamandB.L.Rothschild,Ramseytheory,StudiesinCombinatorics(ed.G.-C.Rota)MathematicalAssociationofAmerica17(1978)80–99.
TheAffairoftheAbove-AverageDriver
In1981O.Svensonsurveyed161SwedishandAmericanstudents,askingthem
toratetheirdrivingabilityandsafetyrelativetotheothersubjects.Forability,69%ofSwedesconsideredthemselvestobeabovethemedian;forsafety,thefigurewas77%.ThefiguresfortheAmericanstudentswere93%forabilityand88%forsafety.HavingpassedtwoAmericandrivingtests,oneofwhichdidnotinvolvegettinginthecar,Icanseewhytheyoverestimatedtheirabilitiestosuchanextent.SeeO.Svenson,Arewealllessriskyandmoreskillfulthanourfellowdrivers?ActaPsychologica47(1981)143–148.Thiseffectoccursformanyothertraits—popularity,health,memory,job
performance,evenhappinessinrelationships.It’snotespeciallysurprising:it’sonewaypeoplemaintaintheirownself-esteem.Andpoorself-esteemcanbeasignofpsychologicalinadequacy—soinordertobehappyandhealthywehaveevolvedtooverestimatehowhappyandhealthyweare.Dunnoaboutyou,butI’mfeelinggreat.
TheBafflehamBurglary
“Thenumbersare4and13,”saidSoames.“Howutterlyamazing.I—”“Youknowmymethods,Watsup.”“Nevertheless,Ifindittotallyremarkablethatyoucandeducetheanswer
fromsuchavagueconversation.”“Hmm.Weshallsee.Theessence,Watsup,isthateachstatementwemake
addsextrainformationthatwebothknow.Andknowwebothknow,andsoon.Supposetheproductofthetwonumbersispandtheirsumiss.Initiallyyouknowp,whereasIknows.Weeachknowthattheotherknowswhatheknows,butwedonotknowwhatthatis.“Sinceyoudonotknowthetwonumbers,pcannotbeaproductoftwo
primes,suchas35.Forthisis5×7,andthereisnootherwaytoexpressitasaproductofnumbersgreaterthan1,soyouwouldimmediatelydeducethetwonumbers.Forsimilarreasonsitcannotbethecubeofaprime,suchas53=125,sincethisfactorisesonlyas5×25.”“Yes,Iseethat,”Ireplied.“Moresubtly,pcannotbeequaltoqmwhereqisprimeandmiscomposite,
providedthatwheneverddividesmandisgreaterthan1,qdisgreaterthan100.”“Yeeeesss…““Forexample,pcannotbe67×3×5,whichfactorisesinthreeways:67×15,
201×5,and335×3.Sincethelasttwousenumbersgreaterthan100,theycanbeignored,leavingonly67and15asthetwonumbers.”“Ah.Quite.”“Now,yourremarkmakesmeawareofthosefacts,butatthatpointIhave
alreadydeducedthesameinformationfrommysum.Infact,sisnotthesumoftwosuchnumbers.Butyouthenbecomeawareofthatfact,becauseItellyou,soyouthenknowwhattoyouisnewinformationabouts.Ofcourse,webothhavetobearinmindthatifs=200thenthenumbersmustbothbe100,andifs=199theyare100and99.”“Ofcourse.”“Onceyouhaveeliminatedtheimpossible…”saidSoames,“allthatremains
turnsouttobesumssequaltooneofthenumbers11,17,23,27,29,35,37,41,47,51,or53.”“Butpreviouslyyouhavemadescathingremarksabout—”“Oh,itworkswellenoughinmathematics,”hesaidairily.“Fortherewecan
beconfidentthattheimpossiblereallyisimpossible.“Now,attherelevantstageinthededuction,webothknowwhatIhavejust
toldyou.Atwhichjunctureyoupromptlyannouncethatyoucandeducethenumbers!SoIquicklyrunthroughallpossiblepairsofnumberswiththosesums,andIfindthattenoftheelevenpossibilitiesforsshareapossibleproductwithadifferentvalueofs.Sinceyouhavetoldmeyounowknowthenumbers,alltencanbeeliminatedfromourinvestigation.Whichleavesonlyonepossiblesum,17,andonlyoneproductnotoccurringfortwodifferentvaluesofs.Namely,52,whichariseswhenwesplit17as4+13,andonlyinthatmanner.Thereforethetwonumbersmustbe4and13.”Icongratulatedhimonhisperspicacity.“SendaBakerStreetIrreducibletoRouladewiththismessage,”heinstructed,
scribblingthenumbersonascrapofpaper.“Hewillhavetwoarrestswithinthehour.”
Malfatti’sMistake
In1930HymanLobandHerbertRichmondprovedthatthegreedyalgorithmisbetterthanMalfatti’sarrangementinsomecases.HowardEvesnoticedin1946thatforanisoscelestrianglewithaverysharpapex,thestackedsolutionhasalmosttwicetheareaoftheMalfattiarrangement.In1967M.GoldbergprovedthatthegreedyalgorithmalwaysdoesbetterthanMalfatti’sarrangement.In1994VictorZalgallerandG.A.Losprovedthatitalwaysproducesthelargestarea.
HowtoStopUnwantedEchoes
M.R.Schroeder,Diffusesoundreflectionbymaximum-lengthsequence.JournaloftheAcousticalSocietyofAmerica57(1975)149–150.
TheEnigmaoftheVersatileTile
Tenshapesformedfromversa-tiles
TheThrackleConjecture
JánosPachandEthanSterling,Conway’sconjectureformonotonethrackles,AmericanMathematicalMonthly118(June/July2011)544–548.
ATilingThatIsNotPeriodic
Howtotileperiodicallyusingthe7-gon
TheTwoColourTheorem
Havingrackedmybrainsforthreehours,IbeggedSoamestorevealhissecret.“Butthenyouwilltellmehowabsurdlysimpleitis.”“Nay!Never!”“Ibegtodiffer,Watsup.Becauseforonceitisabsurdlysimple.”Silence
stretcheduntilherelented.“Verywell.Assumethattheonlyavailablecoloursareblackandgrey,withwhiteforregionsasyetunconsidered.Letusbeginbycolouringoneregionblack(seetopleftfigureonpage294).ThenIchooseoneadjacentregion,andcolouritgrey(topmiddle).ThenIcolouranadjacentregionblack,thenanothergrey,andsoon.”“Iseethatafterthefirstchoice,successivechoicesareforced,”Isaid
hesitantly.“Yes!Thesolution,ifitexists,mustbeunique—saveforinterchangingthe
twocolours.Andyouseethateventuallythewholemapiscoloured,usingonlyblackandgrey.Sointhiscase,atleast,asolutiondoesexist.”“Agreed.ButIdon’tentirelysee—”“Why.Anexcellentremark.Foronce,mydearWatsup,youhavehitthenail
firmlyonitshead,notyourownthumb.Theproblemistoprovethatanysuchchainofcolouringsleadstothesameoutcome,yes?Becausethatway,theprocesscanneverterminatewithasituationforwhichthenextremainingregioncannotbecoloured.”
Thefirstfewstagesincolouringthemap
“IthinkIseethat.”“Itcanbedone,”saidSoames.“Butthereisasimplermethod.Observethat
everytimewecrossacommonboundary,thecolourchanges.Thereforeifwecrossanoddnumberofboundaries,wemustchoosegrey,andifwecrossanevennumberofboundaries,wemustchooseblack.”Inodded.“But…howcanwebesurethatthereisnoinconsistency?”I
blurted.Soamesgaveabriefgrin.“BecausewecantakeahintfromwhatIhavejust
said,andprescribetheexactcolourofeveryregion.Merelycounthowmanycirclescontainagivenpoint—notonanycircumference,ofcourse,becausewedonotcolourthose.Ifthatnumberiseven,colourthepointblack;ifodd,colouritgrey.“Now,crossinganyboundarylineeitheraddsonecontainingcircle,or
subtractsone.Eitherway,oddchangestoevenandeventoodd,sothetwocoloursoneithersideofthatboundaryaredifferent.”
Numberingregionsaccordingtohowmanycirclestheylieinside.Notehowparity(odd/even)changesacrossboundaries.
Theproofwasasclearasdaylight.“Why,Soames—”“Ofcourse,”heinterrupted,withthebaresthintofasmile,“someofthe
circlesmaybetangenttoothers.Butthesamemethodstillapplies,suitablyinterpreted.Onemustavoidcrossingaboundarylineatapointoftangency,andalittlethoughtshowsthatthiscanalwaysbedone.”Well,maybenotquitedaylight,but…yes,Iunderstood.“Itis—”Ibegan;
thenpaused,seeinghisexpression.“Veryclever,”Ifinished.
TheFourColourTheoreminSpace
Fourequalspherescanbearrangedsothateachtouchestheotherthree.Placethreetomakeanequilateraltriangle,touchingeachother,andthenputthefourthontoptofitintothecentraldimple,makingatetrahedron.Asmallersphere,ofexactlytherightsize,cannowbeplacedinthemiddletotouchallfour.Sowehavefivespheres,eachtouchingtheotherfour,andthesemustallhavedifferentcolours.
Fittinginthefifthsphere
TheGreekIntegrator
Answerfirst.Wehavetosolvetheequation .Divideby4πr2toget.Thereforer=3.Nowforthepalimpsest.
Left:AtypicalpageoftheArchimedespalimpsest.Thethirteenth-centuryreligioustextrunsvertically;thefainteroriginalrunshorizontally.Right:Cleaned-uppagewithclearmathematicaldiagrams.
Archimedes’originalmanuscripthasnotsurvived,butthiscopy(nodoubttheculminationofaseriesofcopies)wasmadebyaByzantinemonkaroundAD950.In1229itwasunboundandscrubbed(fairly)clean,alongwithatleastsixothermanuscripts.Thesheetswerefoldedinhalfandusedtowritea177-pageChristianliturgicaltext—adescriptionoftheproceduresforreligiousservices.Inthe1840sConstantinvonTischendorf,aGermanBiblicalscholar,came
acrossthistextinConstantinople(nowIstanbul),noticedfaintmathematicalwritings,andbroughtapageofithome.In1906aDanishscholar,JohanHeiberg,realisedthatpartofthepalimpsestwasaworkofArchimedes.Hephotographedit,publishingsomeextractsin1910and1915.ThomasHeathtranslatedthematerialshortlyafterwards,butitattractedlittleattention.Inthe1920sthedocumentwasinthepossessionofaFrenchcollector;by1998ithadsomehowmadeitswaytotheUSA,becomingthesubjectofacourtcasebetweenChristie’sauctionhouseandtheGreekOrthodoxChurch,whichallegedthatthedocumenthadbeenstolenfromamonasteryin1920.ThejudgeruledinChristie’sfavouronthegroundsthatthedelaybetweentheallegedtheftandthelegalactionhadbeentoolong.Thedocumentwaspurchasedbyananonymousbuyer(reportedbyDerSpiegeltobeAmazon’sfounderJeffBezos)for$2million.Between1999and2008thedocumentwasconservedattheWaltersArtMuseum,Baltimore,andanalysedbyateamofimagingscientiststoenhancethehiddenwriting.Archimedes’methodcanbeexplained(usingmodernlanguageand
symbolism)asfollows.Startwithasphereofradius1,itscircumscribedcylinder,andacone.Ifweplacethecentreofthesphereatpositionx=1ontherealline,thenthecross-sectionalradiusatanyxbetween0and2is ,anditsmassisproportionaltoπtimesthesquareofthis,namelyπx(2–x)=2πx–πx2.Next,consideraconeobtainedbyrotatingtheliney=xaroundthex-axis,
againfor0≤x≤2.Thecrosssectionatxisacircleofradiusx,andhasareaπx2.Itsmassisproportionaltothis,withthesameconstantofproportionality,sothecombinedmassofthesliceofsphereandthesliceoftheconeis(2πx–πx2)=πx2=2πx.Placethetwoslicesatx=–1,distance1totheleftoftheorigin.Bythelaw
ofthelever,theyareexactlybalancedbyacircleofradius1placeddistancextotheright.Nowmoveallslicesofthesphereandtheconetothesamepointx=–1,so
thattheirtotalmassisconcentratedatthissinglepoint.Thecorresponding(andbalancing)circlesallhaveradius1,andareplacedatalldistancesfrom0to2.Theythereforeformacylinder.Itscentreofmassisinthemiddle,atx=1.
Therefore,bythelawofthelever,
WhatArchimedesdid.Top:Sliceasphere,cone,andcylinder(shownincrosssection:sphere=circle,cone=triangle,cylinder=square)likeloavesofslicedbread.Thenthevolumeofasliceofthecylinder(grey)isthesumofthevolumesofthecorrespondingslicesofthesphereandthecone.Theslicesherehavenonzerothickness,whichintroduceserrors.Archimedesthoughtaboutinfinitelythinslices,forwhichtheerrorsbecomeassmallasweplease.Bottom:theweighingargumentthatrelatesthethreevolumes.Slicesatxofsphereandcone,placedat–1,balancesliceofcylinder
placedatx.
massofsphere+massofcone=massofcylinder
andsincemassisproportionaltovolume,
volumeofsphere+volumeofcone=volumeofcylinder.
However,Archimedesalreadyknewthatthevolumeoftheconeisonethirdthatofthecylinder(onethirdareaofbasetimesheight,remember?),sothevolumeofthesphereistwothirdsthatofthecylinder.Thevolumeofthecylinderisthe
areaofthebase(πr2)timestheheight(2r),thatis,2πr3.Sothevolumeofthesphereis ofthis,namely .Archimedesderivedthesurfaceareaofthespherebyasimilarprocedure.Hedescribedtheprocessgeometrically,butit’seasiertofollowtheargument
usingmodernnotation.Consideringthathedidallthisaround250BC,andthathealsodevelopedthelawofthelever,it’sanamazingachievement.
WhytheLeopardGotItsSpots
W.L.Allen,I.C.Cuthill,N.E.Scott-Samuel,andR.J.Baddeley.Whytheleopardgotitsspots:relatingpatterndevelopmenttoecologyinfelids,ProceedingsoftheRoyalSocietyB:BiologicalSciences278(2011)1373–1380.
PolygonsForever
Itlooksasthoughthefigurewillgrowwithoutlimit,butactuallyitremainswithinaboundedregionoftheplane:acirclewhoseradiusisabout8.7.Theratiooftheradiiofacirclecircumscribedaboutaregularn-gonandone
inscribedinitissecπ/n,wheresecisthetrigonometricsecantfunctionandI’musingradianmeasurefortheangle.(Replaceπby180°fordegreemeasure.)Soforeachntheradiusofthecirclecircumscribedabouttheregularn-goninthepictureis
S=secπ/3×secπ/4×secπ/5×…×secπ/n
Wewantthelimitofthisproductasntendstoinfinity.Takelogarithms:
logS=logsecπ/3+logsecπ/4+logsecπ/5+…+logsecπ/n
Whenxissmall,logsecx~x2/2,sothisseriescanbecomparedwith
1/32+1/42+1/52+…+1/n2
whichconvergesasntendstoinfinity.ThereforelogSisfinite,soSisfinite.Thesumofthetermsupton=1,000,000yields8.7asareasonableestimate.Ilearnedaboutthisproblem,andtheanswergivenabove,fromabookreview
byHaroldBoas[AmericanMathematicalMonthly121(2014)178–182].HetraceditbackasfarasMathematicsandtheImaginationbyEdwardKasnerand
JamesNewmanin1940.Hewrites:“Perhapsthisamusingexamplewillbecomepartofthestandardloreifthefiguregetsreproducedinenoughbooks.”I’mtrying,Harold.
TheAdventureoftheRowingMen
SoamesandIfoundtwofurtherarrangements,notcountingreflectionsinthecentrelineasdifferent:
Arrangements0167and0356
“Forallthemechanicalcomplexityoftheproblem,”saidSoames,“itreducesultimatelytomerearithmetic.Wehavetosplitthenumbersfrom0to7intotwosets,eachwithsum14.”“Ifweknowonesuchset,theotherisdeterminedandalsohassum14.”“Yes,Watsup,thatisevident:listthenumbersthatarenotinthefirstset.”“Iagreeitistrivial,Soames,butitimpliesthatwecanworkontheset
containing0,whichamountstoputtingthesternoarontheleft,whichwemayassumebyapplyingareflectionifnecessary.Therebyreducingthenumberofcasestobeconsidered.”“True.”Nowthedeductionsalmostmadethemselves.“Ifthesetalsocontains1,”I
pointedout,“thentheothertwoaddto13,sotheymustbe6and7,giving0167.Ifitdoesnotcontain1butdoescontain2,thentheonlypossibilityis0257.Ifitstarts03therearetwopossibilities:0347and0356.Wecandismissarrangementsthatstart04becauseitisnotpossibletomake10fromtwoofthenumbers5,6,7.Asimilarargumentdisposesof05,06,and07.”“Soyouhavededuced,”Soamessummarised,“thattheonlypossibilities,
excludingleft–rightreflections,are
0167025703560347
Now0257istheGermanrigand0347theItalian.Therearetwoothers:thosethatIhavelaidoutwithmymatchst—”Hesuddenlysatup,rigid.“GreatScott!”“What,Soames?”
“Ithasjuststruckme,nopunintended,Watsup,thatthismatch—”hewaveditatme—“isnotarareearlyCongreve,asIhadimagined,butoneofIrinyi’snoiselessmatches.Whenhischemistryprofessorblewhimselfup,Irinyiwasinspiredtoreplacepotassiumchloratebyleaddioxideintheheadofthematch.”“Ah.Isthatsignificant,Soames?”“Mostassuredly,Watsup.Itcastsanentirelynewlight—againnopun
intended—ononeofourmostbizarreunsolvedcases.”“TheRemarkableAffairoftheUpsideDownTeapot!”Icried.“Youhaveit,
Watsup.Now,ifyournotesrecordwhetherthematchwasdroppedtotheleftortherightofthemummifiedparrot…”Soames’sanalysisisbasedon:MauriceBrearley,‘Oararrangementsinrowingeights’,inOptimalStrategies
inSports(ed.S.P.LadanyandR.E.Machol),North-Holland1977.JohnBarrow,OneHundredEssentialThingsYouDidn’tKnowYouDidn’t
Know,W.W.Norton,NewYork2009.AsSoameswarned,itisaninitialsimplifiedapproachtoahighlycomplex
issue.Bytheway,the1877BoatRacewasadeadheat—theonlyoneinthehistory
oftheevent.
RingsofRegularSolids
JohnMasonandTheodorusDekkerfoundsimplermethodsthanŚwierczkowski’stoprovetheimpossibility.Wheneveryougluetwoidenticaltetrahedronsbytheirfaces,eachisareflectionoftheotherinthecommonface.
Twotetrahedronswithacommonface(shaded).Eachisareflectionoftheotherinthisface.
Startwithonetetrahedron.Ithasfourfaces,sotherearefoursuchreflections;
callthemr1,r2,r3,andr4.Eachreflectionsendseverythingbackwhereitstartedifyoudoittwice,sor1r1=e,whereeisthetransformation‘donothing’.Thesamegoesfortheotherreflections.Soallcombinationsofseveralreflectionsareproductslike
r1r4r3r4r2r1r3r1
wherethesequenceofsubscripts14342131canbeanysequenceformedbythefournumbers1,2,3,4inwhichnonumbereveroccurstwiceinarow.Forexample,14332131isnotpermitted.Thereasonisthatherer3r3isthesamereflectionperformedtwice,soitisthesamease,whichhasnoeffectandcanthereforebedeleted.Ifachainclosesup,onefurtherreflectionappliedtothelasttetrahedroninthe
chainproducesatetrahedronthatcoincideswiththeinitialone.Sowegetanequationlike
r1r4r3r4r2r1r3r1=e
(onlylongerandmorecomplicated)whereestandsfor‘donothing’.Bywritingdownformulasforthefourreflections,andusingsuitablealgebraicmethods,itcanbeprovedthatnosuchequationholds.Fordetails,see:T.J.Dekker,OnreflectionsinEuclideanspacesgeneratingfreeproducts,
NieuwArchiefvoorWiskunde7(1959)57–60.M.ElgersmaandS.Wagon,ClosingaPlatonicgap,Mathematical
Intelligencerinthepress.J.H.Mason,Canregulartetrahedronsbegluedtogetherfacetofacetoforma
ring?MathematicalGazette56(1972)194–197.H.Steinhaus,Problem175,ColloquiumMathematicum4(1957)243.S.Świerczkowski,OnafreegroupofrotationsoftheEuclideanspace,
IndagationesMathematicae20(1958)376–378.S.Świerczkowski,Onchainsofregulartetrahedra,Colloquium
Mathematicum7(1959)9–10.
TheImpossibleRoute
“Asyousorightlysay,youdonotseeit,”saidSoames.“Youknowmymethods:usethem.”“Verywell,Soames,”Ireplied.“Youhavealwaysinstructedmetodiscard
thatwhichisirrelevant.Ishallthereforerepeatmyanalysis,andtoeliminateanyconceivablepossibilityoferrorIshallrepresenttheprobleminitssimplestform.Inumbertheregionsonthemap—likeso.Therearefiveofthem.ThenIdrawadiagram—Ibelieveitiscalledagraph—showingtheregionsandtheirconnectionsinschematicform.”Heremainedsilent,hisexpressionunreadable.“Wemustproceedfromregion1toregion5,leavingbridgeAtothelast.
Startingfrom1,theonlyalternativeistocrossbridgeB,andthenCandDareforceduponus.WemustuseeitherbridgeEorF.LetussaywetakeE.WecannotuseFbecausethattakesustoregion4andwecannotproceedfurther.However,wecannotthenuseA,becausethattakesustoregion1andwecannotproceedfurther.ThesamegoesifweemployFinplaceofE.Irestmycase.”
Left:Watsup’sfiveregions.Right:Graphofconnections.
“Why,Watsup?”“Because,Soames,Ihaveeliminatedtheimpossible.”Heraisedoneeyebrow.
“Sowhatremains,howeverunlikely,”Icontinued,“mustbe—”“Goon.”“ButSoames,nothingremains!Thereforetheproblemhasnosolution!”“Wrong.Ihavetoldyouthatthereareeight.”“Thenyoumusthaveliedabouttheconditions.”
“Ididnot.”“ThenIamstumped.WhathaveImissedout?”“Nothing.”“But—”“Youputtoomuchin,Watsup.Youmadeanunwarrantedassumption.Your
errorwastoassumethatthepathdoesnotleavethemap.”“ButyoutoldmethattheriverscontinuetoflowtotheSwissborders,andwe
arenotpermittedtorecrosstheborder.”“Yes.ButthemapdoesnotdepictthewholeofSwitzerland.Wheredoesthe
rivercomefrom?”“D’oh!”Istruckmyforeheadwithmyhand.“Dough?”“Merelyaninadvertentexpressionberatingmyselfformyownstupidity,
Soames.Not‘dough’.Morealongthelinesof‘D’oh!’”“Iadviseyoutoavoidit,Watsup.Itdoesnotbecomeyou,anditwillnever
catchon.”“Asyousay,Soames.Whatcausedmyoutburstwastherealisationthatwe
cancompletemysecondattemptbyencirclingthesourceoftheriverandpassingoverbridgeA.”“Correct.”“Soregions1and4inmyfigureareactuallythesameregion.”“Indeed.”
Soames’sroute
“That,”Isaidafteramoment,“wasunfair.HowamItoknowthattheriverriseswithinSwissborders?Thesourcewasnotshownonyourmap.”“Because,Watsup,Itoldyouthatthereisatleastoneroutesatisfyingmy
conditions.ItfollowsthatthesourcemustlieinSwitzerland.”Touché.ThenIrememberedthathehadreferredtoeightroutes.“Iseea
secondroute,Soames:interchangebridgesEandF.ButIconfesstheothersixeludeme.”“Ah.YourassertionthatwemustbeginwithbridgeBisnolongervalidwhen
regions1and4aremerged.Letmeredrawyoursimplifiedfigurecorrectly.”
Soames’scorrectedgraph
“IhavedrawnbridgeAasadottedlineasareminderthatwemustleaveAuntillast.Observe:startingfromregion1,thebridgesotherthanAformtwodistinctloops:BCDandEF.Wecantraverseeachloopintwodirections:BCDorDCB,andEForFE.Furthermore,wecanstartwitheitherloopandthentraversetheother.Finally,wemustappendbridgeA.Sothedifferentroutesare
BCD–EF–ADCB–EF–ABCD–FE–ADCB–FE–AEF–BCD–AEF–DCB–AFE–BCD–AFE–DCB–A
“Atotalofeight.”“Iseemyerrorclearlynow,Soames,”Iadmitted.“Youseeyourspecificerror,Watsup,butnottheunderlyinggenerality,which
afflictsallargumentsabouteliminatingtheimpossible.”Ishookmyhead,puzzled.“Whatdoyoumean?”“Imean,Watsup,thatyoudidnotconsiderallpossibilities.Andthereason
was—”
AgainIstruckmyheadwithmyhand,butthistimeIrefrainedfromutteringasound,notwishingtobethebuttofSoames’sscorn.“Iforgottothinkoutsidethebox.”
TheGreatMathematicalProblemsIANSTEWART
‘Ianreallyisunsurpassedasraconteuroftheworldofnumbers.Heguidesusonamind-bogglingjourneyfromtheultra-trivialtotheprofound.’NewScientist
TheGreatMathematicalProblemsdescribesthereallybigproblemsthatdefinemathematics:whytheyexist,whytheymatter,whatdrivesmathematicianstoincrediblelengthstosolvethemandwheretheystandinthecontextofmathematicsandscienceasawhole.Itcontainssolvedproblems–likethePoincaréConjecture,crackedbytheeccentricgeniusGrigoriPerelman,whorefusedacademichonoursandamillion-dollarprizeforhiswork,andoneswhich,liketheRiemannHypothesis,remainbafflingaftercenturies.
Stewartistheguidetothismysteriousandexcitingworld,showinghowmodernmathematiciansconstantlyrisetothechallengessetbytheirpredecessors,asthegreatmathematicalproblemsofthepastsuccumbtothenewtechniquesandideasofthepresent.
March2013
PaperbackISBN9781846683374
eBookeISBN9781847653512
MathematicsofLife
IANSTEWART
‘Asalways,[Stewart]explainscomplicatedmathematicalideasbrilliantly’NewScientist
Anewpartnershipofbiologistsandmathematiciansisusingmathsasneverbefore:tounlockthesecretsoflifeonearth.Theyareuncoveringthehiddentruthsbehindthebehaviourofanimalsandplants–andtheecologicalbalanceofourentireplanet.CelebratedmathematicianIanStewartprovidesexamplesfromthemysteriousoperationsofthebraintothepuzzlingbehaviourofviruses.Inbetween,heinvestigatesthecreationofartificiallifeandtheprobabilityofaliensandanswersintriguingquestionslikehowthetigergotitsstripesandwhymatinglizardsplayrock-paper-scissors.
Together,mathematiciansandbiologistsareansweringsomeofthemostcrucialquestionsinscience:thisbookisaninsider’sviewofwhatthey’redoing–andwherethatknowledgeistakingus.
July2013Paperback
ISBN9781846682056eBook
eISBN9781847653505
ProfessorStewart’sIncredibleNumbersIANSTEWART
ThenewmathematicalbestsellerfromProfessorIanStewart
IanStewartexplorestheastonishingpropertiesofnumbersfrom1to10tozeroandinfinity,includingonefigurethat,ifyouwroteitout,wouldspantheuniverse.Helooksateverykindofnumberyoucanthinkof–real,imaginary,rational,irrational,positiveandnegative–alongwithseveralyoumighthavethoughtyoucouldn’tthinkof.Heexplainstheinsightsoftheancientmathematicians,showshownumbershaveevolvedthroughtheages,andrevealsthewaynumericaltheoryenableseverydaylife.
UnderProfessorStewart’sguidanceyouwilldiscoverthemathematicsofcodes,sudoko,Rubik’scube,music,primesandpi.Youmaybesurprisedtofindyouliveineleven-dimensionalspace,thatofthetwenty-threepeopleonafootballpitchtwoaremorelikelythannottosharethesamebirthday,andthatforty-twoisaveryinterestingnumber.
ProfessorStewart’sIncredibleNumberswilldelighteveryonewholovesnumbers–includingthosewhocurrentlythinktheydon’t.
Availablenowfromtheappstore
17EquationsthatChangedtheWorldIANSTEWART
Auniquehistoryofhumanitytoldthroughitsseventeendefiningequations;fromPythagorastoCalculus
FromNewton’sLawofGravitytotheBlack-Scholesmodelusedbybankerstopredictthemarkets,equations,areeverywhere–andtheyarefundamentaltoeverydaylife.
SeventeenEquationsthatChangedtheWorldexaminesseventeengroundbreakingequationsthathavealteredthecourseofhumanhistory.HeexploreshowPythagoras’sTheoremledtoGPSandSatNav;howlogarithmsareappliedinarchitecture;whyimaginarynumberswereimportantinthedevelopmentofthedigitalcamera,andwhatisreallygoingonwithSchrödinger’scat.
Entertaining,surprisingandvastlyinformative,SeventeenEquationsthatChangedtheWorldisahighlyoriginalexploration–andexplanation–oflifeonearth.
February2012Paperback
ISBN9781846685323eBook
eISBN9781847657695
ProfessorStewart’sHoardofMathematicalTreasuresIANSTEWART
‘AnidealpresentforanyoneaddictedtoSudoku-likepuzzlesandbeginningtowonderwhatmightliebeyond’Spectator
November2010
BPB352ppISBN9781846683466
ProfessorStewart’sCabinetofMathematicalCuriositiesIANSTEWART
‘StewarthasserveduptheinstructiveequivalentofaMichelin-starredtastingmenu’GuardianJuly2010BPB320ppISBN9781846683459