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i

Up-and-down elevator games andPiet Hem's mechanical puzzles

by .Martin Gardner

Elevators, unlike cars, trains, planes,ships and other common modesI of transportation, have been un-duly neglected by recreational mathe-maticians. This month we undertake torectifv the situation by considering fourunusual elevator problems. The firstthree were provided by Donald E.Knuth, a computer scientist at StanfordUniversity and author of a classic seven-volume work in progress titled The Artof Computer Programming. Before dis-cussing two combinatorial problems thatappear for the first time in his third vol-ume (published in January by Addison-Weslev), we consider a well-knownprobability paradox with a startling gen-eralization that Knuth discovered a fewyears ago.

George Gamow and Marvin Stern in-troduced theelevator paradox in the pro-logue to their little book Puzzle-Math(Viking, 1958). Gamow once had anoffice on the second floor of a seven-storybuilding in San Diego and Stern had anoffice on the sixth floor. When Gamowwanted to go up to see

Stern,

he noticedthat in about five cases out of sixthe firstelevator to stop on his floor was goingdown. It seemed as if elevators werebeing manufactured on the roof .andthen sent down the shafts to be stored inthe basement. For Stern the situationwas the opposite. When he wanted to godown to see

Gamow,

about five times

tors is complicated hy cond iabilities. As Knuth puts itof which elevator is first to jsecond floor is partly ...whether it was above us or (an elevator that is belov*.floor when we begin to v*. .. i

out of six the first elevator to arrive was arrive ahead of an elevatoi "'■ .on its wayup. Were elevators being fab- (all other things being t-ijricated in the basement and then sent to 1969 paper [see "Bihlioft .the roof to be carried off by helicopters? 124] Knuth analyzes Gam.. .

The explanation, as Knuth pointed as follows: Consider the *-out later, requires a few idealizing as- elevator's route that starts ..sumptions. Suppose each elevator trav- floor, then goes down to ll*els independently in continuous cycles and up to the second, a t..:Afrom bottomfloor to top and back again, 1/3 of the entire route. I).,moving with constant speed .and with half of this portion thethe same average waiting time on each next at the second floor goifloor. Thus at the timea button is pushed during the other half it » .on any floor we can assume that each going up. Therefore we i.u.elevator is at a random point in its cycle. unbiased portion, since it

For a single elevator, calculating the towardup or down.probability that it is on its way down If there are n

elevator*,

1

when it stops on a given floor is quite distinguishes two cases:easy

Stem,

on the sixth floor, has five 1. No elevator is in thr ufloors below and one above; therefore tion. The probability of llthe probability is 5/6 that the elevator since it is 2/3 for each elr- .'is below him and will be moving up elevator to stop on the m «when it arrives.

Gamow,

on the second be going down,floor, has five floors .above and one be- 2. At least one elevatm

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low'therefore theprobability is 5/6 that biased portion. The pn.the elevator is above him and will reach (2/3)". We can ignore am <his floor on its way down. Gamow and side the unbiased portumStern explained this in their book, but those in the unbiased pollthen they made a slip. If there is more sarily reach the second i

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than one elevator, they wrote, the prob- case the elevator willabilities "of course remain the same." withprobability 1/2.The slip is understandable because the Combining these' rcM.iistatement seems so intuitively true. Ap- ability of (2/3)" +*< "parently Knuth was the first to realize 5i(2/3)» that the firs.

. "that it is not true at .all! Indeed, as the on the second floor winnumber of elevators approaches infinity If there are just two i- .the probability that the first elevator to Gamow's seven-stor\stop on anv floor (except the top or hot- elevator to stop .it t>*torn floors) is going up (or down) ap- be headed

w:"jpreaches exactly 1/2-a rather unex- £ + 2/9-13/18.pected result. Yet the probability (for, than

5/6,

so that ""'catching an up e.ev.i'If there are seven < '■■abilityof an elevator's:be 2,315/4,374, win ■'

uk/b dk/b [uk /b) [dk /b]

0 . 23/20 1/2.Knuth gives the ■:.■

any building by *>■>''tance from a given «-"divided by the dist.u..'bottom floors. For (»»'Stern p is 5/6. The k<all values of p betwe«"■"

2 3

5/23/2

3 23.2

5/2

2 2

3/23/2

2 003/2

■

__ . > —»--.. aa, * : .''.i*'

, -_-__- --p- -- -- i »xiT A 1 T A J 1 I"T1 C\ sav> e second floor) rm,,If 4TI 1^ VI VH Iw\ i I \illf Va every individual elevator. ..,The solution for two ..r i

b = 2

107

5/6 forsll eleva-B next to

re eleva-nal prob-le choiceve on theigent on

3W,

sincesecond

likely to: is above." In hisij" pagesituationon of anic fourth

st floor

tor stopsown, andrext stopall it theot biased

mth now

ised por-(2/3)",

The nextfloor will

a the un-:y is 1 —

out-:e one of"ill neces-st. In thisig down

a prob-r) = * +to arriveng down,rnning in

the firstfloor will"obabilityduly lesslances of

iproved.he prob-vn wouldfar from

niula forthe dis-

e bottoma top and:

1/6,

formla for

is

-i

, The pair of vertical lines indicate theabsolute value of the expressionbetween

j them. The probability approaches 1/2as n, the number of elevators, approach-

i es infinity.Our second elevator problem is from

the third volume of Knuth's series, abook that deals entirely with computer( techniques of sorting and searching forinformation. Like its two predecessors,it is comprehensive in scope, written in aclear, informal style (although at timesit is necessarily terse and technical)and rich in humor, historical data andproblems of great recreational interest.

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On pages 11 through 72, for instance,Knuth brilliantly summarizes .almost ev-erything known about the combinatorialproperties of permutations, a topic thatties in with scores of classic puzzle prob-lems. The book's exercises concern such

j entertaining topics as solitaire cardj games, shuffling, anagrams, snowplows,

the design of tennis tournaments (in-cluding Lewis Carroll's flawed efforts tofind a design that does the best possible! justice to the second-best player), rook

| problems, sorting puzzles, the unsolved-. weight-ranking problem, the Josephusproblem, parking problems, Fibonaccinumbers, the odd "tableaux" of AlfredYoung (which have a curious relevanceto the eightfold way of particle theory)

j and a hundred other things that leadstraight intorecreational mathematics.

Here we are concerned with pages357 through 360, where Knuth regardstheelevator as a model of one-tape com-puter sorting. A building has n floors,

♦ each holding exactly c people. There isI a single elevator that carries at most h

people. We assume that the building is3 full (contains en people). Exactly c per-sons want to go to each floor: c to thefirst

floor,

c to the second floor and soI on. Some people may already be on theirdesired floor, but it is more interestingto assume that all or most are misfitswho want to be on another floor.

The elevator always starts at the bot-tom. It movesup and down, loading andunloading passengers, until each personis where he wants to be. The elevatorthen returns to the first floor. A move-ment of the elevator from

any

floor tothe next floor above or below will becalled a unit trip. The problem is to findan algorithm that willsort all the peoplein a minimum number of unit trips. Thisoperation is equivalent, of course, tominimizing the distance traveled or (as-

\ suming a constant elevator speed) tominimizing the time required for thesorting.

As Knuth points out, the people cor-respond to records that are to be com- Richard M. Karp's algorithm

f 4/12 =' the first

A

5 5~J ..

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