elementary farkle strategy donald e. hooley bluffton university for the miami university mathematics...

Elementary Farkle Elementary Farkle StrategyStrategy

Donald E. HooleyDonald E. HooleyBluffton UniversityBluffton University

for thefor the

Miami University Miami University Mathematics ConferenceMathematics Conference

September 26, 2008September 26, 2008

FarkleFarklePlayPlay

Throw six diceThrow six dice

Keep scoring diceKeep scoring dice

Stop or throw remaining diceStop or throw remaining dice

If all six scoring may continueIf all six scoring may continue

““hot dice”hot dice”

If no score on throw If no score on throw

““farkled” and lose pointsfarkled” and lose points

Standard ScoringStandard Scoring DiceDice ScoreScoreEach 1Each 1 100 100Each 5Each 5 50 50Three 1’sThree 1’s 10001000Three 2’sThree 2’s 200 200Three 3’sThree 3’s 300 300Three 4’sThree 4’s 400 400Three 5’sThree 5’s 500 500Three 6’sThree 6’s 600 600

Scoring VariationsScoring Variations

CombinationCombination ScoreScoreFour of a kindFour of a kind three times tripletthree times tripletFive of a kindFive of a kind five times tripletfive times tripletSix of a kindSix of a kind ten times tripletten times tripletStraightStraight 25002500Three pairsThree pairs 15001500

ref. wikipedia.orgref. wikipedia.org

Farkle AppletFarkle Applet

Ref. Ref. www.keithv.com/dicegame.htmlwww.keithv.com/dicegame.html

PlayPlay

6 4 5 5 5 1 6 4 5 5 5 1

PlayPlay

2 4 4 5 6 5 2 4 4 5 6 5

Play OptionsPlay Options

Example.Example.

1 – 2 – 3 – 3 – 3 – 51 – 2 – 3 – 3 – 3 – 5

Options.Options.

Score three 3’s, throw three leftScore three 3’s, throw three left

Score 1, throw five leftScore 1, throw five left

Score all, throw one leftScore all, throw one left

Score all, stopScore all, stop

Basic ResultsBasic ResultsQuestion. Question.

What are the expected value and probability What are the expected value and probability of farkling for of farkling for nn = 1, 2, 3, 4, 5, 6 dice using = 1, 2, 3, 4, 5, 6 dice using standard scoring?standard scoring?

One dieOne die11 2 3 4 2 3 4 55 6 6

Expected value = (100+50)/6 = 25Expected value = (100+50)/6 = 25Farkling probability = 4/6 = .6667Farkling probability = 4/6 = .6667

Basic Results for Two DiceBasic Results for Two Dice

1111 1212 1313 1414 1515 16162121 2222 2323 2424 2525 26263131 3232 3333 3434 3535 36364141 4242 4343 4444 4545 46465151 5252 5353 5454 5555 56566161 6262 6363 6464 6565 6666

Expected value = 1800/36 = 50Expected value = 1800/36 = 50Farkling probability = 16/36 = .4444Farkling probability = 16/36 = .4444Hot dice probability = 4/36 = .1111Hot dice probability = 4/36 = .1111

MathematicaMathematica Program ProgramInitiate six nested loopsInitiate six nested loops

Find number of each valueFind number of each value

Six, five, four of kindSix, five, four of kind

Two tripletsTwo triplets

One triplet and extraOne triplet and extra

Less than three 1’s and 5’sLess than three 1’s and 5’s

(Straights and three pairs)(Straights and three pairs)

Complete loopsComplete loops

Output results (points, hot dice, farkles)Output results (points, hot dice, farkles)

Standard Scoring ResultsStandard Scoring Results# dice# dice Exp. Val. Exp. Val. PP(farkling)(farkling)

11 25 25 .6667 .6667

22 50 50 .4444 .4444

33 86.8056 86.8056 .2778 .2778

44 141.3194 141.3194 .1574 .1574

55 215.5093 215.5093 .0772 .0772

66 308.8831* 308.8831* .0309 .0309*disagrees with Wikipedia.org value 302*disagrees with Wikipedia.org value 302

Results With All Results With All VariationsVariations

# dice Exp. Val.# dice Exp. Val. PP(farkling) (farkling) PP(hot dice)(hot dice)

11 25 25 .6667 .6667 .3333.3333

22 50 50 .4444 .4444 .1111.1111

33 86.805686.8056 .2778 .2778 .0556.0556

44 145.8333 145.8333 .1574 .1574 .0355.0355

55 235.8218 .0772 235.8218 .0772 .0303.0303

66 452.2891 .0231* 452.2891 .0231* .0779.0779*disagrees with Wikipedia.org value 1/42 = .0238*disagrees with Wikipedia.org value 1/42 = .0238

Elementary Playing Elementary Playing StrategyStrategy

Question.Question.

What is the criterion level to determine throwing of What is the criterion level to determine throwing of remaining remaining nn dice, dice, nn = 1, 2, 3, 4, 5, 6? = 1, 2, 3, 4, 5, 6?

Notation:Notation:

xx = criterion value = criterion value

EE((nn) = expected value of ) = expected value of nn dice dice

PP((f|nf|n) = farkling probability with ) = farkling probability with nn dice dice

PP((hot|nhot|n)= probability of hot dice with )= probability of hot dice with nn dice dice

Elementary Playing Elementary Playing StrategyStrategy

Question.Question.What is the criterion level to determine throwing of What is the criterion level to determine throwing of remaining remaining nn dice, dice, nn = 1, 2, 3, 4, 5, 6? = 1, 2, 3, 4, 5, 6?

Elementary model.Elementary model.Expected gain = [1-Expected gain = [1-PP((f|nf|n)][)][EE((nn) / (1-) / (1-PP((f|nf|n)])]

+ + PP((hot|nhot|n))EE(6) (6) – – PP((f|nf|n))xx

soso[[EE((nn)+)+PP((hot|nhot|n))EE(6)] / (6)] / PP((f|nf|n) = ) = xx

Elementary Playing Elementary Playing StrategyStrategy

Question.Question.What is the criterion level to determine throwing of What is the criterion level to determine throwing of remaining n dice, n = 1, 2, 3, 4, 5, 6?remaining n dice, n = 1, 2, 3, 4, 5, 6?

# dice # dice EE((nn)) PP((f|nf|n) ) PP((hot|nhot|n) Crit. Level) Crit. Level11 2525 .6667 .6667 .3333.3333 263.6088 263.608822 5050 .4444 .4444 .1111.1111 225.5835 225.583533 86.8056 86.8056 .2778 .2778 .0556.0556 402.9981 402.998144 145.8333 145.8333 .1574 .1574 .0355.0355 1028.5233 1028.523355 235.8218 .0772 235.8218 .0772 .0303 3232.2041.0303 3232.204166 452.2891 .0231 452.2891 .0231 .0779.0779 21104.8667 21104.8667

Approximate StrategyApproximate StrategyQuestion.Question.

What is the criterion level to determine throwing of What is the criterion level to determine throwing of remaining n dice, remaining n dice, nn = 1, 2, 3, 4, 5, 6? = 1, 2, 3, 4, 5, 6?

# dice Crit. Level Approx. Strategy# dice Crit. Level Approx. Strategy 11 263.6088263.6088 never never 22 225.5835225.5835 never never 33 402.9981402.9981 400 400 44 1028.5233 1028.5233 1000 1000 55 3232.2041 3232.2041 alwaysalways 66 21104.8667 21104.8667 alwaysalways

““Extra” 5 or 1Extra” 5 or 1Question.Question.

When should player pick up an “extra” 5 or 1 When should player pick up an “extra” 5 or 1 and throw and throw nn+1 dice?+1 dice?

Elementary model.Elementary model.

Expected Gain = - pick up valueExpected Gain = - pick up value

- - PP((f|nf|n+1)[+1)[EE(6-((6-(n+n+1)) / (1-1)) / (1-PP((ff|6-(|6-(n+n+1))]1))]

+ [1-+ [1-PP((f|nf|n+1)][+1)][EE((nn+1) / (1-+1) / (1-PP((f|nf|n+1))]+1))]

+ + PP((hot|nhot|n+1)+1)EE(6)(6)

““Extra” 5 or 1Extra” 5 or 1Question.Question.

When should player pick up an “extra” 5 or 1 When should player pick up an “extra” 5 or 1 and throw and throw nn+1 dice?+1 dice?# dice left# dice left E.G. less “5” E.G. less “1”E.G. less “5” E.G. less “1”

00 -44.6274 -44.6274 -94.6274 -94.6274 11 -26.6654 -26.6654 -76.6654 -76.6654 22 28.5624 28.5624 -21.4376 -21.4376 33 97.7247 97.7247 47.7247 47.7247 44 193.7356193.7356 143.7356143.7356

““Extra” 5’s or 2’sExtra” 5’s or 2’sQuestion.Question.

When should player pick up “extra” two 5’s When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice?or three 2’s and throw all remaining dice?

Model for two 5’s.Model for two 5’s. Expected Gain = - 100Expected Gain = - 100

- - PP((f|nf|n+2)[+2)[EE(6-((6-(n+n+2)) / (1-2)) / (1-PP((ff|6-(|6-(n+n+2))]2))]+ [1-+ [1-PP((f|nf|n+2)][+2)][EE((nn+2) / (1-+2) / (1-PP((f|nf|n+2))]+2))]+ + PP((hot|nhot|n+2)+2)EE(6)(6)

““Extra” 5’s or 2’sExtra” 5’s or 2’sQuestion.Question.

When should player pick up “extra” two 5’s When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice?or three 2’s and throw all remaining dice?

Model for three 2’s.Model for three 2’s. Expected Gain = - 200Expected Gain = - 200

- - PP((f|nf|n+3)[+3)[EE(6-((6-(n+n+3)) / (1-3)) / (1-PP((ff|6-(|6-(n+n+3))]3))]+ [1-+ [1-PP((f|nf|n+3)][+3)][EE((nn+3) / (1-+3) / (1-PP((f|nf|n+3))]+3))]+ + PP((hot|nhot|n+3)+3)EE(6)(6)

““Extra” 5’s or 2’sExtra” 5’s or 2’sQuestion.Question.

When should player pick up “extra” two 5’s When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice?or three 2’s and throw all remaining dice?# dice left E.G. less “5’s” E.G. less “2’s”# dice left E.G. less “5’s” E.G. less “2’s”

00 -76.6654 -76.6654 -121.4376 -121.4376 11 -21.4376 -21.4376 -52.2753 -52.2753 22 47.7247 47.7247 43.7356 43.7356 33 143.7356143.7356 ----

Note: Three 3’s would never give positive E.G.Note: Three 3’s would never give positive E.G.

Summary of Elementary Summary of Elementary Approximate StrategyApproximate Strategy

Throw all remaining ifThrow all remaining ifa) 3 dice and less than 400 pointsa) 3 dice and less than 400 points

b) 4 dice and less than 1000 pointsb) 4 dice and less than 1000 points

c) 5 or 6 dice alwaysc) 5 or 6 dice always

Pick up a 5 or 1 ifPick up a 5 or 1 if

3 or 4 dice remaining3 or 4 dice remaining

Pick up two 5’s or three 2’s ifPick up two 5’s or three 2’s if

2 or 3 dice remaining2 or 3 dice remaining

Strategy VariationsStrategy VariationsExact criterion valuesExact criterion values

compare to estimated strategycompare to estimated strategy

Variable strategiesVariable strategies

depend on depend on

opponent totalsopponent totals

game completiongame completion

player typeplayer type

safety first, risky, changeablesafety first, risky, changeable

Computer SimulationComputer SimulationDefine Define decision vectordecision vector

list of criterion levels for continuing playlist of criterion levels for continuing play

given given

number of dice remainingnumber of dice remaining

current accumulated scorecurrent accumulated score

Simulate turnsSimulate turns

Calculate output statisticsCalculate output statistics

Preliminary Computer Preliminary Computer Simulation ResultsSimulation Results

Decision vectorDecision vector# dice left 6 5 4 3 2 1# dice left 6 5 4 3 2 1criterion level all 4500 1500 500 criterion level all 4500 1500 500 x yx y

Average score for 100,000 turnsAverage score for 100,000 turns yy

200200 300 300 400 400200200 512.188512.188 512.770512.770 510.068510.068

xx 300 300 512.917512.917 512.925512.925 510.283510.283 400400 505.150505.150 505.513505.513 503.254503.254Note: No pickup options in initial simulation program.Note: No pickup options in initial simulation program.

ReferencesReferencesSinger, Daniel. Zilch, Singer, Daniel. Zilch, http://http://

www.cs.duke.ed/~des/other_stuff/zilch.htmlwww.cs.duke.ed/~des/other_stuff/zilch.html. August 25, . August 25, 2008.2008.

Campo, Brian. Review: Farkle Dice by SmartBox Design, Campo, Brian. Review: Farkle Dice by SmartBox Design, http://www.mytodayscreen.com/review-farkle-dice-by-smarthttp://www.mytodayscreen.com/review-farkle-dice-by-smartbox-design/2box-design/2. April 26, 2008. April 26, 2008

Sparks, Heather. Some Farkle probability questions, Sparks, Heather. Some Farkle probability questions, http://http://www.hisparks.com/farkle.pdfwww.hisparks.com/farkle.pdf. August 25, 2008.. August 25, 2008.

Vertanen, Keith. Farkle Dice Game, Vertanen, Keith. Farkle Dice Game, http://www.keithv.com/cs161/project_description.htmlhttp://www.keithv.com/cs161/project_description.html. . August 30, 2008.August 30, 2008.

Wikipedia. Farkle, Wikipedia. Farkle, http://http://www.en.wikipedia.org/wiki/Farklewww.en.wikipedia.org/wiki/Farkle. . August 30, 2008.August 30, 2008.