Farkle

1. Elementary Farkle StrategyDonald E. HooleyBluffton Universityfor theMiami University Mathematics ConferenceSeptember 26, 2008

2. Farkle Play Throw six dice Keep scoring dice Stop or throw remaining dice If all six scoring may continue “hot dice” If no score on throw “farkled” and lose points

3. Standard Scoring Dice Score Each 1 100 Each 5 50 Three 1’s 1000 Three 2’s 200 Three 3’s 300 Three 4’s 400 Three 5’s 500 Three 6’s 600

4. Scoring Variations CombinationScore Four of a kind three times triplet Five of a kind five times triplet Six of a kind ten times triplet Straight 2500 Three pairs 1500 ref. wikipedia.org

5. Farkle Applet Ref. www.keithv.com/dicegame.html

6. Play 6 4 5 5 5 1

7. Play 2 4 4 5 6 5

8. Play Options Example. 1 – 2 – 3 – 3 – 3 – 5 Options. Score three 3’s, throw three left Score 1, throw five left Score all, throw one left Score all, stop

9. Basic Results Question. What are the expected value and probability of farkling for n = 1, 2, 3, 4, 5, 6 dice using standard scoring? One die 1 2 3 4 5 6 Expected value = (100+50)/6 = 25 Farkling probability = 4/6 = .6667

10. Basic Results for Two Dice 1112 13 141516 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 5152 53 545556 61 62 63 64 65 66 Expected value = 1800/36 = 50 Farkling probability = 16/36 = .4444 Hot dice probability = 4/36 = .1111

11. Mathematica Program Initiate six nested loops Find number of each value Six, five, four of kind Two triplets One triplet and extra Less than three 1’s and 5’s (Straights and three pairs) Complete loops Output results (points, hot dice, farkles)

12. Standard Scoring Results # dice Exp. Val. P(farkling) 1 25 .6667 2 50 .4444 3 86.8056 .2778 4 141.3194 .1574 5 215.5093 .0772 6 308.8831* .0309 *disagrees with Wikipedia.org value 302

13. Results With All Variations # dice Exp. Val. P(farkling) P(hot dice) 1 25 .6667 .3333 2 50 .4444 .1111 3 86.8056 .2778 .0556 4 145.8333 .1574 .0355 5 235.8218 .0772 .0303 6 452.2891 .0231* .0779 *disagrees with Wikipedia.org value 1/42 = .0238

14. Elementary Playing Strategy Question. What is the criterion level to determine throwing of remaining n dice, n = 1, 2, 3, 4, 5, 6? Notation: x = criterion value E(n) = expected value of n dice P(f|n) = farkling probability with n dice P(hot|n)= probability of hot dice with n dice

15. Elementary Playing Strategy Question. What is the criterion level to determine throwing of remaining n dice, n = 1, 2, 3, 4, 5, 6? Elementary model. Expected gain = [1-P(f|n)][E(n) / (1-P(f|n)] + P(hot|n)E(6) – P(f|n)x so [E(n)+P(hot|n)E(6)] / P(f|n) = x

16. Elementary Playing Strategy Question. What is the criterion level to determine throwing of remaining n dice, n = 1, 2, 3, 4, 5, 6? # dice E(n) P(f|n) P(hot|n) Crit. Level 1 25 .6667 .3333 263.6088 2 50 .4444 .1111 225.5835 3 86.8056 .2778 .0556 402.9981 4 145.8333 .1574 .0355 1028.5233 5 235.8218 .0772 .0303 3232.2041 6 452.2891 .0231 .0779 21104.8667

17. Approximate Strategy Question. What is the criterion level to determine throwing of remaining n dice, n = 1, 2, 3, 4, 5, 6? # dice Crit. Level Approx. Strategy 1 263.6088 never 2 225.5835 never 3 402.9981 400 4 1028.5233 1000 5 3232.2041 always 6 21104.8667 always

18. “Extra” 5 or 1 Question. When should player pick up an “extra” 5 or 1 and throw n+1 dice? Elementary model. Expected Gain = - pick up value - P(f|n+1)[E(6-(n+1)) / (1-P(f|6-(n+1))] + [1-P(f|n+1)][E(n+1) / (1-P(f|n+1))] + P(hot|n+1)E(6)

19. “Extra” 5 or 1 Question. When should player pick up an “extra” 5 or 1 and throw n+1 dice? # dice left E.G. less “5” E.G. less “1” 0 -44.6274 -94.6274 1 -26.6654 -76.6654 2 28.5624 -21.4376 3 97.7247 47.7247 4 193.7356 143.7356

20. “Extra” 5’s or 2’s Question. When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice? Model for two 5’s. Expected Gain = - 100 - P(f|n+2)[E(6-(n+2)) / (1-P(f|6-(n+2))] + [1-P(f|n+2)][E(n+2) / (1-P(f|n+2))] + P(hot|n+2)E(6)

21. “Extra” 5’s or 2’s Question. When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice? Model for three 2’s. Expected Gain = - 200 - P(f|n+3)[E(6-(n+3)) / (1-P(f|6-(n+3))] + [1-P(f|n+3)][E(n+3) / (1-P(f|n+3))] + P(hot|n+3)E(6)

22. “Extra” 5’s or 2’s Question. When should player pick up “extra” two 5’s or three 2’s and throw all remaining dice? # dice left E.G. less “5’s” E.G. less “2’s” 0 -76.6654 -121.4376 1 -21.4376 -52.2753 2 47.7247 43.7356 3 143.7356 -- Note: Three 3’s would never give positive E.G.

23. Summary of Elementary Approximate Strategy Throw all remaining if a) 3 dice and less than 400 points b) 4 dice and less than 1000 points c) 5 or 6 dice always Pick up a 5 or 1 if 3 or 4 dice remaining Pick up two 5’s or three 2’s if 2 or 3 dice remaining

24. Strategy Variations Exact criterion values compare to estimated strategy Variable strategies depend on opponent totals game completion player type safety first, risky, changeable

25. Computer Simulation Define decision vector list of criterion levels for continuing play given number of dice remaining current accumulated score Simulate turns Calculate output statistics

26. Preliminary Computer Simulation Results Decision vector # dice left 6 5 4 3 2 1 criterion level all 4500 1500 500 x y Average score for 100,000 turns y 200 300 400 200 512.188 512.770 510.068 x 300 512.917 512.925 510.283 400 505.150 505.513 503.254 Note: No pickup options in initial simulation program.

27. References Singer, Daniel. Zilch, http://www.cs.duke.ed/~des/other_stuff/zilch.html. August 25, 2008. Campo, Brian. Review: Farkle Dice by SmartBox Design, http://www.mytodayscreen.com/review-farkle-dice-by-smartbox-design/2. April 26, 2008 Sparks, Heather. Some Farkle probability questions, http://www.hisparks.com/farkle.pdf. August 25, 2008. Vertanen, Keith. Farkle Dice Game, http://www.keithv.com/cs161/project_description.html. August 30, 2008. Wikipedia. Farkle, http://www.en.wikipedia.org/wiki/Farkle. August 30, 2008.