1. Elementary Farkle Strategy Donald E. Hooley Bluffton University for the Miami University Mathematics Conference September 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 DiceScore DiceScore Each 1 100 Each 5 50 Three 1’s1000 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 kindthree times triplet Five of a kindfive times triplet Six of a kindten times triplet Straight2500 Three pairs1500 ref. wikipedia.org

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

6. Play 6 4 5 5 5 1 6 4 5 5 5 1

7. Play 2 4 4 5 6 5 2 4 4 5 6 5

8. Play Options Example. 1 – 2 – 3 – 3 – 3 – 5 1 – 2 – 3 – 3 – 3 – 5Options. 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 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 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 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 1 25.6667 2 50.4444 2 50.4444 3 86.8056.2778 3 86.8056.2778 4 141.3194.1574 4 141.3194.1574 5 215.5093.0772 5 215.5093.0772 6 308.8831*.0309 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 386.8056.2778.0556 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(hot|n)E(6) – P(f|n)x – P(f|n)x so 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 125.6667.3333 263.6088 250.4444.1111 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 1263.6088 never 1263.6088 never 2225.5835 never 2225.5835 never 3 402.9981 400 3 402.9981 400 4 1028.5233 1000 4 1028.5233 1000 5 3232.2041always 5 3232.2041always 6 21104.8667always 6 21104.8667always

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 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 leftE.G. less “5” E.G. less “1” 0 -44.6274 -94.6274 0 -44.6274 -94.6274 1 -26.6654 -76.6654 1 -26.6654 -76.6654 2 28.5624 -21.4376 2 28.5624 -21.4376 3 97.7247 47.7247 3 97.7247 47.7247 4193.7356143.7356 4193.7356143.7356

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 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 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 0 -76.6654 -121.4376 1 -21.4376 -52.2753 1 -21.4376 -52.2753 2 47.7247 43.7356 2 47.7247 43.7356 3143.7356-- 3143.7356-- Note: Three 3’s would never give positive E.G.

23. Summary of Elementary Approximate Strategy Throw all remaining if 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 Pick up a 5 or 1 if 3 or 4 dice remaining Pick up two 5’s or three 2’s if 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 Average score for 100,000 turns y 200 300 400 200 300 400 200512.188512.770510.068 x 300512.917512.925510.283 400505.150505.513503.254 400505.150505.513503.254 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. http://www.cs.duke.ed/~des/other_stuff/zilch.html Campo, Brian. Review: Farkle Dice by SmartBox Design, http://www.mytodayscreen.com/review-farkle-dice-by- smartbox-design/2. April 26, 2008 http://www.mytodayscreen.com/review-farkle-dice-by- smartbox-design/2 http://www.mytodayscreen.com/review-farkle-dice-by- smartbox-design/2 Sparks, Heather. Some Farkle probability questions, http://www.hisparks.com/farkle.pdf. August 25, 2008. http://www.hisparks.com/farkle.pdf Vertanen, Keith. Farkle Dice Game, http://www.keithv.com/cs161/project_description.html. August 30, 2008. http://www.keithv.com/cs161/project_description.html Wikipedia. Farkle, http://www.en.wikipedia.org/wiki/Farkle. August 30, 2008. http://www.en.wikipedia.org/wiki/Farkle