Analysis of Farkle

Introduction

Games involving dice seem to be a theme so far this year.  Last weekend, my nephew introduced me to Farkle, an easy-to-learn game requiring just six dice and paper-and-pencil for scoring.  As usual, rather than simply enjoying playing the game, I wondered about optimal playing strategy and how to evaluate it.

Rules of the Game

The basic rules are pretty simple, although there are many different scoring variants (more on this later).  Two or more players take turns accumulating points by rolling the dice as follows.  A player starts by rolling all 6 dice; he then sets aside one or more of the dice worth some positive number of points, and either:

  1. Ends his turn, adding all points accumulated so far to his running total; or
  2. Re-rolls the remaining dice and sets aside one or more of them, at which point he may again either end his turn or re-roll the new, smaller remainder.

If at any point a roll contains no scoring dice, then the turn ends and the player loses all points accumulated during that turn (a “Farkle”).  Play continues until some player reaches 10,000 or more points, at which point each other player gets one more turn to try to beat his score.  The highest score wins the game.

How are the dice scored?  I will focus on the version of the game by Patch Products that we played (see here for a PDF rule sheet):

  • Two triplets (e.g., 1-1-1-2-2-2) are worth 2500 points.
  • Four of a kind and a pair (e.g., 1-1-1-1-2-2) are worth 1500 points.
  • A straight (1-2-3-4-5-6) is worth 1500 points.
  • Three pairs (e.g., 1-1-2-2-3-3) are worth 1500 points.
  • Six of a kind is worth 3000 points.
  • Five of a kind is worth 2000 points.
  • Four of a kind is worth 1000 points.
  • Three 1s are worth 300 points.
  • Three 2s are worth 200 points.
  • Three 3s are worth 300 points.
  • Three 4s are worth 400 points.
  • Three 5s are worth 500 points.
  • Three 6s are worth 600 points.
  • Individual 1s are worth 100 points each (unless they are used in any of the above scoring).
  • Individual 5s are worth 50 points each (unless they are used in any of the above scoring).

Finally, there is one more interesting rule: if a player is able to score all 6 dice– not necessarily in a single roll– he may continue accumulating points in his turn by re-rolling all of the dice again, setting aside scoring dice, re-rolling the remainder, etc.  As we will see, this rule is what makes Farkle a bit more challenging to analyze.

Computing Playing Strategy

[Edit: Matt Busche provides an excellent analysis of the game.  (Actually, he focuses on a similar game called Zilch, of which Farkle is a simpler variant.)  He even has an online strategy generator, customizable for many different scoring options.  But it was originally missing a couple of things that I was interested in: (1) it did not appear to support the 2500-point “two triplets” scoring option, and (2) beyond just the various expected values, I was also interested in the full probability distribution of possible scores in a turn.  Later edit: Matt has since updated his strategy generator to support both of these features.]

To evaluate Farkle strategy, we need just two functions.  First, the following function (in Mathematica) implements the scoring rules, computing the score of a given subset of dice.  Note the use of -infinity to simplify the convention that including any non-scoring dice always yields a score of zero:

score[dice_] := Total[Tally[dice] /. {
     {{_, 3}, {_, 3}} -> {2500},
     {{_, 4}, {_, 2}} | {{_, 2}, {_, 4}} -> {1500},
     {{_, 1}, {_, 1}, {_, 1}, {_, 1}, {_, 1}, {_, 1}} -> {1500},
     {{_, 2}, {_, 2}, {_, 2}} -> {1500},
     {x_, 6} :> {3000, 3000, 3000, 3000, 3000, 3000}[[x]],
     {x_, 5} :> {2000, 2000, 2000, 2000, 2000, 2000}[[x]],
     {x_, 4} :> {1000, 1000, 1000, 1000, 1000, 1000}[[x]],
     {x_, 3} :> {300, 200, 300, 400, 500, 600}[[x]],
     {x_, 
       2} :> {200, -Infinity, -Infinity, -Infinity, 100, -Infinity}[[
       x]],
     {x_, 
       1} :> {100, -Infinity, -Infinity, -Infinity, 50, -Infinity}[[x]]
     }] /. -Infinity -> 0

Next, we compute the expected value (i.e., the expected final accumulated score for the turn) of rolling n dice with a current accumulated score of s points… and subsequently playing to maximize the expected final score.  This expected value is a probability-weighted sum over all possible rolls of the dice, where for each roll we find the best of all possible “moves,” either stopping with the points accumulated so far, or (recursively) re-rolling some dice.

value[{s_, 0}] := value[{s, 0}] = If[s > $maxScore, s, value[{s, 6}]]
value[{s_, n_}] := value[{s, n}] = Module[{roll, weight, moves},
   Total[Map[
      (
        {roll, weight} = #;
        (* Consider all scoring dice and resulting game states. *)
        moves = DeleteCases[
          Map[{s + score[#], n - Length[#]} &, Subsets[roll]],
          {s, _}
          ];
        (* Pick best move, weighted by probability of this roll. *)
        weight*If[moves === {}, 0,
          Max[Select[First /@ moves, # >= $minScore &], value /@ moves]
          ]
        ) &,
      Tally[Sort /@ Tuples[Range[6], n]]
      ]]/6^n
   ]

There are a few interesting things going on here.  First, Mathematica’s automatic memoization means we can solve the problem “top-down,” so that the code reads essentially like the equations (or the cocktail-napkin pseudocode) describing the problem.

Second, note the $minScore and $maxScore parameters.  The minimum score merely allows handling the situation at the start of the game, where a player must score at least 500 points to “get on the board.”  The maximum score is more interesting, since it addresses the following problem with the recursive algorithm: what is the base case?

If scoring all 6 dice ended a player’s turn– or even if it allowed just a single additional “bonus” roll– then the problem is easy.  But since a player can repeatedly re-roll all 6 dice, as long as he continues to score all of them, there is in principle no limit on the points he can accumulate in a single turn.

However, since we are maximizing expected score, there is some sufficiently large score beyond which a player is no longer willing to risk rolling again and losing it all with a Farkle.  At worst, we can find this maximum score by trial and error; it turns out that this maximum score is 16,750 points!  It is an interesting open question whether we can compute this maximum directly, ahead of time, possibly with some constraints on the scoring options.

Results

What does the resulting strategy look like?  First, the following table simply shows the probability of a Farkle when rolling the given number of dice.  (Note that these probabilities are independent of any particular playing strategy.)

  • Roll 6 dice: P(Farkle) = 5/216 = 0.02315
  • Roll 5 dice: P(Farkle) = 25/324 = 0.07716
  • Roll 4 dice: P(Farkle) = 17/108 = 0.15741
  • Roll 3 dice: P(Farkle) = 5/18 = 0.27778
  • Roll 2 dice: P(Farkle) = 4/9 = 0.44444
  • Roll 1 die: P(Farkle) = 2/3 = 0.66667

Also, before describing the strategy itself, the following plot shows the effect of the strategy, as the distribution of possible scores for a turn.  Note that there is a very long tail of the distribution that is not shown, extending to the maximum possible score of 19,750 points… but the tail not shown makes up less than 0.8% of the total probability.  Interestingly, the overall probability of a Farkle on any given turn is nearly 0.2.

Probability distribution of number of points in a turn. There is a long tail extending to 19,750 points, but scores exceeding 3000 points occur with probability less than 0.8%.

Probability distribution of number of points in a turn. There is a long tail extending to 19,750 points, but scores exceeding 3000 points occur with probability less than 0.8%.

An important note on “optimality”: I am focusing here on the strategy that maximizes expected score, with $minScore set to zero.  This corresponds roughly to most of the turns “in the middle” of a game, i.e., after the first turn with the 500-point minimum, but also before the last turn(s) near the end of the game, where, for example, the real objective is not to maximize expected score, but to maximize the probability of exceeding the other player’s score.  The value function above is easily modified to compute such a strategy, for a specific end-game difference between scores.

Finally, the following table “encodes” the playing strategy, showing the expected final score assuming that the player rolls the given number of dice with the given current score.  For example, rolling all 6 dice at the start of a turn, the overall expected score is 542.063 points.

Score       6          5          4          3          2          1
=======================================================================
    0    542.063    300.378    218.304    182.650    172.474    200.898
   50    582.354    333.255    235.843    193.380    183.621    214.664
  100    623.033    370.062    261.926    205.487    194.933    228.841
  150    664.951    407.619    295.428    224.308    206.296    243.327
  200    708.096    445.427    331.827    254.316    220.526    257.929
  250    751.866    485.066    368.557    290.158    242.646    272.546
  300    795.711    526.678    405.708    326.018    269.822    287.214
  350    839.563    568.757    445.481    361.888    297.045    301.998
  400    883.719    610.847    487.460    397.764    324.302    316.933
  450    928.271    652.946    529.448    433.642    351.573    332.004
  500    973.325    695.049    571.445    469.529    378.846    347.142
  550   1018.699    737.546    613.450    505.431    406.120    362.294
  600   1064.152    781.081    655.457    541.355    433.416    377.448
  650   1109.613    825.039    697.465    577.291    460.769    392.606
  700   1155.075    868.998    739.473    613.230    488.175    407.892
  750   1200.562    912.959    781.484    649.168    515.604    423.388
  800   1246.787    956.921    823.496    685.108    543.036    438.996
  850   1293.543   1000.945    865.509    721.050    570.468    454.626
  900   1340.431   1046.017    907.523    756.992    597.901    470.256
  950   1387.322   1092.079    949.539    792.936    625.334    485.887
 1000   1434.215   1138.144    991.557    828.882    652.767    501.519
 1050   1481.109   1184.212   1033.577    864.830    680.201    517.150
 1100   1528.003   1230.282   1075.603    900.780    707.637    532.784
 1150   1574.897   1276.352   1117.633    936.735    735.074    548.423
 1200   1621.805   1322.422   1159.663    972.698    762.516    564.067
 1250   1668.733   1368.493   1201.694   1008.667    789.976    579.712
 1300   1715.669   1414.564   1243.725   1044.637    817.460    595.383
 1350   1762.605   1460.638   1285.757   1080.607    844.955    611.139
 1400   1809.691   1506.712   1327.790   1116.577    872.451    626.957
 1450   1857.144   1552.787   1369.826   1152.549    899.946    642.775
 1500   1904.598   1598.863   1411.863   1188.521    927.442    658.595
 1550   1952.055   1644.940   1453.900   1224.496    954.938    674.415
 1600   1999.514   1691.020   1495.940   1260.472    982.433    690.235
 1650   2046.975   1737.104   1537.982   1296.449   1009.929    706.056
 1700   2094.438   1783.191   1580.027   1332.428   1037.425    721.877
 1750   2141.900   1829.279   1622.078   1368.410   1064.921    737.698
 1800   2189.363   1875.369   1664.133   1404.398   1092.417    753.519
 1850   2236.826   1921.459   1706.191   1440.396   1119.920    769.341
 1900   2284.290   1967.549   1748.248   1476.401   1147.444    785.162
 1950   2331.755   2013.639   1790.306   1512.408   1174.996    801.023
 2000   2379.220   2059.732   1832.364   1548.416   1202.564    816.976
 2050   2426.916   2105.825   1874.424   1584.423   1230.134    833.002
 2100   2474.942   2151.919   1916.485   1620.432   1257.704    849.046
 2150   2523.073   2198.014   1958.548   1656.441   1285.275    865.090
 2200   2571.204   2244.110   2000.612   1692.453   1312.845    881.135
 2250   2619.337   2290.208   2042.677   1728.465   1340.416    897.179
 2300   2667.471   2336.307   2084.744   1764.479   1367.986    913.224
 2350   2715.605   2382.408   2126.814   1800.495   1395.557    929.269
 2400   2763.740   2428.511   2168.888   1836.512   1423.127    945.314
 2450   2811.875   2474.615   2210.964   1872.536   1450.698    961.359
 2500   2860.009   2520.718   2253.043   1908.568   1478.274    977.404
 2550   2908.144   2566.822   2295.122   1944.607   1505.870    993.449
 2600   2956.279   2612.926   2337.201   1980.648   1533.491   1009.528
 2650   3004.413   2659.029   2379.280   2016.689   1561.126   1025.691
 2700   3052.753   2705.133   2421.359   2052.730   1588.763   1041.921
 2750   3101.394   2751.236   2463.438   2088.771   1616.400   1058.166
 2800   3150.130   2797.340   2505.517   2124.811   1644.038   1074.411
 2850   3198.866   2843.444   2547.595   2160.852   1671.675   1090.657
 2900   3247.602   2889.547   2589.674   2196.893   1699.312   1106.902
 2950   3296.338   2935.651   2631.753   2232.934   1726.949   1123.147
 3000   3345.074   2981.755   2673.832   2268.975   1754.587   1139.392
 3050   3393.809   3027.858   2715.911   2305.016   1782.224   1155.638
 3100   3442.545   3073.962   2757.990   2341.057   1809.861   1171.883
 3150   3491.281   3120.065   2800.069   2377.098   1837.499   1188.128
 3200   3540.017   3166.169   2842.148   2413.139   1865.136   1204.374
 3250   3588.753   3212.273   2884.227   2449.179   1892.773   1220.619
 3300   3637.489   3258.376   2926.306   2485.220   1920.411   1236.864
 3350   3686.225   3304.480   2968.385   2521.261   1948.048   1253.110
 3400   3734.961   3350.583   3010.463   2557.302   1975.685   1269.355
 3450   3783.697   3396.687   3052.542   2593.343   2003.323   1285.600
 3500   3832.433   3442.791   3094.621   2629.384   2030.960   1301.846
 3550   3881.169   3488.894   3136.700   2665.425   2058.597   1318.091
 3600   3929.905   3534.998   3178.779   2701.466   2086.235   1334.336
 3650   3978.641   3581.102   3220.858   2737.507   2113.872   1350.582
 3700   4027.377   3627.205   3262.937   2773.547   2141.509   1366.827
 3750   4076.113   3673.309   3305.016   2809.588   2169.147   1383.072
 3800   4124.849   3719.412   3347.095   2845.629   2196.784   1399.318
 3850   4173.585   3765.516   3389.174   2881.670   2224.421   1415.563
 3900   4222.321   3811.620   3431.253   2917.711   2252.059   1431.808
 3950   4271.057   3857.723   3473.332   2953.752   2279.696   1448.054
 4000   4319.793   3903.827   3515.410   2989.793   2307.333   1464.299
 4050   4368.529   3949.930   3557.489   3025.834   2334.971   1480.544
 4100   4417.265   3996.034   3599.568   3061.875   2362.608   1496.789
 4150   4466.000   4042.138   3641.647   3097.915   2390.245   1513.035
 4200   4514.736   4088.241   3683.726   3133.956   2417.883   1529.280
 4250   4563.472   4134.345   3725.805   3169.997   2445.520   1545.525
 4300   4612.208   4180.448   3767.884   3206.038   2473.157   1561.771
 4350   4660.944   4226.552   3809.963   3242.079   2500.795   1578.016
 4400   4709.680   4272.656   3852.042   3278.120   2528.432   1594.261
 4450   4758.416   4318.759   3894.121   3314.161   2556.069   1610.507
 4500   4807.152   4364.863   3936.200   3350.202   2583.707   1626.752
 4550   4855.888   4410.967   3978.278   3386.243   2611.344   1642.997
 4600   4904.624   4457.070   4020.357   3422.283   2638.981   1659.243
 4650   4953.360   4503.174   4062.436   3458.324   2666.619   1675.488
 4700   5002.096   4549.277   4104.515   3494.365   2694.256   1691.733
 4750   5050.832   4595.381   4146.594   3530.406   2721.893   1707.979
 4800   5099.568   4641.485   4188.673   3566.447   2749.531   1724.224
 4850   5148.304   4687.588   4230.752   3602.488   2777.168   1740.469
 4900   5197.040   4733.692   4272.831   3638.529   2804.805   1756.715
 4950   5245.776   4779.795   4314.910   3674.570   2832.443   1772.960
 5000   5294.512   4825.899   4356.989   3710.610   2860.080   1789.205
 5050   5343.248   4872.003   4399.068   3746.651   2887.717   1805.451
 5100   5391.984   4918.106   4441.146   3782.692   2915.355   1821.696
 5150   5440.720   4964.210   4483.225   3818.733   2942.992   1837.941
 5200   5489.456   5010.313   4525.304   3854.774   2970.629   1854.186
 5250   5538.191   5056.417   4567.383   3890.815   2998.267   1870.432
 5300   5586.927   5102.521   4609.462   3926.856   3025.904   1886.677
 5350   5635.663   5148.624   4651.541   3962.897   3053.541   1902.922
 5400   5684.399   5194.728   4693.620   3998.938   3081.179   1919.168
 5450   5733.135   5240.832   4735.699   4034.978   3108.816   1935.413
 5500   5781.871   5286.935   4777.778   4071.019   3136.453   1951.658
 5550   5830.607   5333.039   4819.857   4107.060   3164.091   1967.904
 5600   5879.343   5379.142   4861.936   4143.101   3191.728   1984.149
 5650   5928.079   5425.246   4904.015   4179.142   3219.365   2000.394
 5700   5976.815   5471.350   4946.093   4215.183   3247.003   2016.640
 5750   6025.551   5517.453   4988.172   4251.224   3274.640   2032.885
 5800   6074.287   5563.557   5030.251   4287.265   3302.277   2049.130
 5850   6123.023   5609.660   5072.330   4323.306   3329.915   2065.376
 5900   6171.759   5655.764   5114.409   4359.346   3357.552   2081.621
 5950   6220.495   5701.868   5156.488   4395.387   3385.189   2097.866
 6000   6269.231   5747.971   5198.567   4431.428   3412.827   2114.112
 6050   6317.967   5794.075   5240.646   4467.469   3440.464   2130.357
 6100   6366.703   5840.178   5282.725   4503.510   3468.101   2146.602
 6150   6415.439   5886.282   5324.804   4539.551   3495.739   2162.848
 6200   6464.175   5932.386   5366.883   4575.592   3523.376   2179.093
 6250   6512.911   5978.489   5408.961   4611.633   3551.013   2195.338
 6300   6561.647   6024.593   5451.040   4647.674   3578.650   2211.583
 6350   6610.383   6070.697   5493.119   4683.714   3606.288   2227.829
 6400   6659.118   6116.800   5535.198   4719.755   3633.925   2244.074
 6450   6707.854   6162.904   5577.277   4755.796   3661.562   2260.319
 6500   6756.590   6209.007   5619.356   4791.837   3689.200   2276.565
 6550   6805.326   6255.111   5661.435   4827.878   3716.837   2292.810
 6600   6854.062   6301.215   5703.514   4863.919   3744.474   2309.055
 6650   6902.798   6347.318   5745.593   4899.960   3772.112   2325.301
 6700   6951.534   6393.422   5787.672   4936.001   3799.749   2341.546
 6750   7000.270   6439.525   5829.751   4972.042   3827.386   2357.791
 6800   7049.006   6485.629   5871.829   5008.082   3855.024   2374.037
 6850   7097.742   6531.733   5913.908   5044.123   3882.661   2390.282
 6900   7146.478   6577.836   5955.987   5080.164   3910.298   2406.527
 6950   7195.214   6623.940   5998.066   5116.205   3937.936   2422.773
 7000   7243.950   6670.044   6040.145   5152.246   3965.573   2439.018
 7050   7292.686   6716.147   6082.224   5188.287   3993.210   2455.263
 7100   7341.422   6762.251   6124.303   5224.328   4020.848   2471.509
 7150   7390.158   6808.354   6166.382   5260.369   4048.485   2487.754
 7200   7438.894   6854.458   6208.461   5296.409   4076.122   2503.999
 7250   7487.630   6900.562   6250.540   5332.450   4103.760   2520.245
 7300   7536.366   6946.665   6292.619   5368.491   4131.397   2536.490
 7350   7585.102   6992.769   6334.698   5404.532   4159.034   2552.735
 7400   7633.838   7038.872   6376.776   5440.573   4186.672   2568.981
 7450   7682.574   7084.976   6418.855   5476.614   4214.309   2585.226
 7500   7731.309   7131.080   6460.934   5512.655   4241.946   2601.471
 7550   7780.045   7177.183   6503.013   5548.696   4269.584   2617.716
 7600   7828.781   7223.287   6545.092   5584.737   4297.221   2633.962
 7650   7877.517   7269.390   6587.171   5620.777   4324.858   2650.207
 7700   7926.253   7315.494   6629.250   5656.818   4352.496   2666.452
 7750   7974.989   7361.598   6671.329   5692.859   4380.133   2682.698
 7800   8023.725   7407.701   6713.408   5728.900   4407.770   2698.943
 7850   8072.461   7453.805   6755.487   5764.941   4435.408   2715.188
 7900   8121.197   7499.909   6797.566   5800.982   4463.045   2731.434
 7950   8169.933   7546.012   6839.644   5837.023   4490.682   2747.679
 8000   8218.669   7592.116   6881.723   5873.064   4518.320   2763.924
 8050   8267.405   7638.219   6923.802   5909.105   4545.957   2780.170
 8100   8316.141   7684.323   6965.881   5945.145   4573.594   2796.415
 8150   8364.877   7730.427   7007.960   5981.186   4601.232   2812.660
 8200   8413.613   7776.530   7050.039   6017.227   4628.869   2828.906
 8250   8462.349   7822.634   7092.118   6053.268   4656.506   2845.151
 8300   8511.085   7868.737   7134.197   6089.309   4684.144   2861.396
 8350   8559.821   7914.841   7176.276   6125.350   4711.781   2877.642
 8400   8608.557   7960.945   7218.355   6161.391   4739.418   2893.887
 8450   8657.293   8007.048   7260.434   6197.432   4767.056   2910.132
 8500   8706.029   8053.152   7302.512   6233.473   4794.693   2926.378
 8550   8754.765   8099.255   7344.591   6269.513   4822.330   2942.623
 8600   8803.501   8145.359   7386.670   6305.554   4849.968   2958.868
 8650   8852.236   8191.463   7428.749   6341.595   4877.605   2975.113
 8700   8900.972   8237.566   7470.828   6377.636   4905.242   2991.359
 8750   8949.708   8283.670   7512.907   6413.677   4932.880   3007.604
 8800   8998.444   8329.774   7554.986   6449.718   4960.517   3023.849
 8850   9047.180   8375.877   7597.065   6485.759   4988.154   3040.095
 8900   9095.916   8421.981   7639.144   6521.800   5015.792   3056.340
 8950   9144.652   8468.084   7681.223   6557.841   5043.429   3072.585
 9000   9193.388   8514.188   7723.302   6593.881   5071.066   3088.831
 9050   9242.124   8560.292   7765.381   6629.922   5098.704   3105.076
 9100   9290.860   8606.395   7807.459   6665.963   5126.341   3121.321
 9150   9339.596   8652.499   7849.538   6702.004   5153.978   3137.567
 9200   9388.332   8698.602   7891.617   6738.045   5181.616   3153.812
 9250   9437.068   8744.706   7933.696   6774.086   5209.253   3170.057
 9300   9485.804   8790.810   7975.775   6810.127   5236.890   3186.303
 9350   9534.540   8836.913   8017.854   6846.168   5264.528   3202.548
 9400   9583.276   8883.017   8059.933   6882.208   5292.165   3218.793
 9450   9632.012   8929.120   8102.012   6918.249   5319.802   3235.039
 9500   9680.748   8975.224   8144.091   6954.290   5347.440   3251.284
 9550   9729.484   9021.328   8186.170   6990.331   5375.077   3267.529
 9600   9778.220   9067.431   8228.249   7026.372   5402.714   3283.775
 9650   9826.956   9113.535   8270.327   7062.413   5430.352   3300.020
 9700   9875.692   9159.639   8312.406   7098.454   5457.989   3316.265
 9750   9924.428   9205.742   8354.485   7134.495   5485.626   3332.511
 9800   9973.164   9251.846   8396.564   7170.536   5513.263   3348.756
 9850  10021.900   9297.949   8438.643   7206.576   5540.901   3365.001
 9900  10070.635   9344.053   8480.722   7242.617   5568.538   3381.246
 9950  10119.371   9390.157   8522.801   7278.658   5596.175   3397.492
10000  10168.107   9436.260   8564.880   7314.699   5623.813   3413.737
10050  10216.843   9482.364   8606.959   7350.740   5651.450   3429.982
10100  10265.579   9528.467   8649.038   7386.781   5679.087   3446.228
10150  10314.315   9574.571   8691.117   7422.822   5706.725   3462.473
10200  10363.051   9620.675   8733.196   7458.863   5734.362   3478.718
10250  10411.787   9666.778   8775.274   7494.904   5761.999   3494.964
10300  10460.523   9712.882   8817.353   7530.944   5789.637   3511.209
10350  10509.259   9758.986   8859.432   7566.985   5817.274   3527.454
10400  10557.995   9805.089   8901.511   7603.026   5844.911   3543.700
10450  10606.731   9851.193   8943.590   7639.067   5872.549   3559.945
10500  10655.467   9897.296   8985.669   7675.108   5900.186   3576.190
10550  10704.203   9943.400   9027.748   7711.149   5927.823   3592.436
10600  10752.939   9989.504   9069.827   7747.190   5955.461   3608.681
10650  10801.675  10035.607   9111.906   7783.231   5983.098   3624.926
10700  10850.411  10081.711   9153.985   7819.272   6010.735   3641.172
10750  10899.147  10127.814   9196.064   7855.312   6038.373   3657.417
10800  10947.883  10173.918   9238.142   7891.353   6066.010   3673.662
10850  10996.619  10220.022   9280.221   7927.394   6093.647   3689.908
10900  11045.355  10266.125   9322.300   7963.435   6121.285   3706.153
10950  11094.091  10312.229   9364.379   7999.476   6148.922   3722.398
11000  11142.827  10358.333   9406.458   8035.517   6176.559   3738.644
11050  11191.563  10404.436   9448.537   8071.558   6204.197   3754.889
11100  11240.299  10450.540   9490.616   8107.599   6231.834   3771.134
11150  11289.035  10496.643   9532.695   8143.640   6259.471   3787.380
11200  11337.771  10542.747   9574.774   8179.681   6287.109   3803.625
11250  11386.507  10588.851   9616.853   8215.721   6314.746   3819.870
11300  11435.243  10634.954   9658.932   8251.762   6342.384   3836.116
11350  11483.979  10681.058   9701.011   8287.803   6370.021   3852.361
11400  11532.715  10727.161   9743.090   8323.844   6397.658   3868.607
11450  11581.452  10773.265   9785.168   8359.885   6425.296   3884.852
11500  11630.188  10819.369   9827.247   8395.926   6452.933   3901.097
11550  11678.924  10865.472   9869.326   8431.967   6480.570   3917.343
11600  11727.660  10911.576   9911.405   8468.008   6508.208   3933.588
11650  11776.396  10957.680   9953.484   8504.049   6535.845   3949.833
11700  11825.132  11003.783   9995.563   8540.089   6563.482   3966.079
11750  11873.868  11049.887  10037.642   8576.130   6591.120   3982.324
11800  11922.604  11095.991  10079.721   8612.171   6618.757   3998.570
11850  11971.341  11142.094  10121.800   8648.212   6646.394   4014.815
11900  12020.077  11188.198  10163.879   8684.253   6674.032   4031.060
11950  12068.813  11234.301  10205.958   8720.294   6701.669   4047.306
12000  12117.549  11280.405  10248.037   8756.335   6729.306   4063.551
12050  12166.286  11326.509  10290.116   8792.376   6756.944   4079.797
12100  12215.022  11372.612  10332.195   8828.417   6784.581   4096.042
12150  12263.758  11418.716  10374.273   8864.458   6812.219   4112.287
12200  12312.494  11464.820  10416.352   8900.499   6839.856   4128.533
12250  12361.230  11510.923  10458.431   8936.540   6867.493   4144.778
12300  12409.967  11557.027  10500.510   8972.580   6895.131   4161.024
12350  12458.703  11603.131  10542.589   9008.621   6922.768   4177.269
12400  12507.440  11649.234  10584.668   9044.662   6950.406   4193.515
12450  12556.176  11695.338  10626.747   9080.703   6978.043   4209.760
12500  12604.913  11741.442  10668.826   9116.744   7005.680   4226.006
12550  12653.650  11787.545  10710.905   9152.785   7033.318   4242.251
12600  12702.386  11833.649  10752.984   9188.826   7060.955   4258.497
12650  12751.123  11879.753  10795.063   9224.867   7088.593   4274.743
12700  12799.859  11925.856  10837.142   9260.908   7116.230   4290.988
12750  12848.596  11971.960  10879.221   9296.949   7143.868   4307.234
12800  12897.333  12018.064  10921.300   9332.990   7171.505   4323.480
12850  12946.070  12064.168  10963.379   9369.031   7199.143   4339.726
12900  12994.808  12110.271  11005.458   9405.072   7226.780   4355.971
12950  13043.546  12156.375  11047.537   9441.113   7254.418   4372.217
13000  13092.283  12202.479  11089.616   9477.154   7282.055   4388.463
13050  13141.021  12248.582  11131.695   9513.195   7309.693   4404.709
13100  13189.758  12294.686  11173.774   9549.236   7337.330   4420.955
13150  13238.496  12340.790  11215.853   9585.277   7364.968   4437.201
13200  13287.234  12386.894  11257.932   9621.318   7392.605   4453.447
13250  13335.971  12432.997  11300.011   9657.359   7420.243   4469.693
13300  13384.709  12479.101  11342.090   9693.400   7447.880   4485.939
13350  13433.447  12525.205  11384.170   9729.441   7475.518   4502.185
13400  13482.185  12571.309  11426.249   9765.482   7503.155   4518.431
13450  13530.923  12617.413  11468.328   9801.523   7530.793   4534.677
13500  13579.661  12663.517  11510.407   9837.564   7558.431   4550.923
13550  13628.399  12709.621  11552.486   9873.605   7586.068   4567.169
13600  13677.137  12755.725  11594.565   9909.647   7613.706   4583.415
13650  13725.875  12801.829  11636.645   9945.688   7641.344   4599.661
13700  13774.613  12847.933  11678.724   9981.729   7668.982   4615.907
13750  13823.351  12894.037  11720.803  10017.770   7696.620   4632.154
13800  13872.091  12940.141  11762.883  10053.812   7724.258   4648.402
13850  13920.834  12986.245  11804.962  10089.853   7751.896   4664.649
13900  13969.577  13032.349  11847.041  10125.895   7779.534   4680.897
13950  14018.320  13078.453  11889.121  10161.936   7807.172   4697.145
14000  14067.063  13124.557  11931.200  10197.977   7834.810   4713.392
14050  14115.806  13170.662  11973.280  10234.019   7862.448   4729.640
14100  14164.549  13216.766  12015.359  10270.060   7890.087   4745.888
14150  14213.292  13262.870  12057.439  10306.102   7917.725   4762.136
14200  14262.035  13308.974  12099.518  10342.144   7945.364   4778.384
14250  14310.778  13355.079  12141.598  10378.186   7973.003   4794.633
14300  14359.524  13401.183  12183.678  10414.227   8000.642   4810.883
14350  14408.275  13447.288  12225.758  10450.269   8028.281   4827.134
14400  14457.026  13493.393  12267.838  10486.311   8055.920   4843.384
14450  14505.777  13539.497  12309.918  10522.353   8083.559   4859.634
14500  14554.528  13585.602  12351.997  10558.395   8111.198   4875.885
14550  14603.279  13631.707  12394.077  10594.436   8138.837   4892.135
14600  14652.030  13677.812  12436.157  10630.478   8166.476   4908.386
14650  14700.781  13723.917  12478.238  10666.520   8194.115   4924.636
14700  14749.533  13770.022  12520.318  10702.562   8221.755   4940.887
14750  14798.286  13816.128  12562.399  10738.604   8249.394   4957.139
14800  14847.039  13862.234  12604.480  10774.647   8277.033   4973.390
14850  14895.793  13908.340  12646.561  10810.689   8304.673   4989.641
14900  14944.546  13954.447  12688.642  10846.732   8332.312   5005.892
14950  14993.300  14000.554  12730.723  10882.775   8359.951   5022.144
15000  15042.054  14046.662  12772.805  10918.818   8387.591   5038.395
15050  15090.808  14092.769  12814.888  10954.862   8415.230   5054.646
15100  15139.562  14138.876  12856.970  10990.907   8442.870   5070.898
15150  15188.316  14184.984  12899.053  11026.953   8470.513   5087.149
15200  15237.070  14231.091  12941.136  11062.999   8498.159   5103.404
15250  15285.823  14277.198  12983.218  11099.045   8525.806   5119.671
15300  15334.600  14323.306  13025.301  11135.091   8553.454   5135.947
15350  15383.428  14369.413  13067.384  11171.137   8581.101   5152.223
15400  15432.255  14415.521  13109.467  11207.183   8608.749   5168.499
15450  15481.083  14461.628  13151.549  11243.229   8636.396   5184.775
15500  15529.910  14507.736  13193.632  11279.275   8664.044   5201.051
15550  15578.738  14553.843  13235.715  11315.322   8691.692   5217.327
15600  15627.566  14599.951  13277.798  11351.368   8719.339   5233.603
15650  15676.394  14646.059  13319.881  11387.414   8746.987   5249.880
15700  15725.225  14692.167  13361.964  11423.460   8774.635   5266.157
15750  15774.055  14738.280  13404.047  11459.506   8802.283   5282.434
15800  15822.886  14784.395  13446.131  11495.553   8829.931   5298.711
15850  15871.717  14830.510  13488.220  11531.599   8857.579   5314.988
15900  15920.547  14876.625  13530.308  11567.646   8885.227   5331.265
15950  15969.379  14922.740  13572.397  11603.692   8912.875   5347.543
16000  16018.211  14968.856  13614.485  11639.739   8940.524   5363.821
16050  16067.045  15014.974  13656.574  11675.785   8968.172   5380.099
16100  16115.879  15061.094  13698.663  11711.832   8995.821   5396.377
16150  16164.713  15107.217  13740.756  11747.879   9023.469   5412.656
16200  16213.548  15153.340  13782.853  11783.927   9051.118   5428.935
16250  16262.385  15199.465  13824.952  11819.979   9078.767   5445.215
16300  16311.224  15245.593  13867.053  11856.033   9106.416   5461.495
16350  16360.064  15291.724  13909.157  11892.091   9134.066   5477.776
16400  16408.906  15337.858  13951.264  11928.150   9161.715   5494.056
16450  16457.748  15383.997  13993.376  11964.213   9189.364   5510.337
16500  16506.591  15430.138  14035.496  12000.279   9217.013   5526.618
16550  16555.433  15476.280  14077.623  12036.357   9244.662   5542.899
16600  16604.276  15522.422  14119.753  12072.454   9272.321   5559.180
16650  16653.119  15568.564  14161.883  12108.565   9300.022   5575.461
16700  16701.961  15614.706  14204.012  12144.676   9327.778   5591.801
16750  16750.804  15660.847  14246.142  12180.787   9355.556   5608.333
16800  16799.646  15706.989  14288.272  12216.898   9383.333   5625.000

References:

Busche, Matthew, Maximizing Expected Scores in the Game of Zilch. [HTML]

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5 Responses to Analysis of Farkle

  1. Andy says:

    I’m confused by the chart. For example, with 250 points and one die left to roll, the EV of rolling is stated as 272.546. However, the EV of rolling the last die by itself is clearly 25, (100+50)/6. So this value should be at least 275. But if we get a 1 or 5, we get to keep rolling, essentially adding the value of starting a turn with zero and 6 dice to what we already have, another 542.063/3.

    • There are a couple of things missing here. First, you seem to be mixing expected *additional* score vs. expected *final* score. That is, 1/6 of the time we roll a 1 for an *additional* 100 points, 1/6 of the time we roll a 5 for an additional 50 points.. but the other 4/6 of the time, we *lose* our initial 250 points.

      Also, if we *do* roll a 1 or 5 and get to keep rolling, we will *not* have the same expected *additional* score of 542.063, since our decisions will get more conservative (since we are risking more than 250 points now, but 350 or 300). If we rolled a 1 and now have 350 points and 6 fresh dice, our new expected *final* score is 839.563 as in the table. If we rolled a 5 and now have 300 points and 6 fresh dice, our new expected score is 795.711. If we rolled anything else with our single die, we lose everything; so our overall expected value is 1/6×839.563 + 1/6×795.711 + 4/6×0, or 272.546.

      • Andy says:

        So if we instead had 300 and 1 die left, we’d stop rolling because 287.214 is less than 300, right? (Assume it’s the first turn of the game so catching up doesn’t matter).

  2. Pingback: Dice puzzle | Possibly Wrong

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