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Zilch is a fun little dice game codified into an online game by Gaby <\/p>\n Beyond the scope of Gaby’s implementation of Zilch, the scoring rules of If you understand conditional probabilities, expectations, and can do a <\/a><\/p>\n I’ve found several blog postings where folks have offered probabilistic <\/a><\/p>\n Zilch is played with two players and six six-sided dice. (Though really Each player takes turns rolling the dice. The dice in a roll can be worth If no dice in a roll score, then the player loses all points accumulated If all dice in a roll score, the player gets to continue his turn with all A player may continue rolling again and again accumulating ever more points If a player ends three consecutive turns with a zilch, they not only lose The game ends when one player has banked a total of 10,000 points and all Scoring is as follows:<\/p>\n <\/a><\/p>\n The strategy presented will maximize your expected Zilch scores, but this My intuition is that when you’re in the lead you should play more <\/a><\/p>\n <\/a><\/p>\n I will start by showing how to maximize the expected points for a For my purposes, a Zilch turn has a state that may be completely defined by 1, 3, 3, 4, 4, 6<\/p>\n The turn state will then advance to (s=100, n=5). You actually have no Suppose with the remaining 5 dice you roll:<\/p>\n 1, 1, 2, 3, 5<\/p>\n Here you have three scoring dice: two 1s and a 5. You now have a choice Note that s includes not just the points taken from this roll, but also all My objective is to find the optimal turn play strategy that defines what to Let E(s, n) be the expected number of additional points you will gain for Suppose we somehow solve for E(s, n) and find that:<\/p>\n
\nVanhegan that can be played at
\nhttp:\/\/playr.co.uk\/<\/a>. Zilch is actually a
\nvariation of the game Farkle which goes by several other names including
\nZonk, 5000, 10000, Wimp Out,
\nGreed, Squelch and Hot Dice1<\/a><\/sup>. I’ve
\nworked out the strategy that maximizes your expected game score and wanted to
\nshare the analysis, my strategy finder software, and the strategy itself.
\nDepending on whether you have zero, one or two consecutive zilches from
\nprevious turns, three successively more conservative turn-play strategies are
\nrequired to maximize your long term average score. Using these three strategies
\nyou rack up an average of 620.855 points per turn, which is the best you
\ncan possibly do.<\/p>\n
\nFarkle vary from venue to venue and the strategies provided here do not
\ngenerally apply, but the analysis and the software do.<\/p>\n
\nlittle algebra, you should be able to follow along. If you’re just here to
\ntake the money and go pound someone in the game, you’ll need to at least read
\nand understand Strategy Formulation<\/a> before you try
\nto interpret the tables<\/a>.<\/p>\nContents<\/h2>\n
\nBackground<\/a><\/div>\nThe rules of Zilch<\/a><\/div>\nLimitations<\/a><\/div>\nAnalysis<\/a><\/div>\nStrategy formulation<\/a><\/div>\nFinding E(s,n) – the Zilch strategy function<\/a><\/div>\nMaximizing expected score across all turns<\/a><\/div>\nResults<\/a><\/div>\nThe optimal strategies<\/a><\/div>\nStrategy T0<\/sub>: playing with no consecutive zilches<\/a><\/div>\nStrategy T1<\/sub>: playing with one consecutive zilch<\/a><\/div>\nStrategy T2<\/sub>: playing with two consecutive zilches<\/a><\/div>\nSoftware<\/a><\/div>\nConclusions<\/a><\/div>\nReferences<\/a><\/div>\n<\/div>\nBackground<\/h2>\n
\nanalyses of various aspects of the game2<\/a>,3<\/a>,4<\/a><\/sup>, but none (that I’ve seen) find the game
\nstrategy that maximizes your expected points. It is possible
\nthat I’m the first to publish these solutions. If not, it was still a fun
\nproblem. I’ve always enjoyed software, algorithms, optimization,
\nand probabilities and this problem delves into all of these areas.<\/p>\nThe Rules of Zilch<\/h2>\n
\nthere’s nothing to stop you playing with more people, but this is not
\nsupported in the online game.)<\/p>\n
\npoints either individually or in combination. If any points are available
\nfrom the roll, the player must set aside some or all of those scoring dice,
\nadding the score from those dice to their point total for the turn. After
\neach roll, a player may either reroll the remaining dice to try for more
\npoints or may bank the points accumulated this turn (though you can
\nnever bank less than 300 points).<\/p>\n
\nthis turn and their turn is ended. This is called a zilch, a sorrowful event
\nindeed.<\/p>\n
\nsix dice. This is called a free roll and is guaranteed to brighten your
\nday.<\/p>\n
\nuntil he either decides to bank those points or loses them all to a zilch.<\/p>\n
\ntheir points from the turn but also lose 500 points from their banked game
\nscore. (This is the only way to lose banked points.) After a triple zilch,
\nyour consecutive zilch count is reset to zero so you’re safe from another triple
\nzilch penalty for at least three more turns.<\/p>\n
\nother players have had a final turn.<\/p>\n\n
\n A set of three 2s is worth 200 points.
\n A set of three 3s is worth 300 points.
\n A set of three 4s is worth 400 points.
\n A set of three 5s is worth 500 points.
\n A set of three 6s is worth 600 points.
\n Each extra die in a set doubles the value of the set. So four 4s are worth 800 points and six 1s are worth 8000.\n<\/li>\n
\nthis is why a 6 die roll is called a free roll: you can’t zilch when rolling
\n6 dice.)<\/li>\n
\nand two 3s you can either count the two 1s for 200 or use all six dice for
\nthree-pair and 1500 points — you can’t use the ones both ways for 1700
\npoints.)<\/li>\n<\/ul>\nLimitations<\/h2>\n
\nis not necessarily the same strategy that will let you reach 10,000 points in
\nthe fewest number of turns; and certainly falls short of giving a complete
\ngaming strategy that will maximize your chances of winning the game
\nparticular, the strategy considers neither your current overall score, nor
\nyour opponent’s score, nor the fact that the game ends when a player reaches
\n10,000 points (after the other player gets a final turn). All that I offer is
\na way to maximize your expected Zilch scores.<\/p>\n
\nconservatively; and when you’re behind you should play more aggressively.
\n(Though I think it a common mistake to be too aggressive too early when behind.)
\nConsider this extreme example. Let’s say you’re currently beating your
\nopponent 7500 to 1500 and it’s your turn. On your turn you rack up 2500
\npoints and are faced with the choice of either banking the 2500 or rolling
\nfive dice to go for more points. The strategy identified here advises you to
\nroll the five dice; but surely in this case it is better to bank the 2500,
\nputting you at the game goal of 10,000 points and forcing your opponent to try
\nto put out 8550 points in a single turn to steal the win away from you.<\/p>\nAnalysis<\/h2>\n
Strategy Formulation<\/h3>\n
\nparticular turn. Because of the three consecutive zilch rule, the strategy
\nthat actually maximizes the average points gained across all turns is
\ndifferent: it is possible to trade off some expected gain in those turns
\nwhere you have either zero or one consecutive zilches to reduce your zilch
\nprobability and more strongly avoid even getting into a turn where you are
\nfacing your third consecutive zilch. I will solve for this more complete
\nstrategy later, but for now let’s stick with maximizing the expected points
\nfor a single turn and just ignore how such a greedy strategy might negatively
\naffect the outcome of subsequent turns.<\/p>\n
\ntwo variables (s, n) where s is the number of points accumulated in the
\ncurrent turn, and n is the number of dice you are about to roll. At the
\nbeginning of a new turn, the turn state is (s=0, n=6). Let’s say for your
\nopening roll you throw:<\/p>\n
\nchoice here: you must always select at least one scoring die and since the 1
\n(worth 100 points) is the only scoring die, you must select it. Furthermore,
\nyou are not allowed to bank less than 300 points so you must roll the five
\nremaining dice.<\/p>\n
\nof turn states that you may enter:<\/p>\n\n
\npoints accumulated in previous rolls during this turn as well. It should be
\nclear that state B is better than A and state D is better than C. Of the
\nremaining three states (B, D and E) it’s not so obvious which is better. You
\nalso have the option of banking from either of states D or E (but not from B
\nsince you don’t have 300 points in that case). Obviously, banking from state
\nD is just plain dumb: if you’re going to bank you’ll do so from state E to
\nbank as many points as you can! That leaves you with four reasonable
\nchoices:<\/p>\n\n
\ndo in all such situations which when followed will maximize the expected
\nnumber of points for the entire turn starting from any given turn state.<\/p>\n
\nthe turn if you (perhaps non-optimally) roll while in state (s, n) but then
\nfollow the optimal turn strategy (which we hope to find) for all subsequent
\ndecisions in the turn. Note that E(s, n) includes not just the expected
\npoints for the upcoming roll, but all the expected points from all subsequent
\nrolls, if any, as dictated by chance and the optimal play strategy.<\/p>\n\n
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