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Yahtzee (aka "Nokia Dice") has the following rules:

The aim of the game is to score a maximum number of points. There are 13 hands on which you can get points, as well as a bonus. 5 dice are used in this game. Order of play for each hand is as follows.

Roll 1

Roll all 5 dice. You can keep any number of dice (0 to 5) and discard the rest. If you keep all the dice, omit roll 2 and 3.

Roll 2

Roll all the discarded dice. You can keep all 5 dice with you or discard any of them (just rolled, or pre-kept). If you keep all the dice, omit roll 3.

Roll 3

Roll all the remaining dice. You must take all the final 5 dice. You may not need all of them for scoring.

Scoring

There are 13 combinations for scoring. You must select a single combination for each hand. If you select a combination that you do not have, you get zero points for that hand. Once a combination is selected, it may not be selected again. A hand has no order, it does not matter which dice was acquired on which roll.

Combinations

a) 1s - Take the sum of all the 1s in your final hand (other dice are ignored)
b) 2s - Take the sum of all the 2s in your final hand (other dice are ignored)
c) 3s - Take the sum of all the 3s in your final hand (other dice are ignored)
d) 4s - Take the sum of all the 4s in your final hand (other dice are ignored)
e) 5s - Take the sum of all the 5s in your final hand (other dice are ignored)
f) 6s - Take the sum of all the 6s in your final hand (other dice are ignored)
g) 3x - If you have a same dice thrice (or more), take the sum of all 5 dice
h) 4x - If you have a same dice four times (or more), take the sum of all 5 dice
i) 2+3 - If you have a full house - 3 of one number and 2 of the other, take a fixed score of 25, irrespective of the numbers used
j) Mini straight - If you have 1234 or 2345 or 3456, as 4 of the 5 dice, take a fixed score of 30
k) Full straight - If you have 12345 or 23456, take a fixed score of 40
l) Jackpot - If all your 5 dice are the same, take a fixed score of 50
m) Chance - Take the sum of all the dice, irrespective of what they are

Bonus

If the sum of points obtained from options a) to f) (over 6 different hands) exceeds 64, collect a bonus of 35 points

Question

I know how to calculate the probability of a combination occurring at any scenario. What confuses me is which combinations to take early in the game, and which ones later. Should I take big ones right at the beginning, or save them for the end. To what extent should 3x and 4x be preferred over the a) to f), due to the bonus, as well as possibility of greater 3x and 4x sums in the future. Any tips or algorithms would be helpful.

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    You may find some useful results on the Internet by knowing this game is actually Yahtzee. Jan 30 '15 at 16:28
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    Also, the strategy for the single player game (maximize your score) is probably going to be significantly different from the strategy for the multi-player game (beat your opponent, but you don't actually care what your score is). Which one are you interested in? Jan 30 '15 at 16:30
  • Technically, this game is very slightly different from Yahtzee. The one defined in this question gives the bonus at scores exceeding 64, instead of scores that are at least 63. It also does not have the chance box to record the score for an arbitrary combination.
    – murgatroid99
    Jan 30 '15 at 16:48
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    My first thought was the game that is superior to Yahtzee in exactly four hundred billion different ways, Kizmet.
    – corsiKa
    Jan 30 '15 at 19:35
  • @murgatroid99 The description claims 13 scoring combinations, but then lists only 12. I'd be prepared to bet a small amount of money what the 13th combination is. Jan 30 '15 at 20:07
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I'm assuming you've read the Wikipedia article, and followed some of those links. If not, read it first.

Yahtzee has been deeply studied by a few people. The best known current techniques involve brute force computations over the entire state space. The state space (the set of all possible games) can be structured to be small enough that every location can be computed.

That said, there is no simple answer to your question. My suggestion to you, is to go to this page, and look at their deep simulation, and test yourself. They have brute force addressed this problem and it's an interesting read.

If you are interested and have a decent math background (undergraduate probability theory course, and an understanding of finite state space Markov Chains) you can read these papers: http://www.cs.loyola.edu/~jglenn/research/optimal_yahtzee.pdf http://www.mimuw.edu.pl/~pan/papers/yahtzee.pdf

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