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Suppose that you have $n \geq 1$ standard 6-sided dice. If all the dice display the same number, we call this an $n$-Yahtzee.

Follow this algorithm:

  1. Roll all the dice. Set aside those dice with the highest mode (Example, for 8 dice if I roll 1 2 4 4 4 5 6 6, set aside the 4s). If two or more are tied, choose one of them. This number will be fixed.
  2. For the remaining dice, roll them and set aside those dice matching the number from step 1.
  3. Repeat step 2 until you have obtained an $n$-Yahtzee.

Question : What is the average number of rolls (depending on $n$) that it takes to achieve an $n$-Yahtzee?

There are many variants of this question I would also like to consider, judging from the interest received on this question.

  • 0
    If you wanted to complicate the calculation you might note that the second roll of your example might produce one 2 and four 3s to go with your saved three 4s: you might then decide to target 3s in future rather than the 4s.2012-12-10
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    Yes, I would like to answer the "non-optimal play" question though.2012-12-11
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    after your first roll the n-yahtzee time is the maximum time to make your point on any of the remaining die, and these times are i.i.d. geometric. I think the problem is no harder than figuring out the distribution of the highest mode, but ....2013-02-20
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    I love it this kind of problems! I will try to solve it, but I confess that I find rather low your bounty...I think that the problem deserves more "MSE-Coins".2013-03-28
  • 0
    @MatemáticosChibchas Originally there was no bounty. I asked this question in December and basically forgot about it until now!2013-03-28

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