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:
- 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.
- For the remaining dice, roll them and set aside those dice matching the number from step 1.
- 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.