On A Roll Again: Analysis of a Dice Removal Game

Abstract

Suppose we have n dice, each with s faces (assume s >= n). On the first turn, roll all of them, and remove from play those that rolled an n. Roll all of the remaining dice. In general, if at a certain turn you are left with k dice, roll all of them and remove from play those that rolled a k. The game ends when you are left with no dice to roll.

For n,s natural numbers greater than zero, such that s >= n, we study the number of turns to finish the game with n dice, each with s faces.

We find recursive and non-recursive solutions its for the expected value and variance, and bounds for both values. Moreover, we show that the number of turns can also be modeled as the maximum of a sequence of i.i.d. geometrically distributed random variables.

Although, as far as we know, this game hasn't been studied before, similar problems have.