On a Roll Again: Analysis of a Dice Removal Game

Suppose we have $n$ dice, each with $s$ faces (assume $s\ge n$). 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 $n,s\in \mathbb{N}\setminus \{0\}$ such that $s\ge n$, let $Y_n^s$ be the random variable for the number of turns to finish the game rolling $n$ dice with $s$ faces. We find recursive and non-recursive solutions for $\mathbb{E}(Y_n^s)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(Y_n^s)$, and bounds for both values. Moreover, we show that $Y_n^s$ 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.