On a Roll Again: Analysis of a Dice Removal Game

Suppose we have $n$ dice, each with $s$ faces (assume $s\geq n$). 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 $s \geq n$, 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 $\mathrm{Var}(Y_n^{s})$, 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.

💡 Research Summary

The paper introduces and rigorously analyses a novel stochastic game that we call the “dice‑removal game”. The game starts with n identical dice, each having s faces (with s ≥ n). In each round the player rolls all dice that are still in play; any die that shows the number equal to the current number of dice k is removed. The process repeats until no dice remain, and the random variable Yⁿˢ denotes the total number of rounds required.

Two complementary probabilistic models are developed. The first treats each die independently. Because at any round a die succeeds (i.e., shows k) with probability p = 1/s, the number of rounds needed for a single die to be removed follows a geometric distribution Geom(p). Consequently, the game‑ending round is the maximum of n i.i.d. geometric variables: Yⁿˢ = max{Z₁,…,Zₙ}. From the order‑statistics identity F_Y(y) =