On a Roll Again: Analysis of a Dice Removal Game
Francesco Camellini, Wissam Ghantous, Andrea M. Lanocita, Layna E. Mangiapanello, Steven J. Miller, Garrett Tresch, Elif Z. Yildirim
TL;DR
The paper studies $Y_n^s$, the number of turns required to finish a dice-removal game where at each turn $k$ dice are rolled and any die showing $k$ is removed, with $n$ dice and $s$ faces and $s\ge n$. It develops two equivalent modeling frameworks: a per-die approach where $Y_n^s$ is the maximum of $n$ i.i.d. Geom$(1/s)$ variables, and a Markovian approach describing absorption time in a finite-state Markov chain, deriving recursive and non-recursive expressions for both the expectation and variance. The main contributions are explicit non-recursive formulas for $\mathbb{E}(Y_n^{s})$ and $\mathrm{Var}(Y_n^{s})$, two recursive formulas for the Markov-based absorption time $\mathbb{E}(T_n^{s})$ and $\mathrm{Var}(T_n^{s})$, and practical bounds showing linear growth in $n$ and $s$ for the mean and quadratic growth for the variance. The results connect maximum-of-geometric distributions with Markovian duration analysis and yield bounds useful for applications in order-statistics contexts, networks, data structures, and bioinformatics where such maxima arise.
Abstract
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.