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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.12262 |
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| _version_ | 1866908840268660736 |
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| author | Camellini, Francesco Ghantous, Wissam Lanocita, Andrea M. Mangiapanello, Layna E. Miller, Steven J. Tresch, Garrett Yildirim, Elif Z. |
| author_facet | Camellini, Francesco Ghantous, Wissam Lanocita, Andrea M. Mangiapanello, Layna E. Miller, Steven J. Tresch, Garrett Yildirim, Elif Z. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Roll Again: Analysis of a Dice Removal Game Camellini, Francesco Ghantous, Wissam Lanocita, Andrea M. Mangiapanello, Layna E. Miller, Steven J. Tresch, Garrett Yildirim, Elif Z. Probability 62G30 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. |
| title | On a Roll Again: Analysis of a Dice Removal Game |
| topic | Probability 62G30 |
| url | https://arxiv.org/abs/2512.12262 |