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arXiv · 2304.08501

A Fair Shake: How close can the sum of $n$-sided dice be to a uniform distribution?

Shamil Asgarli·Michael Hartglass·Daniel Ostrov·Byron Walden
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Abstract

Two possibly unfair $n$-sided dice, both labelled $1, 2, \ldots, n$, are rolled, and the sum is recorded. How should the dice's sides be weighted so that the resulting sum is closest to the uniform distribution on $2, 3, \ldots, 2n$? We answer this question by explicitly identifying the optimal pair of dice. This resolves a question raised by Gasarch and Kruskal in 1999 in a surprising way. We present additional results for the case of more than two possibly unfair $n$-sided dice and for the hypothetical case where the weights on each die are permitted to be negative, but must still sum to one.

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Shamil Asgarli, Michael Hartglass, Daniel Ostrov, Byron Walden. 2023-08-02. A Fair Shake: How close can the sum of $n$-sided dice be to a uniform distribution?. https://arxiv.org/abs/2304.08501

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