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

Abstract

Two possibly unfair $n$-sided dice, both labelled $1,2,\dots ,n$, 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 $2,3,\dots ,2n$? 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.

arXiv comment9 pages