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 \(1, 2, \ldots, n\), 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.