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 labeled 1 , 2 , … , 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 , … , 2 n ? 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.