Sacks of dice with fair totals

Abstract

A fair sack is a finite set of independent dice, not required to be fair and allowed to have any number of sides, for which all totals are equally likely. These have been studied for over 60 years. Most results restrict the possible orders of dice in such a sack and almost no examples were known. Building on a rather different approach due to Gasarch and Kruskal, we give an explicit construction of all such sacks.

15 pages, 3 figures. Substantially revised to add a proof of the Gasarch-Kruskal Theorem, a guide to the constructions with a roster of all fair sacks for total 12, and figures illustrating the steps in the proof of the main theorem. To appear in Amer, Math. Monthly