Sacks of dice with fair totals

TL;DR

This paper provides an explicit construction method for all fair sacks of dice, which are sets of independent dice with possibly unequal sides, where all total sums are equally likely, extending over 60 years of prior research.

Contribution

It introduces a new explicit construction approach for all fair sacks of dice, broadening understanding beyond previous restrictions on die order.

Findings

Explicit construction for all fair sacks of dice

Extension of known results over 60 years

Broadens the class of fair dice sets that can be constructed

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.