Go First Dice for Five Players and Beyond.

Abstract

Abstract Before a game begins, the players need to decide the order of play. This order of play is determined by each player rolling a die. Does there exist a set of dice such that draws are excluded and each order of play is equally likely? For four players the solution involves four 12-sided dice, sold commercially as Go First Dice. However, the solution for five players remained an open question. We present two solutions. The first solution has a particular mathematical structure known as binary dice, and results in a set of five 60-sided dice, where every place is equally likely. The second solution is an inductive construction that results in one one 36-sided die; two 48-sided dice; one 54-sided die; and one 20-sided die, where each permutation is equally likely.