A mathematical puzzle posed around 2010 has resulted in a set of five 60-sided dice that can establish a fair playing order for up to five people. Mathematician Eric Harshbarger of Auburn University and his collaborators spent about 15 years searching for a practical solution, tells Live Science.
The original challenge came from board game designer James Ernest. He wanted dice that players could select and roll once, with everyone having an equal probability of going first. The system also had to work fairly for any subset of players without ties or rerolls.
Avoiding ties was relatively straightforward because each die could carry different numbers. The difficulty was distributing those numbers so every combination of selected dice produced equal probabilities. Early work by Harshbarger and mathematician Robert Ford produced three six-sided dice for three players. Ford later developed a four-player solution using 12-sided dice.
Researchers then discovered a more remarkable property. The dice did not merely select the first player. Every possible ordering of players occurred with equal probability, a characteristic the team calls permutation fairness.
Creating a practical five-player version proved much harder. The possible arrangements numbered around 10^128, making brute-force computation unrealistic. Mathematical patterns and symmetries helped narrow the search, but early solutions required impractical dice with too many sides.
A breakthrough arrived in 2023 when Canadian software engineer Paul Meyer analyzed patterns in the four-player solution and developed a program that found a workable configuration.
The result is five 60-sided dice containing every integer from 1 through 300 exactly once. Any subset of players can select dice and receive equal odds.
Harshbarger later built giant wooden versions from pine, poplar, oak, walnut, and mahogany for Auburn University, turning the solution into a physical demonstration of an easily understood but remarkably difficult mathematical problem.
