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A decades-old challenge from the board‑game world has a neat solution: a mathematician network has produced a set of five specially numbered 60‑sided dice that guarantee a perfectly fair turn order for any number of players, with no ties or rerolls. The result grew from a casual question around 2010 and more than a decade of incremental work, capped by a key computational discovery in 2023.
A simple rule, a complicated demand
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The original idea was straightforward: each player grabs one die, everyone rolls, and the highest roll goes first. But the constraint that made the problem hard was stronger — the fairness had to hold for any subset of players who might show up.
That meant the dice could not just avoid ties; the distribution of numbers across faces had to ensure equal odds whether two people rolled or five. Early attempts solved small cases, but extending those patterns proved stubborn.
Early progress: small sets, surprising properties
After the question was raised at a gaming convention, mathematician Eric Harshbarger began working with collaborators. Within weeks he and Robert Ford found a three‑player solution: three standard six‑sided dice collectively carrying the numbers 1–18 arranged so each player had an equal chance of coming out on top. Ford later derived a fair four‑player set using four 12‑sided dice.
Harshbarger turned those designs into physical sets, etching numbers onto blank 12‑sided dice and selling them to enthusiasts. Retailers later offered the four‑player product for wider distribution.
As the team examined the behavior of these dice, they noticed a deeper feature: the configurations did more than pick a first player. They produced every possible turn order with equal probability. The researchers adopted the term permutation fairness to describe that property — every ordering of the players is equally likely.
When five players became astronomically hard
Scaling up to five players exploded the search space. The number of possible number assignments is so vast that brute‑force checking is impractical — the team describes it as far larger than the number of atoms in the universe. That forced them to look for mathematical shortcuts: symmetries, patterns and other reductions to prune the search.
Even with those tools, many candidate solutions demanded dice that would be impractical to hold or manufacture. The goal remained a set small enough for ordinary players to buy and roll.
The computational key and the final set
In mid‑2023 a Canadian software engineer, Paul Meyer, contacted Harshbarger with a program that exploited patterns found in the four‑player data. His code produced a valid arrangement: five 60‑sided dice — hexecontahedrons — engraved so that every face across the set carries a unique integer from 1 to 300.
Harshbarger checked Meyer’s configuration and confirmed it met the team’s strict conditions: no ties, equal chances for any subset of players, and full permutation fairness. The solution is compact enough to manufacture and use at a game table.
From workshop models to public display
With a working five‑player design in hand, Harshbarger returned to his woodshop to create oversized replicas. He carved five giant 60‑sided sculptures, each from a different wood, and installed them in Auburn University’s new mathematics building, which opened this fall.
- Pine
- Poplar
- Oak
- Walnut
- Mahogany
Harshbarger and his colleagues hope the pieces draw attention to mathematics that is easy to explain but hard to solve. The set of five 60‑sided dice answers a playful question about fairness while illustrating how persistent curiosity, handcrafted experimentation and targeted computation can combine to close a long‑running problem.
