Mathematicians unveil the world’s fairest dice after 15-year quest

During a dinner at a gaming convention around 2010, board game designer James Ernest posed a deceptively simple question to his friend Eric Harshbarger: Could you design a set of dice so that each player in a group, whether two people or any number, could grab one, roll and have a perfectly equal shot at going first? The rule itself was straightforward. The challenge was making that outcome truly fair for every player, regardless of how many people sat at the table, with no ties and no rerolls.

Harshbarger, a mathematician at Auburn University in Alabama, did not have an answer that evening. Yet the question lingered. Over the following 15 years, he and a loose network of collaborators casually worked to crack what eventually became known as the “go first dice” problem.

Today, they have finally arrived at a solution: a set of five 60-sided dice, collectively engraved with every integer from 1 to 300, with no repeats. To commemorate the achievement, Harshbarger constructed five giant wooden replicas of these hexecontahedrons, each carved from a different type of wood. The sculptures are now on permanent display in Auburn’s new mathematics building.

The problem with dice

To grasp why this challenge proved so difficult, it helps to start with the simple part.

“The easy thing is to avoid ties; you just put different numbers on all the dice,” Harshbarger explained. “The problem comes in how you distribute those different numbers across the dice so that the probability is equal not only for the whole set but for any subset.”

That second condition is what makes the puzzle so formidable. Ernest’s request demanded that whether eight players rolled fairly or just three grabbed dice from the bag, the odds remained equal. The numbers on each die had to be arranged so that fairness held regardless of how many people showed up to play.

Harshbarger enlisted his childhood friend Robert Ford, a mathematician at Dalton State College. Within weeks, they had devised a three-player solution: numbers 1 through 18, spread across three standard six-sided dice in precisely the right arrangement. Ford later worked out, entirely by hand, a four-player solution using four 12-sided dice.

By 2012, Harshbarger was presenting talks about the four-player set at math conferences, and The Guardian covered the story. His inbox exploded. He began selling handmade sets from home, purchasing blank 12-sided dice, etching numbers onto each face with a laser cutter in his workshop, and inking every number by hand. At his peak, he was dropping off 30 envelopes at the post office several times a week, shipping sets to customers around the world.

But as the team probed deeper into the mathematics, they discovered something even more remarkable: the configurations determined not only who went first but also the entire turn order, with every possible sequence of players equally likely to appear.

“If four people rolled, the chance that players finished in the order A, B, C, D was just as likely as C, B, D, A, or any of those combinations,” Harshbarger noted. “‘Go first dice’ is actually a bit of a misnomer, because they do much more than just determine who goes first; they actually determine the ordering of all the players.”

This property, which the team calls permutation fairness, became the benchmark they pursued for every set of dice from that point on.

More combinations than atoms in the universe

Solving the four-player case was one thing. Tackling five players was an entirely different beast.

Mathematically, the team knew a five-player same-shape set was feasible. But locating it required searching an incomprehensibly vast space of possible number arrangements.

“We’re talking literally more than the number of atoms in the universe, like 10 to the 128th power combinations,” Harshbarger said. “There’s just no way, even if we had a billion billion years and all the computers, all the AI. We still couldn’t do it today.”

Brute force was out of the question. They needed mathematical shortcuts: symmetries and patterns to shrink the search space. Yet even with these tools, years passed without a practical solution. Every avenue led to the same obstacle: dice too large to hold, with too many sides to manufacture.

“My goal was a set for five players that can actually be manufactured, that you can hold in your hand, that board gamers could buy and roll,” Harshbarger explained. “Even for a problem that is silly and pointless as far as gameplay, just mathematically, can you buy these things and roll them, and they work? You can’t do that with 180-sided dice.”

Then, in mid-2023, Canadian software engineer Paul Meyer emailed Harshbarger out of the blue. He had been studying patterns in Harshbarger’s four-player data and had written a program to exploit them. He did not expect it to work right away.

It did. Meyer discovered a configuration of five 60-sided dice that satisfied every condition.

“I double-checked his work and went, ‘Oh my goodness; we’ve been searching for so long for this. This is amazing,'” Harshbarger recalled.

From workshop to gallery wall

The four-player set had long since been picked up by retailers, including Maths Gear in the U.K. and Math Art Fun in the U.S., so Harshbarger had ceased hand-making those years earlier. Now, the five-player version was real and small enough to manufacture.

But the story had one more twist. Auburn University was constructing a new home for its mathematics department and was seeking sculptural ideas to fill the space. Harshbarger proposed building giant versions of the five new dice out of wood. The department agreed.

He spent months in his wood shop, constructing five oversized dice. Each is crafted from a different type of wood: pine, poplar, oak, walnut or mahogany. The five sculptures now stand in Auburn’s new math building, which opened this fall.

For Harshbarger, the entire endeavor is about making mathematics impossible to walk past.

“When people see giant dice or little dice, they’re fascinated just by the geometry,” he said. “The hope is, they see these things and go, ‘Oh, this is math too, and this is interesting.’ Some of the most fun math problems are ones that are easily understood and not easily answered.”

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