Sitting down for game night often involves a major killjoy: choosing who goes first. Some games set obscure conditions—for instance, the game Arboretum lets the person who watered a plant most recently in real life take the first move. Auburn University lecturer Eric Harshbarger took a different approach, though: he set out on a multiyear quest to mathematically design infallible turn-order dice.
One way to decide the question is to have everyone roll a die and see who gets the highest number. Usually, though, this strategy introduces a key problem: if multiple people roll the same number, they must roll again. Back in 2010 Harshbarger and a group of mathematicians started to develop “Go First Dice,” which guarantee a random turn order with only one roll. He’s even partnered with a small manufacturer to make and sell them.
Now, more than a decade into the work, Auburn University is memorializing Go First Dice with a sculpture of them built by Harshbarger on display in the school’s new science, technology, engineering, and math (STEM) and agriculture complex.
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The Go First Dice come in different versions, depending on the number of players. The version in the sculpture is a set of five 60-sided dice that collectively display every number between 1 and 300 on their faces. Each player in a game rolls one die, and each die is unique. After every player has rolled once, they play in the order of the numbers they rolled, from highest number to lowest. Because there are no ties, there’s no need to reroll.
A mathematical challenge in designing these sets was to tile the numbers across the dice in a way that would make it equally likely for any player to, well, go first. And on top of that, as Harshbarger points out, you want any turn order to be equally likely, too. This means that each player has an even chance of going first, that the remaining players have an even chance of going second, and so on. Harshbarger calls this idea permutation fairness.
“‘Go First Dice’ is actually a bit of a misnomer at this point,” he says. “The dice we’re looking for these days actually have an equal probability for any ordering of the players, and this is much, much harder.”
Achieving “permutation fair” dice for five players was no easy feat, and Harshbarger wanted to commemorate it with a piece for the new complex. He hand carved a three-foot-wide replica of each of the five dice using a different type of wood—pine, poplar, red oak, walnut and mahogany—for each and raced to finish them in time for the complex’s opening.
“I actually made a living building sculptures and mosaics out of LEGO bricks,” Harshbarger says of his time before being a university lecturer. “So I just directly went to the head of the department and said, ‘Listen, if we want these things done, I have this idea for giant 60-sided dice.’”
A few years prior, it wouldn’t have been feasible for him to even carve such a structure, let alone manufacture the dice. Though he and his collaborators had come up with simpler sets of Go First Dice for three and four players, a solution that would achieve permutation fairness for five players eluded them for years. The best Harshbarger had come up with was a set of five 120-sided dice that were so large that they were almost too unwieldy and spherical to make a reality—he likens them to tennis balls.
To see why permutation fairness is so difficult, consider designing a pair of “Go First Coins” for two players: One coin has faces “1” and “4,” and the other has “2” and “3.” If each player flips one of the coins, each will have exactly a one-in-two chance of getting a higher number than the other and going first.
Though that sounds easy enough, the difficulty compounds quickly when more players are added. Remember that permutation fairness means every turn order has an even chance of happening. So a three-player set of dice not only needs to provide an even chance of each player going first but, after selecting the first player, it must give the two remaining players an even chance of going second. By the time you get to five-player dice, you essentially need not to ensure that each player goes first a fifth of the time and that all the nested problems under the surface are satisfied as well.
Mathematician Barry Cipra describes this compound difficulty. “One property is that you can use the dice not only to decide who goes first but also who goes second, third, fourth and fifth—again, with each possible result occurring with equal probability,” he says. “The other property is that if you have fewer than five players, they can each choose a die at random, and the result of their rolls still assigns equal probability to each possible ... result.”
Beyond the tricky math, Harshbarger also faced the problem of making the five-player Go First Dice feel like, well, a set of dice. Not all numbers of faces can be made into a regular die-shaped form. Although unwieldy sets were known to be possible, these weren’t easy to manufacture into rollable dice, to sell to enthusiasts or, for Harshbarger, to carve into nice-looking sculptures. This is where five-player Go First Dice sat for a while.
Then, in 2023, Google software engineer and math enthusiast Paul Meyer learned about the project from a video on the YouTube channel Numberphile and realized he had the know-how to make some progress. “I play a lot of board games,” Meyer says. “I have some math background.... I’m a software engineer.” With these skills combined, he got to work trying to build an algorithm to find a solution.
Go First Dice fall within the field of combinatorics, the mathematics of counting. In combinatorics, computers have been invaluable for searching through possible solutions to counting problems, but for this one, the scale becomes unmanageable too quickly. Meyer estimates the number of five-player dice combinations to crawl through totals around 10200, more than the number of atoms in the observable universe.
Meyer found patterns for Go First Dice by hand, hoping to scale up the patterns for four-player sets to get to five players. This work paid off, and he was able to cut down the number of possibilities to “maybe 100 billion, or up to a trillion different configurations,” he estimates. Within about 10 minutes of running his algorithm on these relatively few possibilities, he found an answer. That answer is what now stands tall at Auburn University.
While finding a simpler set of five-player dice with 30 sides hasn’t been ruled out yet, both Harshbarger and Meyer say they would be shocked if it could be done. So with the five-player setting fairly settled and immortalized, Harshbarger has turned his focus to the final frontier: six players (seven and beyond are provably infeasible). “We feel there’s a decent chance that six 120-sided dice could be found,” he says. “I don’t think there’s much of a market there..., but it would be a nice mathematical novelty.”