Rolling dice for high scores is a common way to decide who moves first in a board game, but ties can demand one or more repeats. One group of mathematicians and computer scientists found this minor inconvenience simply unacceptable and began a hunt for a solution that lasted over a decade.
It was over dinner with a friend in 2012 that at Auburn University, Alabama, first discussed the idea of so-called Go First dice 鈥� a theoretical set of dice where every player could roll one die and have an equal chance of winning, but also a guarantee of not tying with anyone else.
It wasn鈥檛 long before he and Robert Ford at Dalton State College, Georgia, had that allowed two, three or four players to throw one of the dodecahedrons and get a statistically fair result 鈥� and a winner 鈥� every time.
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Harshbarger admits that it was really a solution looking for a problem, but that the idea of finding a set that worked with five players came up almost immediately. 鈥淧ractically speaking, it doesn鈥檛 matter, but mathematically, I think it鈥檚 a really beautiful problem,鈥� he says.
Harshbarger became a kind of steward for the problem in the years since, operating a website to track discoveries as about 20 computer scientists and mathematicians from around the world drifted in and out of work on the problem.
Theoretical sets of five Go First dice were discovered fairly soon after the set of four dice, but there were always problems: some had an unworkable number of varying sides, well into the hundreds. These worked on paper, but were too finicky to build and roll in practice. And solutions with varied numbers of sides weren鈥檛 just less elegant, but were also likely to lead people to suspect they were unfair, even if that wasn鈥檛 the case.
鈥淭his certainly got the ball rolling,鈥� says Harshbarger.
Better solutions gradually emerged, such as a set of matching 180-sided dice by Harshbarger himself, and a set of 120-sided dice from an Australian researcher. But it was Paul Meyer, a professional software engineer, who recently came on board and used a combination of seeking mathematical patterns in previous solutions and brute-force computer search to arrive at a workable design: a set of five 60-sided dice, which satisfied all the mathematical criteria but, crucially, could also be made and used in the real world.
鈥淭hey鈥檙e quite round, and they take their time to stop when you roll them, but they do work,鈥� says Harshbarger.
鈥淚鈥檝e rolled dice a lot to see who goes first when playing board games, so it really spoke to me. But when I saw that math problem, the way it was laid out, I was thinking, 鈥極h, this is really just a computational search problem, I bet I could write an algorithm鈥�,鈥� says Meyer. 鈥淚 didn鈥檛 really expect to solve it.鈥�
The search space Meyer had to work with was gargantuan 鈥� Harshbarger says there are more possible designs of five dice than there are atoms in the universe 鈥� but similarities and symmetries in previous solutions allowed him to reduce the space. His search wasn鈥檛 comprehensive; there could be smaller dice out there that satisfy the criteria, yet to be found.
There is also the question of a hunt for six fair Go First dice, where the search space balloons from the already unimaginably large once more. The researchers already know that a set with 360 sides works, but it is impractical in the real world, and Harshbarger thinks that quantum computers or AI might be needed to crack the problem.
But despite the elusive result working perfectly on paper, at the London School of Economics and Political Science, who wasn鈥檛 involved in the work, is still unconvinced that the reality of these 60-sided dice for five players will match the theory.
鈥淚t is definitely not very practical: rolling standard dice and re-rolling them among tied winners is easier than using special 60-side dice where even being fair is questionable,鈥� says von Stengel. 鈥淲ill they land on each side with equal chance 鈥� how precise and hard-wearing does the die need to be manufactured?鈥�
