Science1 publisher2 min readPublished
Five 60-sided dice settle a five-player turn order in a single roll
Eric Harshbarger and a group of mathematicians spent more than a decade hunting for five dice whose 300 faces make all 120 five-player turn orders equally likely. Auburn has now put a hand-carved set on display.
The Scientist · Science desk
What happened
- Harshbarger and a group of mathematicians began work in 2010 on dice that would produce a random turn order from a single roll, with no reroll needed when two players match.
- Auburn University is memorializing the result with a sculpture Harshbarger built, now on display in the school's new science, technology, engineering, math and agriculture complex.
- Harshbarger's goal, which he calls permutation fairness, is that every ordering of the players is equally likely, not merely that each player has an equal shot at first.
- He hand carved a three-foot-wide replica of each die in a different wood, pine, poplar, red oak, walnut and mahogany, racing to finish before the complex opened.
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Why it matters
- constraint The guarantee lives in the numbering and assumes each of the 60 faces comes up one time in 60. Mould tolerance and mass distribution can erode fairness that the design itself cannot enforce.
- capability A small manufacturer already makes and sells the sets, so five people who want a fair order in one roll can buy the object instead of waiting for someone to build it.
- decision Versions differ by the number of players, so anyone buying a set has to match it to the size of their table; the fairness that was worked out applies to the count the set was designed for.
Five 60-sided dice have 300 faces between them, and the set uses every integer from 1 to 300 exactly once [15]. No number appears twice, so two players cannot roll the same value, and play runs from the highest roll down to the lowest with no reroll [5]. The two-player version of the idea fits on a pair of coins: one reading 1 and 4, the other 2 and 3, which leaves each flipper a one-in-two chance of the higher number [13].
Getting first place right is the small half of the problem. Five players can be ordered 120 ways [16], and five 60-sided dice land in 60^5, or 777,600,000, equally likely combinations [17]. Fairness across orderings means exactly 6,480,000 of those combinations have to produce each of the 120 orders, which is 777,600,000 divided by 120 [18]. The condition nests: once the first player is fixed, each of the four who remain has to take second place a quarter of the time, and so on down [24]. All of that has to come out of where 300 numbers sit on 300 faces. Sets for three and four players came comparatively easily; the five-player case held out for years [8].
Eric Harshbarger, a lecturer at Auburn University [2], has been on it since 2010, and the account in Scientific American describes the work as more than a decade old [22].
"'Go First Dice' is actually a bit of a misnomer at this point," Harshbarger said. "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." [7]
His best earlier five-player set used 120-sided dice, so large and so close to spherical that he likens them to tennis balls [9]. Cutting each die to 60 faces halves the numbers in the set, from 600 to 300 [19]. A few years before the 60-face solution, according to Scientific American, carving such a shape would not have been feasible for him, let alone manufacturing it [23].
The sculpture happened because Harshbarger had built large objects before. "I actually made a living building sculptures and mosaics out of LEGO bricks," he said [11].
What to watch
- A published five-player construction using fewer than 60 faces per die would show the 60-face set is not the floor.
- Whether the method extends to six players, and how many faces per die that takes before a mold becomes impractical.
- Whether the full numbering of the five sculpture dice is published, so anyone can check the fairness claim independently.