In their paper, posted online in late November 2022, a key part of the proof involves showing that, for the most part, it doesn’t make sense to talk about whether a single die is strong or weak. Buffett’s dice, none of which is the strongest of the pack, are not that unusual: If you pick a die at random, the Polymath project showed, it’s likely to beat about half of the other dice and lose to the other half. “Almost every die is pretty average,” Gowers said.
The project diverged from the AIM team’s original model in one respect: To simplify some technicalities, the project declared that the order of the numbers on a die matters—so, for example, 122556 and 152562 would be considered two different dice. But the Polymath result, combined with the AIM team’s experimental evidence, creates a strong presumption that the conjecture is also true in the original model, Gowers said.
“I was absolutely delighted that they came up with this proof,” Conrey said.
When it came to collections of four or more dice, the AIM team had predicted similar behavior to that of three dice: For example, if A beats B, B beats C, and C beats D, then there should be a roughly 50-50 probability that D beats A, approaching exactly 50-50 as the number of sides on the dice approaches infinity.
To test the conjecture, the researchers simulated head-to-head tournaments for sets of four dice with 50, 100, 150, and 200 sides. The simulations didn’t obey their predictions quite as closely as in the case of three dice but were still close enough to bolster their belief in the conjecture. But though the researchers didn’t realize it, these small discrepancies carried a different message: For sets of four or more dice, their conjecture is false.
“We really wanted [the conjecture] to be true, because that would be cool,” Conrey said.
In the case of four dice, Elisabetta Cornacchia of the Swiss Federal Institute of Technology Lausanne and Jan Hązła of the African Institute for Mathematical Sciences in Kigali, Rwanda, showed in a paper posted online in late 2020 that if A beats B, B beats C, and C beats D, then D has a slightly better than 50 percent chance of beating A—probably somewhere around 52 percent, Hązła said. (As with the Polymath paper, Cornacchia and Hązła used a slightly different model than in the AIM paper.)
Cornacchia and Hązła’s finding emerges from the fact that although, as a rule, a single die will be neither strong nor weak, a pair of dice can sometimes have common areas of strength. If you pick two dice at random, Cornacchia and Hązła showed, there’s a decent probability that the dice will be correlated: They’ll tend to beat or lose to the same dice. “If I ask you to create two dice which are close to each other, it turns out that this is possible,” Hązła said. These small pockets of correlation nudge tournament outcomes away from symmetry as soon as there are at least four dice in the picture.
The recent papers are not the end of the story. Cornacchia and Hązła’s paper only begins to uncover precisely how correlations between dice unbalance the symmetry of tournaments. In the meantime, though, we know now that there are plenty of sets of intransitive dice out there—maybe even one that’s subtle enough to trick Bill Gates into choosing first.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.