Looking at three dice, if you know that A beats B and B beats C, that seems like evidence that A is the strongest situations where C beats A should be rare. Do you have to contrive such examples carefully, or can you pick dice randomly and have a good shot at finding an intransitive set? What mathematicians didn’t know until recently was how common intransitive dice are. Mathematicians came up with the first examples of intransitive dice more than 50 years ago, and eventually proved that as you consider dice with more and more sides, it’s possible to create intransitive cycles of any length. “It’s not intuitive at all that should even exist,” said Brian Conrey, the director of the American Institute of Mathematics (AIM) in San Jose, who wrote an influential paper on the subject in 2013. Mathematicians say that such a set of dice is “intransitive.”
If Gates had chosen first, then whichever die he chose, Buffett would have been able to find another die that could beat it (that is, one with more than a 50% chance of winning).īuffett’s four dice (call them A, B, C and D) formed a pattern reminiscent of rock-paper-scissors, in which A beats B, B beats C, C beats D and D beats A. Gates had recognized that Buffett’s dice exhibited a curious property: No one of them was the strongest.
But after Gates examined the dice, he returned a counterproposal: Buffett should pick first.
Buffett offered to let Gates choose first, so he could pick the strongest die. These weren’t standard dice - they had a different assortment of numbers than the usual 1 through 6. Each would select one of four dice belonging to Buffett, and then they’d roll, with the higher number winning. As Bill Gates tells the story, Warren Buffett once challenged him to a game of dice.
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