Non-Transitive Dice (Efron Dice) by Bradley Efron

These first came to the public’s attention via Martin Gardner’s Mathematical Games column in Scientific American (December, 1970).  There, he discussed a set of four Efron Dice which were discovered by and named after, statistician Bradley Efron, for which: A > B > C > D > A … in the sense that each die beats the next one listed with a probability 2/3.

Effect: Ask your opponent to select any one of the four dice. You select another and both dice are thrown at the same time, a predetermined number of times, to see who gets the highest number on each throw, and hence wins the throw. In a game of “Best of Ten Throws” you will almost always have more wins. Invite the player to choose another dice – perhaps your “winner” – leaving you to select another for yourself and play again. Again you will win. Whichever die your opponent selects, your choice, in a longer run of ten or more throws, will always win.

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