Nontransitive Dice Set For Three Players
M. Oskar van Deventer introduced a set of seven dice (all faces with probability 1/6) as follows:
- A: 2, 2, 14, 14, 17, 17
- B: 7, 7, 10, 10, 16, 16
- C: 5, 5, 13, 13, 15, 15
- D: 3, 3, 9, 9, 21, 21
- E: 1, 1, 12, 12, 20, 20
- F: 6, 6, 8, 8, 19, 19
- G: 4, 4, 11, 11, 18, 18
One can verify that A beats B,C,E; B beats C,D,F; C beats D,E,G; D beats A,E,F; E beats B,F,G; F beats A,C,G; G beats A,B,D. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,
- G beats A,B; F beats A,C; G beats A,D; D beats A,E; D beats A,F; F beats A,G;
- A beats B,C; G beats B,D; A beats B,E; E beats B,F; E beats B,G;
- B beats C,D; A beats C,E; B beats C,F; F beats C,G;
- C beats D,E; B beats D,F; C beats D,G;
- D beats E,F; C beats E,G;
- E beats F,G.
Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents' dice.
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—Ralph Waldo Emerson (1803–1882)
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—Bible: Hebrew, Deuteronomy 17:14,15.
“The players have often mentioned it as an honour to Shakespeare, that in his writing, whatsoever he penned, he never blotted out [a] line. My answer hath been, “Would he had blotted a thousand.””
—Ben Jonson (c. 1572–1637)