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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Famous quotes containing the words dice, set and/or players:
“The dice of God are always loaded. The world looks like a multiplication-table, or a mathematical equation, which, turn it how you will, balances itself.”
—Ralph Waldo Emerson (1803–1882)
“One set of messages of the society we live in is: Consume. Grow. Do what you want. Amuse yourselves. The very working of this economic system, which has bestowed these unprecedented liberties, most cherished in the form of physical mobility and material prosperity, depends on encouraging people to defy limits.”
—Susan Sontag (b. 1933)
“Will you see the players well bestowed? Do you hear, let them
be well used, for they are the abstracts and brief chronicles of the time. After your death you were better have a bad epitaph than their ill report while you live.”
—William Shakespeare (1564–1616)