appearing exactly twice, or to a set of icosahedral (20-sided) dice, with each number appearing four times. This eliminates the need for a "reroll" face. This solution was discovered by Jon Chambers, an Australian Pre-Service Mathematics Teacher. Four players A four-player set has not yet been discovered, but it was proved that such a set would require at least 19 dice. Intransitive 4-sided dice Tetrahedra can be used as dice with four possible results. Set 1 A: 1, 4, 7, 7 B: 2, 6, 6, 6 C: 3, 5, 5 ,8 P(A > B) = P(B > C) = P(C > A) = The following tables show all possible outcomes: In "A versus B", A wins in 9 out of 16 cases. In "B versus C", B wins in 9 out of 16 cases. In "C versus A", C wins in 9 out of 16 cases. Set 2 A: 3, 3, 3, 6 B: 2, 2, 5, 5 C: 1, 4, 4, 4 P(A > B) = P(B > C) = , P(C > A) = 9/16 Intransitive 12-sided dice In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. The points on each of the dice result in the sum of 114. There are no repetitive numbers on each of the dodecahedra. Miwin’s dodecahedra (set 1) win cyclically against each other in a ratio of 35:34. The miwin’s dodecahedra (set 2) win cyclically against each other in a ratio of 71:67. Set 1: Set 2: Intransitive prime-numbered 12-sided dice It is also possible to construct sets of intransitive dodecahedra such that there are no repeated numbers and all numbers are primes. Miwin’s intransitive prime-numbered dodecahedra win cyclically against each other in a ratio of 35:34. Set 1: The numbers add up to 564. Set 2: The numbers add up to 468. See also Blotto games Freivalds' algorithm Nontransitive game Rock paper scissors Condorcet's voting paradox References Sources External links MathWorld page Ivars Peterson's MathTrek - Tricky Dice Revisited (April 15, 2002) Jim Loy's Puzzle Page Miwin official site (German) Open Source nontransitive dice finder Non-transitive Dice by James Grime mgf.winkelmann Miwins intransitive Dodekaeder'' Maths Gear Conrey, B., Gabbard, J., Grant, K., Liu, A., & Morrison, K. (2016). Intransitive dice. Mathematics Magazine, 89(2), 133-143. Awarded by Mathematical Association of America Timothy Gowers' project on intransitive dice Probability theory paradoxes Dice