Intransitive dice are well known in mathematical circles, constituting a paradox which runs counter to conventional transitive logic. For example, three dice designated A, B, and C each have numbers disposed on their six sides. The numbers are not the standard 1 through 6, but rather such numbers as:
Die A--1, 1, 4, 4, 4, 4; PA1 Die B--3, 3, 3, 3, 3, 3; and, PA1 Die C--2, 2, 2, 2, 5, 5.
If die A is rolled against die B, die A will produce a higher number (win) more than fifty percent of the time. Similarly, if die B is rolled against die C, die B will win more than fifty percent of the time. One would reasonably suspect then that since die A beat die B, and die B beat die C, that die A should also beat die C. This would be a transitive relationship. Surprisingly however, if die C is rolled against die A, die C will also win more than fifty percent of the time, thereby exhibiting the intransitive nature of the set of dice.
U.S. Pat. No. 5,133,559 shows several sets of intransitive dice which are applied to a casino dice game. The fifteen sets of three intransitive dice which posses the highest winning percentages are disclosed. The fifteen sets are used in the play of a casino game in which either single die or double die contests are offered.
Funkenbusch, W. W. in "Sheep Fleecing Dice", Journal of Recreational Mathematics, vol 15(3), 1982-1983, pp. 194-198, describes the application of several sets of intransitive dice to various wagering games.