Efron's Dice

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A set of four nontransitive Dice such that the probabilities of A winning against B, B against C, C against D, and D against A are all 2:1. A set in which ties may occur, in which case the Dice are rolled again, which gives Odds of 11:6 is

$\begin{figure}\begin{center}\BoxedEPSF{EfronsDiceModified.epsf scaled 700}\end{center}\end{figure}$

See also Dice, Sicherman Dice

References

Gardner, M. ``Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference.'' Sci. Amer. 223, 110-114, Dec. 1970.

Honsberger, R. ``Some Surprises in Probability.'' Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 94-97, 1979.