Maximally Nontransitive Dice

Abstract

We construct arbitrarily large sets of dice with some remarkable nontransitivity properties. In a sense made precise later, each set exhibits all possible pairwise win/loss relationships by summing different numbers of rolls. The proof of this fact relies on an asymptotic formula for the difference between the median and mean of sums of multiple rolls of dice. This formula is a consequence of a suitable Edgeworth series (an asymptotic refinement of the central limit theorem), for which we give a detailed sketch of a proof in the final section.

MSC:

• Primary 60C05

ACKNOWLEDGMENTS

We thank Richard Arratia, Persi Diaconis, Anthony Gamst, Larry Goldstein, Fred Kochman, and Sandy Zabell for interesting and useful conversations.

Additional information

Notes on contributors

Joe Buhler

JOE BUHLER received his Ph.D. from Harvard University, and taught at Reed College before becoming the Deputy Director of the Mathematical Sciences Research Institute (MSRI) in 1999, and then the Director of the Center for Communications Research, San Diego, in 2004, succeeding Al Hales. One of his avocations is juggling, including both the practice thereof, as well as writing mathematical papers on juggling with Ron Graham.

Ron Graham

RON GRAHAM received his Ph.D. from the University of California, Berkeley, and worked at AT&T Bell Labs, where he was the Director of Information Sciences and then Chief Scientist, until he joined the faculty at the University of California, San Diego, in 1999, where he is currently Distinguished Research Professor. He served as MAA President from 2003–2004.

Al Hales

AL HALES received his Ph.D. from Caltech and was a Postdoctoral Researcher at Cambridge University and Harvard University before he joined the Mathematics Department at the University of California, Los Angeles in 1966. He served as the Department Chair, UCLA during 1988–1991, and became the Director of the Center for Communications Research, San Diego, 1992.