Maximally Nontransitive Dice. Issue 5 (28th May 2018)

• Record Type: Journal Article
• Authors: Buhler, Joe; Graham, Ron; Hales, Al
• Is Part Of: American Mathematical Monthly. Volume 125:Issue 5(2018)
• Page Start: 387
• Page End: 399
• Publication Date: 2018-05-28
• Subjects: Primary 60C05; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2018.1427392
• Languages: English
• ISSNs: 0002-9890

Abstract

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.