Carl B. Allendoerfer Awards

“Intransitive Dice,” Mathematics Magazine, Volume 89, Number 2, April 2016, pages 133–143. Intuition suggests that transitivity should hold in matters of strength. This intuition fails spectacularly in an example described by Martin Gardner in 1970 and originally due to Bradley Efron a few years earlier. The example consists of four six-sided dice labeled A–D, with all faces having numbers belonging to {1, 2, 3, 4, 5, 6}, such that A beats B with probability 2/3, B beats C with probability 2/3, C beats D with probability 2/3, and D beats A with probability 2/3. The fundamental question asked in “Intransitive Dice” is: How rare is this? In other words, given a random set of dice, how likely is it that one could put them into a cycle that is intransitive? The question is a tantalizing one, and the authors deftly move from the concrete to the abstract in their search for the answer. The authors first look for intransitive triples of dice. The frequency of ties in the setting of six-sided dice may leave one unsatisfied, so the authors go on to consider triples of n-sided dice, allowing numbers in {1, . . . , n} and imposing the condition that the sum of the numbers on each die be n(n+1)/2. The authors conjecture that as n grows, the probability of a tie goes to 0, while the probability of an intransitive triple goes to 1/4. After giving some computational evidence for these conjectures, they prove that as the number of sides grows, the probability of an intransitive cycle when there are no ties is 1/4. The authors conclude by returning to the four-dice setting of the original example and taking n-sided dice where the sum of all numbers on each die is n(n+1)/2. They present both heuristic and computational evidence that as n grows the probability of an intransitive cycle approaches 3/8. Even more provocatively, they conjecture that for k such dice, the limiting value approaches an expression in k that in turn goes to 1 as k grows. As they write, “. . . our intuition that intransitive dice are rare and that larger sets are even rarer is completely unfounded. They are common for three dice and almost unavoidable as the number of dice grows.” It’s not exactly common for high school students to participate in a Math Circle that leads to a published article in a mathematics journal, but in the case of “Intransitive Dice” we have just that. The authors of this article hit on all of the important modes of mathematical research. They collect data, generalize patterns, look for conjectures, and even prove theorems. All of this is tied together in a fun article that touches on many areas of mathematics and keeps the reader engaged to the end.