I've talked before about non-transitive dice. We say that given a pair of dice X and Y, X beats Y if more than half the time when the pair is rolled X has a larger number face up than Y. It turns out one can construct dice A, B and C such that A beats B, B beats C, but C in fact beats A. This is a neat and weird property.

During a recent discussion I used non-transitive dice as an example of a counter-intuitive aspect of mathematics, I was pointed to an even weirder variant. Consider the following set of dice: A has sides (5,5,5,2,2,2), B has sides (4,4,4,4,4,1) and C has sides (6,3,3,3,3,3).

Here A beats B, B beats C and C beats A. But here's the really cool part: Let's say I roll two copies of A, two copies of B or two copies of C. Now things actually reverse! That is, a pair of Bs beats a pair of As and a pair of As beats a pair of Cs and a pair of Cs beats a pair of Bs.

This is a much more sensitive property than just non-transitive dice. Most sets of non-transitive dice will not have this property. We can also describe this sensitivity in a more rigorous fashion. Suppose we have a strictly increasing function f(x). That is, a function such that f(x) is greater than f(y) whenever x is greater than y. Now suppose we take a set of non-transitive dice and relable each value x with f(x). Then they will still be non-transitive. But, given a set of non-transitive, reversable dice, reversibility is not necessarily preserved by the f mapping. This reflects the much more sensitive nature of the reversible dice.

Here's a question I have so far been unable to answer: Is it possible to make a set of die which do an additional reversal? That is, is there a set of dices such rolling three copies the dice results in another reversal direction?

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## 4 comments:

Intuitively, I think the answer is yes. I'm playing with A=(12,12,2) B=(8,8,8) C=(16,3,3). If that doesn't work, I can always use 6 or 12 or 18 sides and re-jigger a single side til it works.

I just saw Moneyball, about the Oakland A's use of statistical analysis to produce a team with just as many wins as the Yankees with one-third the salary. It reminded me of this problem- the Yankees kept beating the A's in the post-season because they used all that extra to buy players in July at exorbitant prices because performance in April, May and June is a more reliable predictor of performance in September than last season's stats. I was especially reminded of this because the A's drafting strategy involves avoiding top-of-the-line players for more under-ranked above-average players. If you'll humor me, 162 dice As will "beat" 162 dice Ys, but 5 dice Ys will "beat" 5 dice As.

A=(12,12,12,12,1,1)

B=(10,10,10,7,7,7)

C=(18,18,4,4,4,4)

do a triple reversal.

This can probably be generalized to a construction of an "n-reversible" set of dice {A,B,C} with the following properties:

For all odd k<=n, k dice of type A beat k dice of type B, which in turn beat k dice of type C, which, counter-intuitively, beat k dice of type A.

For all even k<=n, k copies of B beat k copies of A, which beat k copies of C, which beat k copies of B.

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