My latest fascination: A set of three dice A,B,C is said to be non-transitive if, in a contest between any pair, probabilistically, A will beat B over half the time, B will beat C over half the time, and, oddly enough, C will beat A over half the time.

There are several examples of this using only the numbers 1 through 6, but the dice have things like 4 sides all with 3s, which seems like cheating. So, we are instead going to look at dice where every side of every die has a different number. It is clear (at least to me), that the numbers may be consecutive integers.

An example (using 6-sided dice): in the book Mathematical Fallacies, Flaws, and Flimflam, we find

A: 18, 9, 8, 7, 6, 5

B: 17, 16, 15, 4, 3, 2

C: 14, 13, 12, 11, 10, 1.

In this labeling, we see that A beats B 21/36 of the time. B beats C 21/36 of the time. C beats A 25/36 of the time.

I don’t like this. Why? Well, it isn’t fair! So I set out to find a set of fair, non-transitive dice. After a while:

A: 18, 14, 11, 7, 4, 3

B: 17, 13, 10, 9, 6, 2

C: 16, 15, 12, 8, 5, 1

A beats B beats C beats A with uniform probability 19/36 (as close to being 1/2 as it can be, by the way). This, I like. I then turned myself to the following question. What about dice with other than 6 sides? 5-sided:

A: 15, 11, 7, 4, 3

B: 14, 10, 9, 5, 2

C: 13, 12, 8, 6, 1

A beats B beats C beats A 13/25 of the time (again, barely over 1/2). 3-sided:

A: 9, 4, 2 9, 5, 1

B: 8, 6, 1 OR 8, 4, 3

C: 7, 5, 3 7, 6, 2

A beats B beats C beats A 5/9 of the time (again with the 1/2).

These were all done with a particular construction. I won’t go into that here. I will mention that it doesn’t work for 4-sided dice. This bugs me. I even went so far as to conjecture no fair non-transitive triple of tetrahedral dice existed. Oops:

A: 12, 10, 3, 1

B: 9, 8, 7, 2

C: 11, 6, 5, 4

A beats B beats C beats A 9/16 of the time.

This definitely does not work with 2-sided dice (otherwise known as coins), but I will at this point guess that for n>2, three n-sided dice exist which are both non-transitive and fair. My next goal is to explore whether or not we can construct fair non-transitive dice where the probability is farther from 1/2.

Thanks for reading.

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