Challenge Level

*This problem follows naturally from* X-Dice*, although it may be attempted independently*.

A company wishes to produce a set of three different 6-sided dice coloured Apple Green, Bright Pink and Cool Grey, called $A, B$ and $C$ respectively.

They are to be made with the following properties:

1. The faces are to be numbered using only whole numbers 1 to 6.

2. Some of the numbers 1 to 6 can be left out or repeated as desired on each dice.

3. Apple Green is expected to beat Bright Pink on a single roll

4. Bright Pink is expected to beat Cool Grey on a single roll.

5. Cool Grey is expected to beat Apple Green on a single roll.

Invent a set of such dice.

What is the probability that each of your dice wins or loses over each of the other dice?

Is it possible to create a totally fair set of such dice with $P(A> B) = P(B> C) = P(C> A)$?

NOTES AND BACKGROUND

These dice exhibit a property called non-transitivity. Other examples of non-transitivity are found in voting systems. You can read about this in our article Transitivity.