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## 'Dicey Dice' printed from http://nrich.maths.org/

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 theĀ 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. You might also
wish to try the related problem

A Dicey Paradox.