Difficulty level:
Junior high school.

Intransitive Dice (or Nontransitive Dice)

When the three dice below are rolled:
In most rolls, Die A has a higher number than Die B.
In most rolls, Die B has a higher number than Die C.
In most rolls, Die C has a higher number than Die A.
Die A with sides marked 3, 5, 7. Die B with sides marked 2, 4, 9. Die C with sides marked 1, 6, 8.

This can happen because the dice do not have to be each higher than another all at the same time. The times that A is higher than B overlap some with the times that B is higher than C and some with the times that C is higher than A. This allows the total times that A is higher than B to be more than half the rolls. The same happens for B beating C and C beating A. Below is a more detailed examination.

Mathematicians call these intransitive dice or nontransitive dice.

Explanation

Each of the three numbers on a die comes up in one-third of the rolls. (The sides of the dice not visible in the image are copies of the visible sides.) Two dice together have nine combinations of two numbers that can come up. The tables below show those combinations.

A versus B
A beats B in 5 of 9 rolls.
Die B
2 4 9
D
i
e
A
3 A B B
5 A A B
7 A A B
B versus C
B beats C in 5 of 9 rolls.
Die C
1 6 8
D
i
e
B
2 B C C
4 B C C
9 B B B
C versus A
C beats A in 5 of 9 rolls.
Die A
3 5 7
D
i
e
C
1 A A A
6 C C A
8 C C C
Each table shows the nine combinations of two numbers that can be rolled. Each square contains A, B, or C to show which die wins that roll. For example, in the first table, five squares show Die A winning, with rolls 3-2, 5-2, 5-4, 7-2, and 7-4.

So A usually beats B, B usually beats C, and C usually beats A.

Other arrangements of numbers can increase some of the probabilities a little, but I chose this arrangement to display because the probability between each pair of dice is the same. It comes from Nathaniel Hellerstein by way of Ivars Peterson's "Math Trek," MAA Online, December 22, 1997.

You can buy intransitive dice and other curiousities at Grand Illusions.

Simulation

         one time.          times.
Last Roll Competing Pair Wins Rolls Win Percentage
Die A   A beat B?   0 0  
Die B   B beat C?   0 0  
Die C   C beat A?   0 0  

Copyright 1997 by Eric Postpischil.