# Non-Transitive Dice

If A > B and B > C, then it must be that A > C, right? Not always.

A 10-sided die (with an average roll of 5.5) will, on average, beat a 6-sided die (average roll: 3.5), which will, on average, beat a 4-sided die (average roll: 2.5). And it is of course true that a 10-sided die will, on average, beat a 4-sided die, which demonstrates transitivity. But non-transitive dice do not follow this rule.

The dice pictured above (available here) are designed so that Red will beat Blue most of the time, and Blue will beat Green most of the time, so you might expect that Red would beat Green on average, but in fact Green beats Red! This is true even though each die has an average roll of 3.5, the same as a normal die. Here are the faces of the dice:

• Red: 4 4 4 4 4 1
• Blue: 3 3 3 3 3 6
• Green: 2 2 2 5 5 5

Comparing the faces makes it easier to understand the outcome of the various matchups:

• Red vs. Blue: Most of the time Red will roll a 4 against Blue’s 3 and win; Blue wins with a 6, but Blue only has one 6.
• Blue vs. Green: Green will roll a 2 half the time and thus lose no matter what Blue rolls, plus whenever Blue rolls a 6, Blue will win.
• Green vs. Red: Green will roll a 5 half the time and thus win no matter what Red rolls, plus whenever Red rolls a 1, Green will win.

You can use non-transitive dice to challenge a friend to a game that you will always win. Simply have your opponent choose a die, then you choose a die, and play best out of ten rolls. You know which die beats which, so you will always win, even on subsequent rounds when your friend may choose the die that you just won with.

Other non-transitive games include Rock Paper Scissors and Rock Paper Scissors Lizard Spock.

This video explains a larger set of 5 non-transitive dice invented by James Grime: