Nontransitive Dice

Sounds pretty scientific. So what exactly are nontransitive dice?

First I'll begin by defining transitive.

Here's an obvious fact: if x > y and y > z, then x > z, right?
Or to give an example in plain english, lets say I have more money than Herbert. Also, Bill Gates has more money than I do. Therefore, Bill Gates also has more money than Herbert.

In each of these cases, the conclusion follows because the greater-than relation ">" is transitive.

Now lets look at a nontransitive example from real life. Let's say Lee, Danny, and Chris decide to have a chess tournament. Danny wins against Lee, Lee wins against Chris, and Chris wins against Danny. Who is the best chess player? There is no clear victor because the method of measuring the best player (i.e. the "best-player" relation) is not transitive.

So that brings us to the main topic: nontransitive dice. Lets say we have four dice. Each die has six sides, and each die is numbered as follows:

Numbers on die
A 01, 02, 03, 09, 10, 11
B 00, 01, 07, 08, 08, 09
C 05, 05, 06, 06, 07, 07
D 03, 04, 04, 05, 11, 12

Now lets say we are going to have a little contest. You pick any one of these four dice. Then I will pick one. We each roll our die and whoever gets the highest number wins. (If there is a tie, we roll again to determine the winner.) Does this sound fair? Think about it.

It turns out this contest is not fair at all. No matter what die you pick, I can pick one that will beat yours about 66% of the time!
For the dice listed above, A beats B, B beats C, C beats D, and D beats A!

If this sounds wrong, it is probably because you assumed that the relation of "winning" was transitive between pairs of dice. However, this is not the case.

For an example, lets say you choose die B. To gain the advantage, I will choose die A. The 36 possible results of us rolling the dice are shown in the chart below. X indicates a win for me, O indicates a win for you, and T indicates a tie.

Your roll (die B)
My roll (die A)
01 X T O O O O
02 X X O O O O
03 X X O O O O
09 X X X X X T
10 X X X X X X
11 X X X X X X

So, out of 34 possible combinations (we don't count the tie games) with each set of rolls equally likely, I will win 22 times - that's 64.7%. Pretty good odds for me, aren't they? A similar chart can be constructed for any pair of dice, and in particular, if you choose the dice as listed above, the probability chart will show that you will win about 66% of the time.

Here are two more sets of four dice that also work. Once again, A beats B, B beats C, C beats D, and D beats A.

Numbers on die
A 00, 00, 04, 04, 04, 04
B 03, 03, 03, 03, 03, 03
C 02, 02, 02, 02, 06, 06
D 01, 01, 01, 01, 05, 05

Numbers on die
A 02, 03, 03, 09, 10, 11
B 00, 01, 07, 08, 08, 08
C 05, 05, 06, 06, 06, 06
D 04, 04, 04, 04, 12, 12

And for convience, I have created some gif files of these dice that may be folded and taped together. Here they are:

Set 1

Set 2

Set 3


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