# You pick first. I pick second.

Three dice. You choose one, I choose another. The higher number wins. Which one do you want?

A: 2, 2, 4, 4, 9, 9
B: 1, 1, 6, 6, 8, 8
C: 3, 3, 5, 5, 7, 7

## The best die depends on its opponent.

A beats B more often. B beats C more often. And C beats A more often. Whatever you choose, another die has an advantage against it.

In each of these matchups, the stronger die wins 20 of the 36 possible face pairings. Its chance is 5/9, or about 55.6% on each roll. The other die can still win a short match.

All three have a mean of 5. But the mean adds values, while this game counts wins. A towering 9 still earns just one point when it beats a 1.

## Open all 36 possibilities

Each face of the first die can meet each face of the second. All 36 pairings are equally likely with fair dice and independent rolls. Cells marked ✓ are wins for the row die.

### A / B

| / |1|1|6|6|8|8|
|---|---|---|---|---|---|---|
|2|✓|✓|·|·|·|·|
|2|✓|✓|·|·|·|·|
|4|✓|✓|·|·|·|·|
|4|✓|✓|·|·|·|·|
|9|✓|✓|✓|✓|✓|✓|
|9|✓|✓|✓|✓|✓|✓|

### B / C

| / |3|3|5|5|7|7|
|---|---|---|---|---|---|---|
|1|·|·|·|·|·|·|
|1|·|·|·|·|·|·|
|6|✓|✓|✓|✓|·|·|
|6|✓|✓|✓|✓|·|·|
|8|✓|✓|✓|✓|✓|✓|
|8|✓|✓|✓|✓|✓|✓|

### C / A

| / |2|2|4|4|9|9|
|---|---|---|---|---|---|---|
|3|✓|✓|·|·|·|·|
|3|✓|✓|·|·|·|·|
|5|✓|✓|✓|✓|·|·|
|5|✓|✓|✓|✓|·|·|
|7|✓|✓|✓|✓|·|·|
|7|✓|✓|✓|✓|·|·|

## Try it at a real table

Put number stickers on three ordinary dice, with two faces for each value. Let a friend choose first. After a few rounds, swap the order of choosing and explain your strategies. The advantage comes from the matchup. No loaded dice required.

## How it works

A mathematical model of ideal dice. Each face has probability 1/6. The computer chooses its stronger die before rolling; it does not change results after seeing them. The simulation uses a pseudorandom generator and can be replayed from the same address. Your observed score is not the exact probability.

NRICH, University of Cambridge, Non-Transitive Dice. We label the dice A, B and C and calculate probabilities by enumerating every face pairing. This property is called nontransitivity.

The NRICH page also includes students’ attempted solutions. Our calculation uses the die faces and a complete enumeration.

[NRICH](https://nrich.maths.org/problems/non-transitive-dice)