← Back to Paradoxes

Intransitive Dice

When A beats B, and B beats C, but C beats A!

The Efron Dice

These four dice have a remarkable property: there is NO "best" die! Each die beats another more than 50% of the time, creating a cycle like rock-paper-scissors.

Die A
4
4
4
4
0
0
Die B
3
3
3
3
3
3
Die C
6
6
2
2
2
2
Die D
5
5
5
1
1
1
Click two dice to battle them!
?
-
VS
?
-

Dominance Cycle

Each arrow shows which die wins more often. Notice the CYCLE!

Battle Statistics

Monte Carlo Simulation

Run simulations for all matchups

The Paradox Explained

In normal comparisons, if A > B and B > C, then A > C. This is called transitivity. But probability doesn't have to be transitive!

The Math Behind the Magic

Let's analyze Die A (faces: 4,4,4,4,0,0) vs Die B (faces: 3,3,3,3,3,3):

• When A rolls 4 (probability 4/6 = 2/3): A wins
• When A rolls 0 (probability 2/6 = 1/3): B wins

P(A beats B) = 2/3 ≈ 66.7%

Now Die B vs Die C (faces: 6,6,2,2,2,2):

• C rolls 6 (prob 2/6): C wins
• C rolls 2 (prob 4/6): B wins (3 > 2)

P(B beats C) = 4/6 ≈ 66.7%

But Die C vs Die A:

• C rolls 6 (prob 2/6): C always wins
• C rolls 2 (prob 4/6): C wins when A rolls 0 (prob 2/6)

P(C beats A) = 2/6 + (4/6 × 2/6) = 2/6 + 4/18 = 10/18 = 5/9 ≈ 55.6%

The cycle completes! A→B→C→A, each winning about 2/3 of the time.

The Buffett Hustle

"Warren Buffett once tried to trick Bill Gates with intransitive dice. He offered Gates first pick of any die, then he would pick second. Gates, being Gates, asked to see the dice first... and immediately realized the trap. No matter which die Gates picked, Buffett could choose one that beat it!"

Why It Matters

Intransitive dice reveal that:

Create Your Own!

The simplest intransitive set uses just 3 dice:

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

A beats B (5/9), B beats C (5/9), C beats A (5/9)