Die A
?
VS
Die B
?
Select a match and click Roll!
Rolls
0
Left Wins
0
Right Wins
0
Left Win %
--

Die A (Red)

0, 0, 4, 4, 4, 4

Die B (Blue)

3, 3, 3, 3, 3, 3

Die C (Green)

2, 2, 2, 2, 6, 6

Die D (Orange)

1, 1, 1, 5, 5, 5

🎲 The Paradox

Die A beats B. Die B beats C. Die C beats D. So A beats D? NO! Die D beats A! This violates our intuition about "better than" being transitive. It's like Rock-Paper-Scissors with dice!

📊 The Probability

Each die beats the next with probability 2/3 (67%)!

A→B→C→D→A
All with P = 2/3

This is the maximum possible for a 4-cycle (Trybuła, 1961).

💼 Buffett vs Gates

Warren Buffett offered Bill Gates a dice game with these dice, letting Bill choose first. Bill became suspicious and insisted Warren pick first—saving himself from the trap! The trick: always pick the die that beats your opponent's!

🔬 Mathematical Insight

This paradox shows that the relation "wins more than half the time" is not transitive. Martin Gardner featured Efron's dice in Scientific American (1970), sparking widespread interest in nontransitive games.