An interactive probability curiosity · No internet needed
A miniature laboratory of chance

The Dice That Refuse a Champion

Four six-sided dice. None is loaded in the ordinary sense: every face has an equal chance to appear. And yet each die beats another die two-thirds of the time, in a perfect circle. Who should choose first?

01 / Your move

Choose your die first

Pick any die. The house will respond with a die that beats yours 2 out of 3 times.
02 / The experiment

Your matchup

?
You
Choose a die
—
VS
?
House
Waiting…
—
Choose a die to reveal the odds.Exact probabilities, not estimates

Every possible face pairing is counted equally.

Waiting for your choice.
Your wins
0
House wins
0
Ties
0
Recent rounds →
03 / The secret

A circle with no winner

Each arrow means “beats with probability ⅔.” Follow the arrows all the way around—and you're back where you started.

Abeats B
Bbeats C
Cbeats D
Dbeats A
A → B → C → D → A·⅔ on every edge
Why does this work? Show the short proof.

A beats B: A rolls 4 on four faces; B always rolls 3. Probability 4/6 = ⅔.
B beats C: C rolls 2 on four faces; B always rolls 3. Probability 4/6 = ⅔.
C beats D: C rolls 6 with probability ⅓ (always winning), or rolls 2 with probability ⅔ while D rolls 1 with probability ½. Total ⅓ + (⅔)(½) = ⅔.
D beats A: D rolls 5 with probability ½ (always winning), or rolls 1 with probability ½ while A rolls 0 with probability ⅓. Total ½ + (½)(⅓) = ⅔.

04 / All possible matchups

The complete probability table

Each square gives the chance that the row die beats the column die. Green is favorable; pink is unfavorable. The diagonal is omitted.

vsABCD
Not every pair is a ⅔–⅓ contest. For instance, B and D are evenly matched. These are the classic Efron's dice, an example of nontransitive probability.
Built as a tiny, self-contained interactive experiment. All calculations and simulations run locally in your browser.Every face has probability 1/6. No hidden advantage in the random rolls.