A circle, not a ladder
A’s high faces help it beat B. B’s middle faces help it beat C. C’s faces beat A’s low faces often enough to close the loop. “Usually beats” does not put the dice in one best-to-worst order.
A small probability playground
A beats B beats
C beats A.
Three fair dice. Every one has an opponent it usually beats—and another it usually loses to. Pick a pair and see the circle for yourself.
Each arrow means “wins 5 times out of 9, on average.”
Tap an arrow to explore its matchup.
Tap an arrow to explore its matchup.
A
224499
B
116688
C
335577
Choose your matchup
vs
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Die A—
Die BThe exact odds are ready. Your sample is still empty.
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0TIES
0RIGHT WINS
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Every possibility, counted once
The 36-outcome table
Each physical face is equally likely. Repeated numbers still occupy separate faces, so each cell has probability 1/36.
The mean doesn’t settle it
Each die has mean 5. A mean describes an average roll; the chance of beating a particular opponent depends on all the paired outcomes.
A sample can disagree
Even the favoured die can lose a short run. The table gives exact probabilities under independent fair rolls. The buttons make random samples; they do not change or prove those probabilities.
Where the dice come from · method and sources
These face sets appear in the University of Cambridge’s NRICH Non-Transitive Dice problem. Here, A is its green die, B its red die, and C its blue die. The explanation and interactive presentation are by Excelsior. The familiar dice construction is not claimed as a new discovery.
This page enumerates all 36 ordered face pairs for the selected dice. A strict win means the left value is greater than the right value. Selecting the same die on both sides means two independent copies, so ties are possible. Simulated rolls select each face independently using the browser’s random-number generator. Counts reset when the matchup changes.
For the broader mathematics, see Conrey, Gabbard, Grant, Liu and Morrison, Intransitive Dice. Everything here runs in your browser; no account or wager is involved.