Non-Transitive Dice

If die A tends to beat die B, and die B tends to beat die C, surely die A tends to beat die C — that's just how "bigger than" works for numbers. It's not how it works for dice. Non-transitive dice are a set where "beats more than half the time" forms a cycle instead of an ordering: A beats B, B beats C, and C beats A, all with probability greater than $1/2$. There is no "best" die in the set. This is a favorite interview question precisely because it breaks an assumption people don't realize they're making.

A concrete set of three dice

One standard example uses these face values:

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

Each die has faces averaging around 5, so none is "bigger" in the usual sense — the trick is entirely in how the values are distributed relative to each other.

Worked example: checking A vs. B

Roll A and B and compare, face by face, all 36 equally likely outcomes. A beats B whenever A's face exceeds B's:

• A = 2: beats B = 1 (2 of B's 6 faces are 1s: both). Loses to B = 6, 6, 8, 8 (4 faces). So A=2 beats B on 2 of 6 rolls of B.
• A = 4: beats B = 1, 1 (2 faces), loses to 6, 6, 8, 8 (4 faces). Same: 2 of 6.
• A = 9: beats every face of B (1, 1, 6, 6, 8, 8 — all less than 9). 6 of 6.

Each face of A occurs with probability $1/3$ (two of six faces share each value), so

$$P(A > B) = \frac{1}{3}\left(\frac{2}{6}\right) + \frac{1}{3}\left(\frac{2}{6}\right) + \frac{1}{3}\left(\frac{6}{6}\right) = \frac{1}{3}\left(\frac{2+2+6}{6}\right) = \frac{10}{18} \approx 55.6\% .$$

So A beats B more often than not. By the same style of calculation (left as the natural next check), B beats C with the same $10/18$ probability, and — the surprising closer — C beats A with $10/18$ as well. The cycle is complete: A > B > C > A, each with the same edge, and no die is uniformly best.

Why this matters beyond the puzzle

The interview point isn't the specific numbers — it's recognizing that pairwise comparison relations don't have to be transitive, and assuming they are (because "greater than" for real numbers is) is a silent, unjustified assumption. This shows up in real settings: voting systems can produce cyclic majority preferences (Condorcet's paradox — a majority prefers A to B, B to C, and C to A), and in trading, a "which strategy is better" comparison based on head-to-head backtests over different time windows can similarly fail to produce a consistent ranking if the comparisons aren't all measured on the same footing.