Published 13 September 2026

Sicherman dice

Can two fair six-sided dice have different labels from ordinary dice but exactly the same distribution of their sum? The Sicherman pair does. One die has faces 1,2,2,3,3,4; the other has 1,3,4,5,6,8. Roll them independently and every sum from two through twelve has the ordinary two-dice probability.

This equality concerns the sum. It does not say the individual dice, ordered face pairs, maximum, or doubles have their ordinary distributions. A game that uses any of those extra details may behave differently.

Compare the complete distribution

Select Sum of both dice. The two bar series coincide in height: counts out of 36 are 1,2,3,4,5,6,5,4,3,2,1. Filled and open bars sit beside each other so neither hides the other. Repeated labels count as separate equally likely faces; they must not be deduplicated before counting outcomes.

For sum seven, the alternative pairs of values are 1+6, 2+5, 3+4, and 4+3. There is one face carrying 1, two carrying 2, two carrying 3, and one carrying 4 on the first die. These supply 1 + 2 + 2 + 1 = six ordered face pairs, hence probability 6/36. Sums two and twelve each have one producing face pair.

Why every sum matches

Encode an ordinary die by the polynomial D(x) = x + x² + x³ + x⁴ + x⁵ + x⁶. Multiplication collects ordered face pairs: the coefficient of x to the power s in D(x)² counts outcomes with sum s. Dividing those coefficients by 36 gives probabilities.

The factorization D(x) = x(1 + x)(1 + x + x²)(1 − x + x²) allows factors to be distributed differently between two dice. The first alternative polynomial is A(x) = x(1 + x)(1 + x + x²) = x + 2x² + 2x³ + x⁴. The second is B(x) = x(1 + x)(1 + x + x²)(1 − x + x²)² = x + x³ + x⁴ + x⁵ + x⁶ + x⁸.

Their product is D(x)². Each polynomial has nonnegative integer coefficients summing to six, so each describes six equally likely faces. The product identity proves equality for all sums at once. It establishes this construction; a uniqueness proof would require checking the permissible factorizations under the chosen constraints.

A matching sum can hide different information

Switch to Equal face values. Ordinary dice have six doubles among 36 face pairs. The unshifted alternative has four: one pair 1,1; two pairs 3,3; and one pair 4,4. Its doubles probability is 1/9 rather than 1/6. There are no alternative 2,2; 5,5; or 6,6 outcomes.

Switch to Larger face value. The second alternative die can show eight, impossible for a standard die. Equality of one statistic’s distribution therefore leaves room for differences in another statistic. Summing discards information about how each total was produced.

Shift labels without changing a sum

The shift control adds the same integer to every first-die face and subtracts it from every second-die face. Each individual ordered pair keeps its sum, so the sum distribution is preserved directly. At either nonzero supported shift, a face becomes zero. This leaves the usual strictly positive-label version of the puzzle, and the display says so.

Neither the polynomial proof nor the enumeration establishes that a physical manufactured die is fair. The model assumes independent uniform face selection. Changing face weights would require multiplying weighted probability polynomials instead.

Reference

George Sicherman’s account documents the face labels and distinguishes sums from doubles. Continue to non-transitive dice to see another reason that a die’s behavior depends on the game statistic being compared.