Solution
The Other Pair of Dice
Show the problem again
Two cubes have each of their six faces labelled with a positive integer (repeats are allowed on a die). When both are rolled, every total occurs with exactly the same probability as with two standard dice numbered 1 to 6. However, the pair is not two standard dice. It turns out such a pair exists and is unique. What is the largest number written on any face? Give an integer.
Worked solution
The answer is 8. Encode a die by the polynomial $\sum x^{\text{face}}$. A standard die is $x(1+x)(1+x+x^2)(1-x+x^2)$, so the pair must factor $x^2(1+x)^2(1+x+x^2)^2(1-x+x^2)^2$ into two polynomials $A, B$ with nonnegative coefficients, $A(1) = B(1) = 6$ and $A(0) = B(0) = 0$. Evaluating at $x=1$ (factors give 2, 2, 3, 3, 1, 1) forces each die to take one $x$, one $(1+x)$ and one $(1+x+x^2)$. Only the factors $(1-x+x^2)$ remain free: one each gives standard dice, while both on one die gives $x(1+x)(1+x+x^2) = x + 2x^2 + 2x^3 + x^4$ and $x(1+x)(1+x+x^2)(1-x+x^2)^2 = x + x^3 + x^4 + x^5 + x^6 + x^8$. These are the Sicherman dice, faces 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. The largest face is 8.
Source: Sicherman dice, found by George Sicherman and popularized by Martin Gardner in Scientific American. Statement written for AxiomIQ.