The Other Pair of Dice

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.

• combinatorics

Source: Sicherman dice, found by George Sicherman and popularized by Martin Gardner in Scientific American. Statement written for AxiomIQ.