Are Stupid Dice Necessary?

SummaryA pair of 6-sided dice cannot be relabeled to make the sums 2, 3, … 12 equally likely. It is possible to label seven, 10-sided dice so that the sums 7, 8, …, 70 occur equally often. We investigate such relabelings for pq-sided dice, where p and q are distinct primes, and show that these relabelings usually involve stupid dice, that is, dice with the same label on every face.