Prove Equal Probability Impossible with Relabeled Dice Faces

##### Scenario Interview assessing candidate's understanding of probability distributions with a dice-configuration puzzle used by a gaming analytics team. ##### Question You have two fair six-sided dice. Can you relabel or configure the faces so that when both dice are rolled, the sums 1 through 12 each appear with equal probability? If yes, provide the new face labels; if no, give a rigorous proof of impossibility. ##### Hints Equal probability requires each of the 12 sums to occur exactly 3 times out of 36. Analyze face-pair counts and parity constraints.