I stared at this for way too long before saying anything useful.
First, clarify that 'relabel' means assigning arbitrary integers to the faces, not necessarily 1-6. Then, use a generating function or probability argument to show that equal probability for sums 1-12 is impossible, because the sum of two dice has a triangular distribution and relabeling cannot flatten it to uniform.
Pro tip: A quick sanity check: the sum of two dice has 36 equally likely outcomes, so each sum 1-12 would need exactly 3 outcomes. But the minimum sum is the sum of the two smallest face values, and the maximum is the sum of the two largest; the number of ways to achieve these extremes is always 1, not 3, unless faces are duplicated—which still fails. This simple counting argument often impresses interviewers.
Confirm that relabeling means assigning any integers to the faces, and that the dice are fair and independent. State that we need each sum from 1 to 12 to have probability 1/12.
There are 36 equally likely outcomes when rolling two dice. For each sum to have probability 1/12, each sum must occur exactly 3 times.
The minimum possible sum is the sum of the two smallest face values (one from each die). This sum can occur in only one way (both dice show their minimum). Similarly, the maximum sum occurs in only one way. Thus, the minimum and maximum sums each occur exactly once, contradicting the requirement of 3 occurrences.
Since the minimum and maximum sums cannot occur 3 times, it is impossible to have a uniform distribution over sums 1-12. Therefore, no relabeling can achieve the goal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.