When you roll a standard pair of 6-sided dice, there are 11 possible outcomes, with a certain probability of occurrence for each possible result (the chance of getting a total of 8 is 5/36, for example). This is called the probability distribution.

Now, it turns out that you can take 2 6-sided dice and put different positive integers (whole numbers) on them such that the probability distribution is the same. That is, you can write a completely different set of numbers on the 2 dice but still have the same likelihood of rolling any total as if the dice were regular. It turns out that the only way to do this is to put 1, 2, 2, 3, 3, 4 on one die and 1, 3, 4, 5, 6, 8 on the other (clearly I'm allowing repeats of numbers, and if you think about it, repeats must occur).

The two problems I have relate to pairs of dice with more than 6 sides. The first problem can be done by trial and error, but the second one most certainly cannot be done that way, and would require an advanced method of solving.

Find the *2* possible relabelings for a pair of 8-sided dice.
Find the only possible relabeling for a pair of 35-sided dice.

Hint 1

Use generating functions. (Sub-hint: practice on the 6-sided case, where you already know the answer.)

Want another hint? Click here.

Click here for the complete solution.