(;) Semicolon Math: Nonstandard Dice

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Nonstandard Dice (ID: 19)

Problem status: not solved

Let \(B\) be the set of (distinct) unordered pairs of fair 16-sided dice with a positive integer written on each face such that the likelihood that the roll a sum of \(k\) is the same as the likelihood of rolling a sum of \(k\) in a pair of fair standard 16-sided dice with faces numbered 1-16, for all \(2 \le k \le 32\). Each element of \(B\) can be represented as an unordered pair of 16-tuples.

Let \(y\) be the maximum difference between the max value of one die with the max value of the other die in any given pair of dice in \(B\).

Find \(|B|y\).

For instance, if we had the same problem but with fair 6-sided die, then \[B = \{\{(1,2,3,4,5,6), (1,2,3,4,5,6)\}, \{(1,2,2,3,3,4), (1,3,4,5,6,8)\}\} \] where the 6-tuple \((1,2,2,3,3,4)\) represents a cube with faces labeled 1, 2, 2, 3, 3, and 4. So \(|B| = 2\) and \(y = \max(8 - 4, 6- 6) = 4\) in this case.

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