Suppose we roll two ordinary, 6-sided dice, $f_i=s_i=\{1,2,3,4,5,6\} $. All $36$ possible rolls with a pair of dice (first dice - $f_i$, second dice $s_i$):
$ \begin{array}{ccccccc} f_i/s_i & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 6 & 7 & 8 & 9 & 10 & 11 & 12 \end{array} $
So we have $1$ variant to roll the sum $2, 2$ variants to roll $3, 3$ variants to roll $4, ..., 2$ variants to roll $11, 1$ variant to roll $12.$
Is there some another pair $(f_i, s_i)$ of $6$-sided dice with different numbers on its sides, but with the same set of variants to roll the same sums $(2,\ldots,12)$? And all $f_i>0, s_i>0$. And how much?
$f_i$ and $s_i$ in such pair can be different.
Special upd $f_i \in \mathbb{N}$, $s_i \in \mathbb{N}$.