2021 AMC 12B Problems/Problem 19
Solution
Suppose the dice have and
faces, and WLOG
. Since each die has at least
faces, there will always be
ways to sum to
. As a result, there must be
ways to sum to
. There are at most nine distinct ways to get a sum of
, which are possible whenever
. To achieve exactly eight ways,
must have
faces, and
. Let
be the number of ways to obtain a sum of
, then
. Since
,
. In addition to
, we only have to test
, of which both work. Taking the smaller one, our answer becomes
.