Solution that fits a generalized pattern:
[ 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 ]
x [ 1 3 5 7 9 11 11 13 13 15 15 17 17 19 19 21 23 25 27 29 ]
Reached by hand:
Began with 2-face, 4-face, 6-face and 8-face dice.
2 faces (a coin, with no alternative to the original spot pattern): [1 2] x [1 2]
4 faces (easy): [1 2 2 3] x [1 3 3 5]
6 faces (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]
8 faces (not too tough, based on those above): [1 2 2 3 3 4 4 5] x [1 3 5 5 7 7 9 11]
20 faces was straightforward after 8 faces refined the the pattern established by 2, 4 and 6 faces.