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, along with the assumption that the intervals on the dice would be symmetric because the resulting distribution is symmetric. (The intervals' symmetry should be either like ABBA and CDDC or like ABCD and DCBA.)
2 faces (essentially 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 pattern established by 2, 4 and 6 faces.