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A 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 ]

Began by looking at what would be equivalent puzzles for pairs of 2-sided, 4-sided, 6-sided and 8-sided dice, along with the assumption that the intervals between their sides' numbers would be symmetric because the resulting distribution is symmetric.

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 sides (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]

4 sides (easy): [1 2 2 3] x [1 3 3 5]

6 sides (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]

8 sides (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 sides was straightforward after 8 sides refined the pattern suggested by 2, 4 and 6 sides.


**Addendum.**   In the terms of [another solution](httphttps://puzzling.stackexchange.com/a/25889/18129), this solution neatly breaks down to $$ \begin{array}{ll} A = [1,2] & B=[1,11]\\ C=[0,1,2,3,4,5,6,7,8,9] & D=[0,2,4,6,8,10.12,14,16,18] \end{array} $$ which clearly shows what underlies the straightforward even-sided-dice pattern better than how I understood it.

A 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 ]

Began by looking at what would be equivalent puzzles for pairs of 2-sided, 4-sided, 6-sided and 8-sided dice, along with the assumption that the intervals between their sides' numbers would be symmetric because the resulting distribution is symmetric.

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 sides (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]

4 sides (easy): [1 2 2 3] x [1 3 3 5]

6 sides (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]

8 sides (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 sides was straightforward after 8 sides refined the pattern suggested by 2, 4 and 6 sides.


**Addendum.**   In the terms of [another solution](http://puzzling.stackexchange.com/a/25889/18129), this solution neatly breaks down to $$ \begin{array}{ll} A = [1,2] & B=[1,11]\\ C=[0,1,2,3,4,5,6,7,8,9] & D=[0,2,4,6,8,10.12,14,16,18] \end{array} $$ which clearly shows what underlies the straightforward even-sided-dice pattern better than how I understood it.

A 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 ]

Began by looking at what would be equivalent puzzles for pairs of 2-sided, 4-sided, 6-sided and 8-sided dice, along with the assumption that the intervals between their sides' numbers would be symmetric because the resulting distribution is symmetric.

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 sides (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]

4 sides (easy): [1 2 2 3] x [1 3 3 5]

6 sides (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]

8 sides (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 sides was straightforward after 8 sides refined the pattern suggested by 2, 4 and 6 sides.


**Addendum.**   In the terms of [another solution](https://puzzling.stackexchange.com/a/25889/18129), this solution neatly breaks down to $$ \begin{array}{ll} A = [1,2] & B=[1,11]\\ C=[0,1,2,3,4,5,6,7,8,9] & D=[0,2,4,6,8,10.12,14,16,18] \end{array} $$ which clearly shows what underlies the straightforward even-sided-dice pattern better than how I understood it.
added reference to Mike Earnest's method, changed "faced" to "sided"
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A 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 ]

ReachedBegan by hand:looking at what would be equivalent puzzles for pairs of 2-sided, 4-sided, 6-sided and 8-sided dice, along with the assumption that the intervals between their sides' numbers would be symmetric because the resulting distribution is symmetric.

Began by looking at what would be equivalent puzzles for pairs of 2-faced, 4-faced, 6-faced and 8-faced dice, along with the assumption that the intervals between their faces' numbers would be symmetric because the resulting distribution is symmetric.

TheThe intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 facessides (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]

4 facessides (easy): [1 2 2 3] x [1 3 3 5]

6 facessides (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]

8 facessides (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.

20 sides was straightforward after 8 sides refined the pattern suggested by 2, 4 and 6 sides.


**Addendum.**   In the terms of [another solution](http://puzzling.stackexchange.com/a/25889/18129), this solution neatly breaks down to $$ \begin{array}{ll} A = [1,2] & B=[1,11]\\ C=[0,1,2,3,4,5,6,7,8,9] & D=[0,2,4,6,8,10.12,14,16,18] \end{array} $$ which clearly shows what underlies the straightforward even-sided-dice pattern better than how I understood it.

A 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 by looking at what would be equivalent puzzles for pairs of 2-faced, 4-faced, 6-faced and 8-faced dice, along with the assumption that the intervals between their faces' numbers would be symmetric because the resulting distribution is symmetric.

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 faces (essentially a pair of coins with no alternative to the standard 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.

A 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 ]

Began by looking at what would be equivalent puzzles for pairs of 2-sided, 4-sided, 6-sided and 8-sided dice, along with the assumption that the intervals between their sides' numbers would be symmetric because the resulting distribution is symmetric.

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 sides (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]

4 sides (easy): [1 2 2 3] x [1 3 3 5]

6 sides (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]

8 sides (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 sides was straightforward after 8 sides refined the pattern suggested by 2, 4 and 6 sides.


**Addendum.**   In the terms of [another solution](http://puzzling.stackexchange.com/a/25889/18129), this solution neatly breaks down to $$ \begin{array}{ll} A = [1,2] & B=[1,11]\\ C=[0,1,2,3,4,5,6,7,8,9] & D=[0,2,4,6,8,10.12,14,16,18] \end{array} $$ which clearly shows what underlies the straightforward even-sided-dice pattern better than how I understood it.
clarified what 2-face, 4-face, etc, mean
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humn
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SolutionA 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 withby looking at what would be equivalent puzzles for pairs of 2-facefaced, 4-facefaced, 6-facefaced and 8-facefaced dice, along with the assumption that the intervals on the dicebetween their faces' numbers would be symmetric because the resulting distribution is symmetric. ( 

The intervals' symmetry shouldcould be either either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.)

2 faces (essentially a coin,pair of coins with no alternative to the originalstandard 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.

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.

A 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 by looking at what would be equivalent puzzles for pairs of 2-faced, 4-faced, 6-faced and 8-faced dice, along with the assumption that the intervals between their faces' numbers would be symmetric because the resulting distribution is symmetric.  

The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.

2 faces (essentially a pair of coins with no alternative to the standard 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.

added note about symmetry
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"the the" --> "the"
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humn
humn
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