I found this here:
Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same.
Here is one of 20 736 solutions with the sum 14.
You find more at the German magazine "Bild der Wissenschaft 3-1980".
Now I have three (Question 1 moved here) questions:
Is $14$ the only possible face sum? At least, in the example given, it seems to related to the fact, that on every face two dice-pairs show up, having $n$ and $7-n$ pips. Is this necessary? Sufficient it is...
How do they get $20736$? This is the dimension of the related group and factors to $2^8\times 3^4$, the number of group elements, right?
i. I can get $2^3$, by the following: In the example given, you can split along the $xy$ ($yz,zx$) plane and then interchange the $2$ blocks of $4$ dice. Wlog, mirroring at $xy$ commutes with $yz$ (both just invert the $z$ resp. $x$ coordinate, right), so we get $2^3$ group lements. $$ $$ ii. The factor $3$ looks related to rotations throught the diagonals. But without my role playing set at hand, I can't work that out. $$ $$ iii. Would rolling the overall die around an axis also count, since back and front always shows a "rotated" pattern? This would give six $90^\circ$-rotations and three $180^\circ$-rotations, $9=3^2$ in total. $$ \\ $$ Where do the missing $2^5\times 3^2$ come from?
Is the reference given, online available?
And to not make tehshrike sad again, here's the special question for $D4$:
What face sum is possible, so that the sum of the points on each side is the same, when you pile up 4 D4's to a pyramid (plus the octahedron mentioned by Henning) and how many representations, would such a pyramid have?