I found this here:
Sum Problem
Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same.
$\hskip2.7in$
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?
EDIT
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?
Thanks