As part of a puzzle, you have to stack 8 little $1\times 1\times 1$-cubes so that they form one big $2\times 2\times 2$-cube. Now I want to check all possible solution to the puzzle and therefor I'm looking for an efficient way to sum up all possible positions for the little cubes. You may turn the little cubes and rearrange them.

A first step might be to look how many arrangements we can make. My guess is $$\frac{8!24^8}{24}=8!24^7.$$ We have $8!$ possible positions for the cubes and each of them has 24 orientations. Dividing by 24 for the possible orientations for the big cube gives me my guess.

This number still seems too much to me.

Once we have the number of arrangements, we need an efficient way to order all posibilities so that I can check them with the computer.



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