Suppose I have 36 black blocks of dimensions 1x2x3. I can stack them 2 across, 3 deep and 6 high to make a nice looking cube of dimensions 6x6x6. I then proceed to paint the surface of this cube red. Now, when I look at the individual blocks, some faces are red and some have remained black.
In how many ways can I reassemble the large 6x6x6 cube so that its surface still appears all red?
You can choose to count rotations separately or as the same if you would like. I'm not so much looking for a final answer as a process. This is a very simple version of a more general problem I'm trying to solve. Even a nudge in the direction of some good literature would be of help. I've never encountered a counting problem like this before.