Given $4$ different colours, we want to paint the four sides of a pyramid in such a way as to sides that share an edge do not have the same colour. We're allowed to use the same colour more than once otherwise.
I drew a pyramid viewed from the top, to help me visualize.
The solution to this problem is $4\cdot3\cdot2\cdot2+4\cdot3\cdot1\cdot3$. However, I can't quite understand why. I was hoping anyone would provide an intuitive explanation.
Best Answer
First label the sides by their (inter)cardinal directions. We have NE, NW, SE and SW.
NW and SE either have the same color, or different colors. This splits us off into two cases:
Add them together, and you have the answer.