A The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
Solution: There is just one coloring; it's forced: Without loss of generality, suppose that the corners are colored in order, R, G, B, P, as shown below.

Notice that vertex $X$ cannot be $\mathrm{R}$, B, or B; it must therefore be colored P. Likewise, every other vertex on the top has only one possible color, since each top vertex is part of a two faces which collectively use three different colors on the bottom level. So there is only one solution, namely, starting from vertex $X$ and going counterclockwise (as seen from above): P, R, G, B.