Problem:
A cube of cheese is cut along the planes and . How many pieces are there?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The cut separates the cheese into points with and those with . Similarly for the other cuts. Thus, which piece a point is in depends only on the relative sizes of its coordinates, . For instance, all points with are in the same piece. Since there are ways to order and , there are pieces.
Since the planes and intersect along the line , they divide all of space into just pieces, not the way most sets of planes do. (Imagine forming the configuration by rotating a single half plane around that line; each rotation to the next position sweeps out one more piece.) Since the line is a major diagonal of the cheese, in each of the pieces of space there is some point of the cheese. Thus the cheese is divided into pieces also.