21. The diagram shows a $3 \times 4 \times 5$ cuboid consisting of 60 identical small cubes. A termite eats its way along the diagonal from $P$ to $Q$. This diagonal does not intersect the edges of any small cube inside the cuboid. How many of the small cubes does it pass through on its journey?
A 8
B 9
C 10
D 11
E 12
Solution
C

We are told that the diagonal from $P$ to $Q$ does not intersect any internal edges. So, while it goes from $\mathrm{P}$ to $\mathrm{Q}$, it moves from one small cube to another by passing through a face. It will have to pass through at least 3 faces in order to get from the base layer up to the top layer, through 4 faces to get from the left to the right and 2 faces to get from the front to the back. The termite starts in the small cube at $P$ and then must pass through another $3+4+2$ new small cubes to reach $Q$. In total, therefore, it must pass through 10 small cubes.