Solution:

The 1" × 1" × 1" cubes that do not have any pink paint on them will be at the core of the wooden block. This core will be 1" × 6" × 10", and will contain 60 cubes.

Out of a total of 288 cubes, there are 60 cubes without any pink paint on them. Therefore, the remaining 228 cubes will have one, two or three sides with pink paint on them (depending on whether they were at the face, edge or corner of the wooden block).

Food for thought:

What if the wooden block was in the shape of a cube (a" × a" × a") rather than a cuboid? Can you show that the number of cubes with paint on them is then given by a³ − (a − 2)³?