One of our multi-stage problems, that get harder as you work through them. And then a challenge to find a generic solution.
Firstly consider a three-inch cube that is completely painted on all six faces. This cube is sawed into one-inch cubes. Some of the resulting one-inch cubes will have paint on three faces, some on two faces, some on one face, and some with no paint at all.
How many of each are there?
If this was easy enough, consider a four-inch cube. This is sawed into one-inch cubes. How many of these have paint on three, two, one and no faces?
You can keep going with this with bigger and bigger cubes, but perhaps it would be more fun to work out the general formula for each of the four types of one-inch cubes for an n-inch cube.
Answer at 9.00 on Monday