mathschallenge.net

Problem

A wooden cube has each of its six faces painted either white or grey.
For example,

How many different cubes can be made?

Solution

Let us consider the number of grey faces painted.

4 grey faces will have 2 white faces and so is equivalent to 2 grey faces, similarly 5 grey faces is equivalent to 1 grey face and 6 grey faces is equivalent to no grey faces.

So there are 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10 ways to paint a cube with two colours.

How many ways can you paint a cube with three colours?