My rather artistic friend is painting the faces of a cube: on each
face she uses one of her two favourite colours: either red or green.
How many different ways can she paint the cube?
Suppose now we declare that two paintings are in the same "rotation class" if there exists a rotation of the cube sending one
painting to the other. (So for example, the painting which has the
top face red, and the rest green, is in the same rotation class as the
painting which has the bottom face red, and the rest green.)
How many different rotation classes are there?
Now repeat all the above using the three colours red, green, blue
to paint the cube, instead of two.