Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?

Cube Paths

Age 11 to 14 Challenge Level

The diagram shows a skeletal view of a 2 by 2 by 2 cube with one
route 'down' the cube.

How many routes are there on the surface of the cube from A to
B?
(No `backtracking' allowed, i.e. each move must be away
from A towards B.)