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Card Trick 2

Can you explain how this card trick works?

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Painting Cubes

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?

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Lesser Digits

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

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.)

Cube Paths diagram