### 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?

### Tri-colour

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

### Permute It

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

# How Many Dice?

##### Stage: 3 Challenge Level:

Look at a die. Had you noticed before that 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7?

I'll call a die which has this property a standard die.

I don't know when this standard convention was adopted, but I know that 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 ?

A mathematician friend from New Zealand, Prof Derek Holton, tells us that one of the standard dice is used in the western world and the other one is used in the east. Can any one confirm this with an example?

Suppose we relax the condition that opposite faces sum to 7. How many different dice can we make now?