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# How Many Dice?

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### Painting Cubes

### Cube Paths

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Age 11 to 14

Challenge Level

- Problem
- Student Solutions

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?**

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?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?