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### Number and algebra

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# Painting Cubes

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### Cube Paths

### How Many Dice?

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Age 11 to 14

Challenge Level

- Problem
- Student Solutions

A large rose-tree stood near the entrance of the garden: the roses growing on it were white, but there were three gardeners at it, busily painting them red....

"Would you tell me please," said Alice, "why you are painting those roses?"

Five and Seven said nothing, but looked at Two. Two began in a low voice, "Why, the fact is, you see, Miss, this here ought to have been a red rose-tree, and we put a white one in by mistake; and, if the Queen was to find it out, we should all have our heads cut off, you know."

Imagine you have some wooden cubes. You also have six paint tins each containing a different colour 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? Remember that two cubes are different only when it is not possible, by turning one, to make it correspond with the other. |

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

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 ?