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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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Cube Paths

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

Musical Chairs

Stage: 3 Short Challenge Level: Challenge Level:1

Suppose that Gill's bottom was firmly in contact with the Hard Chair. Then the possible combinations would be:

  Hard Chair Comfy Chair
1 Gill, Nurse, Me -
2 Gill, Me, Nurse -
3 Gill, Nurse Me
4 Gill, Me Nurse
5 Gill Me, Nurse
6 Gill Nurse, Me

So there are $6$ possibilities if Gill sits on the Hard Chair, so there are $2\times 6 = 12$ possibilities in total.

This problem is taken from the UKMT Mathematical Challenges.