### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### 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?

### Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

# Weekly Problem 1 - 2010

##### Stage: 3 Short Challenge Level:

One can see the greatest number of cubes when looking at three faces at once. We count the cubes on each face, giving $3\times 11^2=363$ cubes, but have to subtract from this the cubes along the three edges that have been counted twice, and then add back for the cube at the corner for which three faces are visible. The final quantity is $363-(3\times 11)+1=331$ cubes.

This problem is taken from the UKMT Mathematical Challenges.

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