### 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?

# Nine Colours

##### Stage: 3 Challenge Level:

If you have 27 small cubes, 3 each of nine colours, can you make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour?

In the picture, the top face and the left face have one of each colour.

Unfortunately the third face has two greens, two blacks, no reds and no browns, so this is not a valid solution.

You might like to explore this problem using cubes. If you don't have any cubes, you could record your work on squared paper by drawing and colouring each layer.

This problem features in Maths Trails - Working Systematically, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details, please see our publications page .