### 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?

### How Many Dice?

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 ?

# Tic Tac Toe

##### Stage: 3 Challenge Level:
The ideas in this problem have been expanded and made into another, more detailed investigation. See Marbles in a Box.
 In the game of Tic Tac Toe (i.e. noughts and crosses) there are 8 distinct winning lines. Investigate how many distinct winning lines there are in a game played on a 3 by 3 by 3 board, with 27 cells. A winning line connects 3 cells.