### 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:

We received only one correct solution to this problem.

There are only 49 winning lines, which no doubt many of you will find hard to believe. Duplication of lines obviously helped muddy the approaches that were used with this problem.

Laura of West Flegg GM Middle School used an approach which classified the type of winning line possible with accompanying sketches :

9 vertical lines, 18 horizontal and 22 diagonal lines.
(Of the diagonal lines 6 go from front to back, 6 go from left to right, 6 go from top to bottom while four lines connect the 8 corner cells.)

a total of 49 lines
(27 of length 1, 18 of length Root 2 and 4 of length Root 3)