### All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. 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?

# Boxed In

##### Age 11 to 14Challenge Level

Catherine of Mount School, York and Joel of ACS Barker, Singapore realised that it is not necessary to calculate the lengths of the edges of the cuboid and they sent very similar solutions. This is Catherine's solution:

If the sides of the cuboid are x, y, and z and the areas of the rectangular faces are p, q and r then:
p = xy, q = yz and r = zx
It follows that pqr = (xy)(yz)(zx) = x 2 . y 2 . z 2 = (xyz) 2
So the volume = xyz = sqrt (pqr) = sqrt (3 x 12 x 25) = sqrt 900 = 30

Students from West Flegg Middle School, Norfolk and Madras College, St Andrew's and Russell Lower School, Ampthill also found the correct answer.