### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### Chess

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

# All in the Mind

##### Age 11 to 14Challenge Level

Many of you correctly guessed that the shape outlined by the water would be a hexagon. You provided us with several different explanations.

Jimmy, from Lady Mary Primary School, made a cube out of blue-tac and - with the help of his teacher! - cut it in half:

I held the cube from one vertex, and stood it on the opposite vertex. Then I measured the height of the cube standing like that. It was 2.4cm. Half of 2.4cm is 1.2cm, so half the cube was above that height and the other half was below. Then I cut it in a horizontal line at 1.2cm from the bottom of the cube. The cross-section I made was a hexagon.

Tilly thought about what happens when you submerge the cube so that more than half or less than half is in the water.

If you just submerge the very tip of the cube in water, you get a triangle shape with the point of the triangle pointing up. If you submerge more and more you get bigger triangles, still with the point pointing up. If you start with all but the tip of the cube in the water, you get a triangle shape with the point of the triangle pointing down. If you submerge more you get bigger triangles, also with the tip pointing down.
When they meet at the middle there's a triangle with the tip pointing up and a triangle with the tip pointing down, so it must be a six pointed star or a hexagon. It is not a six pointed star so it must be a hexagon.

And James had yet another reason!

The water touches all six sides and it does the same thing to all of them, because if you rotate the cube the same thing still happens. So the water must make a regular six-sided shape: a hexagon!

Thank you, Jimmy, Tilly and James!