### 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?

### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

### Paving Paths

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

# All in the Mind

##### Stage: 3 Challenge 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!