### 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?

### 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?

# Farmers Field

##### Stage: 3 Challenge Level:

Phil of Westwood High School, Leek, Staffordshire wrote:

The problem called farmers fields in the 15+ challenges can be proven wrong, the shortest wall isn't always a straight line.Consider this field: -

If the left side is slightly larger than the right side then the shortest distance to build the wall could be to use a curve as shown by the red line (A).To use straight lines would involve using three if placed as shown by the green lines (B), or two straight lines could be used as shown by the purple line (C).

To use one straight line the shortest route would be as indicated by the blue line (D).It is obvious from looking that when the difference is very small that a single straight line won't be the shortest difference.