### 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?

### 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?

# Concrete Wheel

##### Age 11 to 14 Challenge Level:

Well done to everyone who correctly worked out that the third solution in the hint is correct. From 20 feet above the ground the bottom of the wheel appears horizontal, so it covers the top right corner almost immediately after covering the top left corner.
Thanks to Julian (Hastingsbury School) who worked out how long the window would be covered for.

If you consider the radius that touches the ground, the angle this makes with the radius 20 feet above the ground is 0.7 degrees, so the wheel turns 1.4 degrees whilst covering the window.

The wheeil travels 100pi miles each time it completes a whole turn, taking pi hours. So the wheel will cover the window for (1.4 / 360) pi hours = 0.0122 hours = 44 seconds. The angle of the tangent to the circle where it first touches the window is less than 1 degree to the horizontal, and so it appears to be coming down horizontally as we expected above!