### 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?

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### 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.

# Concrete Wheel

##### Stage: 3 Challenge Level:
Well done toeveryone 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. This can be seen by using the interactivity and making the wheel very large by pressing the up arrow:
This text is usually replaced by the Flash movie.

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!