### Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

### Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

# The Perforated Cube

##### Age 14 to 16Challenge Level
A large cube made from 125 smaller cubes, 5 by 5 by 5, is 'perforated' by removing cubes.

Any cube may be removed so imagine some uncoloured, transparent cubes propping up anything that needs it.

If a 'perforated cube' has the three views (projections) below, what is the most and the least cubes possible to make a shape that has these three projections ? In other words what could be the most and the least cubes left from the original 125 ?