### 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...

### Rotating Triangle

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

# Efficient Packing

##### Stage: 4 Challenge Level:

How efficiently can you pack disks of the same size with no overlap? Imagine attempting to cover a 1m square with 10cm diameter disks with no overlap. What percentage of the area of the square can you actually cover using this obvious packing for disks?

How much more efficiently would you be able to pack 1cm diameter disks into the 1m square? Could you make an estimate for the efficiency of packing disks of diameter 1mm?

• As a harder extension, make an estimate of the number of 1cm diameter balls that would be able to fit into a 1m cubed box.

#### Notes and background

Whilst it might seem relatively simple, the problem of 'shape packing' is often very difficult mathematically to solve with certainty for many shapes. Intuitive visualisation often works just as well as a strict mathematical analysis, and often is the only sensible possibility with packing together complicated shapes.

You might like to consider situations in which efficient shape packing is relevant in the physical world.