### 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?

# Steel Cables

#### If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...

Cables can be made stronger by compacting them together in a hexagonal formation.

Here is a 'size 5' cable made up of 61 strands:

How many strands are needed for a size 10 cable?

How many for a size n cable?

Once you've had a go at the problem, click below to see the diagrams some 15 year old students produced when they worked on it.
Do these diagrams give you any ideas for how you could work out the number of strands needed?

Group 1

Group 2

Group 3

Group 4

The work that these students did using their diagrams is given on the Getting Started page, if you would like a bigger hint.

Which of the four approaches makes the most sense to you?

Can you think of any other approaches?

Notes and Background

Hexagonal packings are often chosen for strength or efficiency. To read more about packings, take a look at the Plus articles Mathematical Mysteries: Kepler's Conjecture and Newton and the Kissing Problem