### 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. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

### Rotating Triangle

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

# Making Tracks

##### Stage: 4 Challenge Level:

For most students this problem will be outside their ordinary mathematical experience.
There is no frequently-used technique which may be just applied in this new situation.

Talk helps enormously in a problem like this. Questions to prompt thinking include :
• What difference can you think of between the front and the rear wheels ?
• What is the connection between the front and rear wheels which, for example, wouldn't be possessed by two uni-cyclists going along the path ?

These suggestions from the hint page may be useful :
• Find a bicycle and experiment.

• Trace a curve on paper and imagine a cyclist following that curve with the front wheel, and then consider the possibilities for the back wheel ?

Above all this is a good problem not to rush .

Leave the way into this problem left open and allow students to bring along fresh thoughts from time to time. The emerging solution will be much more satisfying and the process a much better experience of real new mathematics emerging.