### 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?

# One Out One Under

##### Stage: 4 Challenge Level:

For the first part, try starting with only four cards and compare the resulting final card with the final card when you start with eight cards.

Notice anything ?

What might be a good number of cards to try next ?

When you are ready to look for a general result, be systematic and list your results, for example : seven cards and the final card is 6, eight cards and the final card is . . . , nine cards . . . ., ten cards . . .

Remember to collect results in the other direction away from seven cards. Six cards and the final card was . . . etc.

If you've been careful you will see a clear pattern.

First describe the pattern. Then try to account for it.