### 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?

# One Out One Under

##### Age 14 to 16 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.