### 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:

Jack from Wolgarston High School gave this a lot of thought, here are some more ideas about the bike tracks :

We know that there's an important difference between a back wheel and a front wheel : the front wheel can point in any direction, just turn the handle bar, but the back wheel always points exactly forward towards the front wheel, or more exactly, towards the point where the front wheel has contact with the ground.

So we could test to see if a track could have been made by that back wheel :

From the track being tested draw out a line that seems to be the wheel's direction at that point, and then continue that direction line until it hits the other track - that's where the front wheel would have needed to be.

Do that for lots of points along the 'tested back wheel track', and if that was a genuine back wheel track the distance along any of those direction lines to the front wheel contact point would be a constant length.

That's because, although the front wheel can turn the contact point with the ground is the same however you angle the handle bar.

If the line first tested is unsuccessful the other track ought to be the back wheel, but it's a good idea to do the test and check !

The straight line from a point on the back wheel track which shows the direction of the back wheel at that point is called the 'tangent'.