### 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?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Cushion Ball Interactivity

##### Age 14 to 16Challenge Level

This interactivity is designed for the problem Cushion Ball.

The interactive diagram below has two labelled points, A and B. What is the shortest path from A to B if you bounce off one cushion? In the diagram, you can click on the "Show" buttons to draw the four possible paths from A to B. Which is the shortest? You may move A and B around by clicking on them.

What is the shortest path from A to B using exactly two cushions? The interactive diagram below shows the eight possible paths from A to B. How would you calculate the shortest path?