### Rotating Triangle

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

### Doodles

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

# Russian Cubes

##### Stage: 4 Challenge Level:

I want some cubes painted with three blue faces and three red faces.

How many different cubes can be painted like that?

James (using blue) and Cath (using red) paint the faces of a cube in turn so that the six faces are painted in order "blue then red then blue then red then blue then red".

Having finished one cube, they begin to paint the next one.

Show that, even though she always goes second, Cath can choose the faces she paints in a way that will ensure that both cubes are identical.