### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

# Cosines Rule

##### Age 14 to 16Challenge Level

(a) Prove that a triangle with sides $3$, $7$, $8$ contains a $60$ degree angle.

(b) Three points $A$, $B$ and $C$ lie in this order on a line, and $P$ is any point in the plane. Use the Cosine Rule to prove that:

${AP^2\over AB.AC}+{CP^2\over CA.CB} = 1 + {PB^2\over BA.BC}.$