### 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 16 Challenge Level:

Why do this problem?
The first part is an easy application of the cosine rule. The second part requires some algebraic skills and gives practice in proving a formula.

Possible approach
Use as practice. The problem does not require learners to decide on the method to use.

Key questions
Look at the formula to be proved. Write down the cosine formula for the triangles in the diagram using the same notation. Can you see how to manipulate the formulas you have established to give the formula to be proved?