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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 is the $60$ degree angle opposite the side of median length?

For the second part all you need is the cosine formula and some algebraic manipulation to get the required result.