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

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Russian Cubes

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

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Picture Story

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

Angle Trisection

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
We haven't received any solutions to this one yet. Here's a copy of one of the diagrams from the Hint, and a suggestion.

You need to show that the angle has been trisected, which would mean angles CBR, RBQ and QBP are equal. Congruent triangles may help you to show this!