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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?

Take Three from Five

Age 14 to 16 Challenge Level:

Start with the problem What Numbers Can We Make?

Think of a simpler problem:
If you choose 2 from 3 numbers, you can always select 2 numbers that add up to an even number. Why?