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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?

Take Three from Five

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?