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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?

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

Circle Box

Age 14 to 16
Challenge Level

Four circles of diameter 1 unit fit easily enough in a square of side 2 , but what is the smallest square into which we can fit 3 circles of diameter 1 unit?

Three Circles in a Square