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I want some cubes painted with three blue faces and three red faces.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?