Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Here is a hint for this tough nut. Just think about the angle between a chord and a tangent to a circle and think about cyclic quadrilaterals.