Doodles

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?

Three Balls

Age 14 to 16 Challenge Level:

As $OP$ is a radius and $PT$ is a tangent, the angle between them is a right angle.
So the triangle $OPT$ is right angled, and the hypotenuse is the diameter of the circle, so $P$ lies on the circle.
In the same way, $Q$ lies on the circle.