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?

Parallel Universe

A right-angled triangle ABC is right-angled at B.

An equilateral triangle is constructed on BC.

A line QD is drawn, where Q is the midpoint of AC.

Prove that AB is parallel to QD.