### 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?

# Zig Zag

##### Age 14 to 16 Challenge Level:

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Congratulations Soh Yong Sheng of Raffles Institution, Singapore and James of Hethersett High School for your solution. The image

can be further expanded into a diagram containing 3 squares of which two are the same size.

The two smallest quadrilaterals produced are squares as their sides are all of the same length equal to the shortest side of the triangle. The dots will form one line containing the diagonals of the three squares as the sides of the squares are all perpendicular or parallel to each other.

Here is another way of looking at it.