### 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?

# Towering Trapeziums

##### Stage: 4 Challenge Level:

$OGH$ is an isosceles right-angled triangle:

Lines $AB$, $CD$, $EF$, and $GH$ are parallel.

Suppose the area of the smallest triangle $OAB$ is one square unit.
• If lines $OC$ and $AB$ have the same length, calculate the area of trapezium $ABDC$.
• If lines $OE$ and $CD$ also have the same length, calculate the area of trapezium $CDFE$.
• If lines $OG$ and $EF$ also have the same length, calculate the area of trapezium $EFHG$.
Suppose that the chain of trapezia continued.

What would be the area of the $n^{th}$ trapezium in the chain?