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Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
Look at the picture below showing different odd sizes of triomino.
Odd triomino growing
Can you see a way to build round the red triomino to make the orange one?
To build round the orange one to make the grey one?
Can you continue to make a size $7$ triomino? And then a size $9$?