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# Russian Cubes

How many different cubes can be painted like that?

Show that, even though she always goes second, Cath can choose the faces she paints in a way that will ensure that both cubes are identical.

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Age 14 to 16

Challenge Level

I want some cubes painted with three blue faces and three red faces.

How many different cubes can be painted like that?

James (using blue) and Cath (using red) paint the faces of a cube in turn so that the six faces are painted in order "blue then red then blue then red then blue then red".

Having finished one cube, they begin to paint the next one.

Show that, even though she always goes second, Cath can choose the faces she paints in a way that will ensure that both cubes are identical.

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

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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