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We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Russian Cubes

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.