Consider a cube and paths along the edges of the cube. Mark one vertex red. Colour other vertices red if they can be reached from a red vertex by travelling along an even number of edges of the cube. Colour vertices blue if they can be reached by travelling along an odd number of edges from a red vertex. Is it possible to have vertices which are both red and blue at the same time (call these redblue vertices)? Now do the same for a tetrahedron.

Do the same for other solids, for example the octahedron, dodecahedron and icosahedron, and prisms with different cross sections. Remember the paths must be along the edges of the solids. Make a table to show your results. What property does the solid need if it is to have redblue vertices?