To solve this tough nut, start by colouring one vertex red. In the cube there are different paths from the red to the blue vertex along the edges but all these paths have an odd number of edges so this is a blue vertex and not a redblue one. By contrast the vertex at the top of the tetrahedron is redblue because it can be reached by going along one edge (an odd number) and also by going along two edges (an even number). Carry on colouring vertices red if they can be reached from a red vertex by travelling along an even number of edges and blue if they can be reached by travelling along an odd number of edges from a red vertex. Do the same for other solids. If a solid has any redblue vertices are all its vertices then redblue? What property does the solid need if it is to have redblue vertices?