Rubik's Cube could be coloured in such a way as to emphasise the corners or edges, rather than the faces as the standard colouring does; but neither of these alternative colourings has ever become popular. The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Each corner has three possible orientations, although only seven (of eight) can be oriented independently; the orientation of the eighth (final) corner depends on the preceding seven, giving 3 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, restricted from 12! because edges must
Halved cube graph in the original graph. For instance, the halved cube graph of order four may be formed from an ordinary three-dimensional cube by keeping the cube edges and adding edges connecting pairs of vertices that are on opposite corners of the same square face. The halved cube graph formula_2 of order 3 is the complete graph "K", the graph of the tetrahedron. The halved cube graph formula_3 of order 4 is "K", the graph of the four-dimensional regular polytope, the 16-cell. The halved cube graph formula_4 of order five is sometimes known as the Clebsch graph, and is the complement of