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
Partial cube graph theory. A "benzenoid graph" is a graph consisting of all vertices and edges lying on and in the interior of a cycle in a hexagonal lattice. Such graphs are the molecular graphs of the benzenoid hydrocarbons, a large class of organic molecules. Every such graph is a partial cube. A Hamming labeling of such a graph can be used to compute the Wiener index of the corresponding molecule, which can then be used to predict certain of its chemical properties. A different molecular structure formed from carbon, the diamond cubic, also forms partial cube graphs. Partial cube In graph