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
Prince Rupert's cube by the two opposite points, as in the case of Prince Rupert's problem. If the aspect ratio is not constrained, the rectangle with the largest area that fits within a cube is the one that has two opposite edges of the cube as two of its sides, and two face diagonals as the other two sides. Alternatively, one may ask for the largest formula_4-dimensional hypercube that may be drawn within an formula_5-dimensional unit hypercube. The answer is always an algebraic number. For instance, the problem for formula_6 asks for the largest cube within a four-dimensional hypercube. After Martin Gardner posed