Editing Professor's Cube (section)

==Permutations==
There are 98 pieces on the exterior of the cube: 8 corners, 36 edges, and 54 centers (48 movable, 6 fixed).

Any [[permutation]] of the corners is possible, including odd permutations, giving [[factorial|8!]] possible arrangements. Seven of the corners can be independently rotated, and the orientation of the eighth corner depends on the other seven, giving 3<sup>7</sup> (or 2,187) combinations.

There are 54 centers. Six of these (the center square of each face) are fixed in position. The rest consist of two sets of 24 centers. Within each set there are four centers of each color. Each set can be arranged in 24! different ways. Assuming that the four centers of each color in each set are indistinguishable, the number of permutations of each set is reduced to 24!/(24<sup>6</sup>) arrangements, all of which are possible. The reducing factor comes about because there are 4! (or 24) ways to arrange the four pieces of a given color. This is raised to the sixth power because there are six colors. The total number of permutations of all movable centers is the product of the permutations of the two sets, 24!<sup>2</sup>/(24<sup>12</sup>).

The 24 outer edges cannot be flipped due to the interior shape of those pieces. Corresponding outer edges are distinguishable, since the pieces are mirror images of each other. Any permutation of the outer edges is possible, including odd permutations, giving 24! arrangements. The 12 central edges can be flipped. Eleven can be flipped and arranged independently, giving 12!/2 × 2<sup>11</sup> or 12! × 2<sup>10</sup> possibilities (an odd permutation of the corners implies an odd permutation of the central edges, and vice versa, thus the division by 2). There are 24! × 12! × 2<sup>10</sup> possibilities for the inner and outer edges together.

This gives a total number of permutations of
:<math> \frac{8! \times 3^7 \times 12! \times 2^{10} \times 24!^3}{24^{12}} \approx 2.83 \times 10^{74}</math>

The full number is precisely 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 possible permutations<ref name="Cubic Circular">[http://www.jaapsch.net/puzzles/cubic3.htm#p18 Cubic Circular Issues 3 & 4] [[David Singmaster]], 1982</ref> (about 283 [[Names of large numbers|duodecillion]] on the [[names of large numbers|long scale]] or 283 trevigintillion on the short scale).

Some variations of the cube have one of the center pieces marked with a logo, which can be put into four different orientations. This increases the number of permutations by a factor of four to 1.13×10<sup>75</sup>, although any orientation of this piece could be regarded as correct. By comparison, the number of atoms in the [[observable universe]] is estimated at 10<sup>80</sup>. Other variations increase the difficulty by making the orientation of all center pieces visible. An example of this is shown below.