Pocket Cube - Permutations

Permutations

Pocket Cube in different forms. From top (to bottom):
i. Solved pocket cube.
ii. Scrambled pocket cube.
iii.Pocket cube with one side tilted.

Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). There is nothing identifying the orientation of the cube in space, reducing the positions by a factor of 24. This is because all 24 possible positions and orientations of the first corner are equivalent due to the lack of fixed centers. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. The number of possible positions of the cube is

The maximum number of turns required to solve the cube is up to 11 full turns, or up to 14 quarter turns.

The number f of positions that require n full twists and number q of positions that require n quarter turn twists are:

n	f	q
0	1	1
1	9	6
2	54	27
3	321	120
4	1847	534
5	9992	2256
6	50136	8969
7	227536	33058
8	870072	114149
9	1887748	360508
10	623800	930588
11	2644	1350852
12	0	782536
13	0	90280
14	0	276