Rubik's Cube Research Paper

Permutations
The interdependence of the components of the distinct, interlinking components of the Rubik’s can yield a massive number of permutations on such a scale. This occurs because of the numerous ways that these components could be arranged and oriented in, including the displacement and orientation of components in unsolvable cubes which consequent in a whole new “world” of different permutations.
As there are 3 distinct, integral components in a Rubik’s Cube, they all must consider when identifying the number of existing of permutations. These 3 integral components are represented by the center, edge, and corner pieces. Starting with the center pieces, they are not calculated in this process because their position is forever fixed

The 43 quintillion ways to arrange the cube drastically increases when center pieces are taken into consideration. With marker pens, one can mark each center piece of each face with four colored marks on each edge, each corresponding to the color of the adjacent face, producing what is known as a “supercube”. Consequently, one can solve the Rubik’s cube but still the markings on the centers rotated, enhancing the level of difficulty.
Marking the cube’s center pieces enhances its difficulty because this modification increases the set of distinguishable possible configurations. Since there are 6 center pieces and 4 possible rotations of piece, we can orient the center pieces in 46/2(2,048) ways. We divide by two because when the cube is in its solved state, apart from the orientation of the center pieces, so there will always be an even number of center pieces requiring a quarter turn, producing an even permutation (a permutation obtained by the modification of an even number of elements). Consequently, the orientations of the center pieces increases the total number of permutations from 43,252,003,274,489,856,000 (4.3×1019) to 88,580,102,706,155,225,088,000