Today in school I wondered if i could make an equation to figure out the number of combinations possible on any rubix cube. Only considering the puzzles that are actually cubes, and have an even distribution of pieces (no 3x4x3's) I came up with p=(6c)! x (3e)! x 8! x 2^(3e) x 3^8/(3e) Spoiler A little ironic how a cube has 6 sides, and if you add all the numbers in the equation and divide by 6, it equals 6. (6+3+8+2+3+3+8+3=36) Where p= # of possible combinations, c= # of center pieces on one side (is zero if the cube is 3x3x3), and e= # of edges on one side. The problem I ran into was when looking at the the equation for a 3x3x3 (12! x 8! x 2^12 x 3^8/12) is it did not include center pieces, due to the center pieces on a 3x3x3 not moving. (while you can move them, it doesnt effect the cube as they will still be in their relevant position to the other centers, moving the red center will not move it away from being adjacent to the green and blue centers, or opposite of the orange) This is my issue, that I dont know how to include the centers, so when making the equation I made above, i treated them like other pieces by using a factorial, without having to compensate for differentiating colors. So I looked at a 4x4x4. There are 4 centers on each side, but can each move to each side, and be in any combination on each side. There are 6 sides, each with 4 centers. So 24 centers, which can be moved to any of the center slots. Same with the 5x5x5, 9 centers on each of the six sides, so 54 centers with the ability to go into any of the 54 slots. I haven't researched this at all, but I thought it was interesting, and would also like to hear if I'm wrong or not haha.
Might be interesting to see if your '36 divide by 6=6' is actually a square root, although i don't know if you can have a rubicks cube with more than 6 sides (cba to google that on my phone). Everything else made perfect sense though, nice work.