Number of combinations to the Rubik's Cube and variations
On this page I have some formulas that I came up with for the number of combinations to the n x n x n rubik's cube, and two variations. I had actually seen the formula for the number of combinations to the n x n x n cube on Richard Carr's webpage before trying to figure out how it works. I spent some time trying to figure out how he came up with it, and the first formula below is my version of the formula found on Richard Carr's page. Using what I learned from figuring out how to prove his formula, I came up with two more formulas. The second formula shows the number of combinations to the n x n x n supercube, where every single piece is distinct from ever other one. The third formula is the number of combinations to the n x n x n super-supercube. This is also referred to as the most general form of the Rubik's cube in three dimensions. It's consists of a group of n3 pieces arranged into a cube, which is able to turn like an n x n x n supercube. However all the pieces on the outside, and inside, part of the cube must be returned to their original locations and orientations in order for the cube to be solved. Any slice turned moves exactly n2 pieces, 4n-4 pieces on the outside part of the cube and (n-2)2 on the inside of the cube. If you are interested in how I came up with these formulas then you can e-mail me. Basically I just used the ideas necessary about even and odd permutations within specific orbits of the cube. An orbit is defined as all the places on the cube that a certain type of piece is allowed to occupy.
Number of combinations to the n x n x n cube
Number of combinations to the n x n x n supercube: The cube where the positions of the centers are noticeable. An example would be an n x n x n cube with pictures on each face.
Number of combinations to the n x n x n super-supercube. An n x n x n super-supercube is an n x n x n supercube, inside of which is an (n-2) x (n-2) x (n-2) supercube, inside of which is an (n-4) x (n-4) x (n-4) supercube, etc.. Turning an inner slice of the outermost n x n x n supercube will affect some or all of the internal supercubes. Per Kristen Fredlund has written a program to simulate the 4x4x4, 5x5x5, 6x6x6, and 7x7x7 super-supercubes. You can download the program from the files section of the Yahoo! Group.