Rubik's cube permutations
I suck with really big numbers, so that's where you guys come in :)
I basically want to know the number of permutations a 1000x1000x1000 Rubik's cube has, as well as a 5x5x5x5x5 Rubik's cube. Yes, a 5-dimensional one.
I've been reading these a formula's, and they aren't that comlicated, they just involve huge numbers, and that's where my brain shuts down.
You can read about 3x3x3, 4x4x4, and 5x5x5 permutations on Wikipedia. I will provide links for all. I will also provide a link about the anatomy of an n-dimensional Rubik's cube as well as the permutations for a 3x3x3x3, 4x4x4x4, and 5x5x5x5 Rubik's cube, similar in the same manner Wikipedia does.
n-dimensional Rubik's anatomy: http://www.gravitation3d.com/magiccube5d/anatomy.html
3x3x3x3, 4x4x4x4, 5x5x5x5: http://www.superliminal.com/cube/permutations.html
Thank you guys for reading! Hope you enjoy computing :)
Here's a place to start:
It should be a finite index subgroup of a direct product (running over the orbits of the "cubies") of wreath products, as for the 3x3x3 cube. Before looking up the answer, I suggest you try working it out yourself following the excellent explanation of the 3x3x3 case at http://unapologetic.wordpress.com/category/rubiks-cube/
Sorry Chris, the math there is beyond me. I've tried figuring this out many times before. I enjoy CRGreathouse's post because that gave me a formula in which to figure out an NxNxN cube, although I'm not sure how to run the script in Maple. I used Maple and converted the script to Java which is far easier for me to read and it compiles after a tweaking session, but I don't know what variables do what :\
At least I'm a bit farther. I was just hoping someone could say "here's a formula for an NxNxN and one for NxNxNxNxN" or a formula for calculating the exact figures (1000x1000x1000 and 5x5x5x5x5). Maybe even a formula for N-sided I-dimensional. Apparently it's not that easy.
I always make the assumption that because I suck at math, everyone else is amazing at it and this is child's play to them. Figuring out these formula's is like me trying to comprehend Graham's number.
Thanks for your effort so far though!
C. R. Greathouse?
This isn't hard if you know what a wreath product is. That isn't hard if you know what a permutation group is. If you're curious, try Neumann, Stoy, and Thompson, Groups and Geometry, Oxford University Press, 1994 which is a really readable and wonderful book with very few prerequisites. The 19th and last chapter computes the number of elements in the group of the 3x3x3 Rubik's cube; a similar computation is given by John Armstrong in the webpage I cited. Once you understand this, you can make a start on the 4x4x4x4 cube and so on.
(Well, it might help to know something about SO(4) and so on in order to make sure you have the right generators of the permutation group whose size we are trying to compute--- you didn't say but I assume you are trying to compute the nxn..xn analog of the group of permutations of the set of "facelets" in the 3x3x3 cube obtained by iterating quarter turns of the six faces, so six generators.)
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