(Please see the edit at the end of this post before reading.)
For 4 years, I have been studying the nxnxn Rubik's cube as one of my hobbies.
For those who know about how to use commutators to solve a cube (which allows a person to not have to resort to using algorithms/cube moves which one must memorize), I believe I have actually proven that any even permutation of the nxnxn Rubik's cube, where n > 1, can be solved with at most 2 commutators.
In addition, those who know about conjugates, I believe I have also proved that all permutations of the nxnxn cube (both odd and even permutations) can be solved with one conjugate.
If you don't know what these terms mean or if you know what they mean in mathematics but not for Rubik's cubes, don't worry. I list definitions and prerequisite information at the beginning of the document.
I sometimes mention the "Cube Laws" in the document. This page explains what I mean by "cube law." Laws of the cube
Here is my document. Commutator and Conjugate Theory
I have posted this on speedsolving.com here, but so far I have gotten very little feedback from them. Unless mentioned otherwise, all of the content in the document is my original work (the images were made from the program cubetwister).
Even before you all read the document (or a portion of it), assuming that the statements I mention are true (and my proofs are correct), has this been known before in the Abstract Algebra world? What real world applications can there be? I've read that commutators are in quantum mechanics...
Lastly, I did a couple of real examples in preceding posts of that thread. Besides the fully detailed 5x5x5 example in post #42, in post #35, I gave a 2 commutator solution to a random 3x3x3 even permutation scramble. In post #38 is a 1 commutator solution to the same 3x3x3 scramble. As you can see in the 1 commutator solution (click the link to bring you to an online cube applet and click the play button to see the animation), I could not solve that 3x3x3 scramble completely with 1 commutator, but almost. (My paper claims that a scramble like that cannot be solved with 1 commutator because of the edge orientation case.)
I have actually constructed a wrong proof for the statement called "Theorem 2" in the PDF (I was given counterexamples thanks to some guys who share my interest in Rubik's cubes). So it's possible that all even permutations of the nxnxn Rubik's cube can be generated/solved with only one commutator. If you follow the thread I linked to earlier (the one with the 5x5x5 example), you will see that NOW I cannot yet find one position which cannot be generated with one commutator. So it is possible that all even permutations of the nxnxn cube can be solved with one commutator, but it is not yet proven (or disproven).
In short, disregard "Theorem 2" in the PDF and the first page (the front cover). Everything else is correct to my knowledge, although some of the restrictions are unnecessary for what will be my one commutator proof, but are completely acceptable for proving "Theorem 3" in the PDF.
Last edited by cmowla (2013-03-17 05:09:11)