Rubik's Cube theory: Why is 1260 the highest order? [EN]

This video gives an introduction to Rubik's cube theory and proves that 1260 is the highest possible order in the Rubik's cube group. It is a followup to my previous video (   • Rubik's Cube theory: Why R U2 D' B D'...  ) where it was shown that R U2 D' B D' has order 1260.

We first look how to model the sequences of moves as permutations and then study cycle decompositions and signatures. We then reduce the claim (that 1260 is the highest order) to a problem in number theory. Finally, we solve this problem with some Python code.

Sections:

• 00:00 intro
• 02:18 maps
• 08:25 permutations
• 11:34 composition of permutations
• 14:01 order of a permutation
• 15:21 Rubik's cube group
• 18:33 reduction to edges and corners
• 26:08 cycles
• 31:34 examples of cycle decompositions
• 41:08 partitions
• 44:06 formula for the order
• 47:43 number theory problem
• 51:56 signature of a permutation
• 55:24 theorem about signatures on Rubik's cube
• 58:17 formula for the signature
• 01:00:05 reduction step
• 01:03:36 coding
• 01:14:41 running the code
• 01:18:00 all the possible orders
• 01:21:13 bigger cubes

Links:

• Signature of a permutation: https://en.wikipedia.org/wiki/Parity_...
• Code that verifies that 1260 is the highest possible order: https://gist.github.com/ScriptRaccoon...
• Orders of the elements in Rubik's cube group: https://oeis.org/A338883
• Determine the highest order of an element of a Rubik's Cube group: https://math.stackexchange.com/questi...
• On Rubik's group: https://people.kth.se/~boij/kandexjob...