I tried to write a program to count the number of repeated sequence(s) to make NxNxN (for N=2 to 20) rubik cube back to it's initial state/placement. I solved it by counting the length of cycle of each tiles and calculate the lcm of all length. When I print the cycle length of each tiles, I never saw cycle length of tiles greater than 24.
Is it always true that if we do repeated sequence(s), the cycle length of tiles of NxNxN (for all N) always less than or equal to 24? Is there a prove or a way to prove this? Or is there a way to count the maximum cycle length of tiles from a repeated sequence(s)?
Best Answer
Yes, each tile belongs to an orbit of no more than $24$ positions under the action of the Rubik's cube motions, no matter how large the cube.
For your larger question, about how many times we must repeat an action to get back where we started (the order of an element in the Rubik's cube group), see this older question.