For a given Rubik's cube algorithm $A$ let $\mathfrak C(A)$ be the number of times, we have to repeat algorithm $A$ to get back to where we've started. For example if $A=RUR'U'$ then $\mathfrak C(A)=6$ The question is:
What is the greatest value of $\mathfrak C(A)$, we can achieve?
This question is equivalent to following maximization problem:
Maximize $LCM(a_1,a_2,…a_i,b_1,b_2,…b_j)$ under the following conditions:
$$~a_1,a_2,…a_i,b_1,b_2,…,b_j \in \mathbb N_+~$$
$$a_1+a_2+…+a_i=8$$
$$b_1+b_2+…+b_j=12$$
$$2 \ | \ (a_1+a_2+…+a_i+b_1+b_2+…+b_j)$$
Where LCM is Least Common Multiple function.
Thanks for all the help.
Longest Rubik’s cube algorithm – maximization problem
gcd-and-lcmoptimizationrubiks-cube
Best Answer
Wikipedia's article on the Rubik's cube group says that the largest order of any element in the group is 1260. For instance, $RU^2D'BD'$ is one such move.