A question that I believe remains unanswered is
Starting from the solved position, how many Singmaster moves must be done such that each of the $43252003274489856000$ configurations are roughly equally likely to occur?
That is,
What is the Markov chain mixing time on the Cayley graph of the Rubik's Cube Group with generators $\langle F, F', F^2, B, B', B^2, L, L', L^2, U, U', U^2, D, D', D^2\rangle$?
We can formalize this as the variation distance mixing time, e.g. finding the smallest $t$ such that $|\Pr(X_t \in A) - \pi(A)| \leq 1/4$ for all words of length $t$ or less and for all subsets $A$ of the Rubik's Cube group $G$.
It is in this context that Diaconis and Bayer showed that $7$ dovetail shuffles is sufficient to mix up a deck of 52 playing cards.