# Shortest paths when randomly scrambling a Rubik's cube

I saw a previous question about local maxima for the number of moves in a Rubik's cube solution, and I wondered what is known about the distribution of shortest paths when randomly scrambling a Rubik's cube.

1. If I were to pick a (solvable) configuration at random, how far is it to the solved state?
2. How many random moves does it take to sufficiently scramble the cube?

For the second question, I'm thinking of it as the mixing time, like for the Diaconis 7 shuffle result for a deck of cards. (Perhaps this is the wrong terminology?)

This is related to another question of mine about adversarially scrambling a Rubik's cube.

Edit: I accepted Marzio's answer, though I still wonder about the mixing time. Google revealed some results for particular finite groups, but I didn't see anything about Rubik's cube. Maybe one of these is applicable / can be ported over?

• @LucasCook: but as commented by Tsuyoshi I didn't answer your second question. It probably involves applying randomly a generator for the group $G$, but I don't know if it's enough to pick uniformly at random a generator (simple random walk) and how many times. Perhaps the question can be generalized to Cayley graphs and re-asked here or on cs.stackexchange.com (I don't think it's research level). – Marzio De Biasi May 10 '12 at 8:21