Is there a sequence of moves that can be repeated over and over again which can solve any legal position the Rubik's Cube? If so what is it, and if there's more than one, what's the shortest? If not, prove that it's impossible.
The sequence can be of any length but cannot be broken up into multiple pieces to be executed depending on Rubik's cube patterns; it must, when repeated from start to finish, end in a perfectly solved Rubik's cube.
Answer
The only way you can have a solve-all sequence is if you have a sequence of moves that goes through all 43 quintillion configurations of the Rubik's Cube. In order to do this, you need to draw a transition graph between all the states of the Rubik's Cube and find a Hamiltonian cycle through them.
This sequence of moves doesn't necessarily have to be 43 quintillion moves long - a simple sequence of 4 moves can produce a cycle of 1,260 configurations as seen in mdc32's answer, and in general a sequence of symbols in the group will produce a cycle of configurations much longer than the cycle itself. However, the sequence will still be very long, simply because 43 quintillion moves is still a lot.
Micah provided a link to a page that did construct such a Hamiltonian cycle in a comment. I haven't been able to make head or tail of its notation (or to figure out how to count the number of moves from the descriptions of the cosets), but it looks like the sequence of moves that is required is billions of moves long, which is still definitely outside of the realm of plausibility for memorization.
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