The question is asking what is the shortest sequence of moves that will have the cube solved at some point during the sequence. The most obvious such sequence would be one that has every solution to every position followed immediately by its inverse. This would be incredibly long, somewhere in the hundreds of quadrillions of moves. Is it even possible to find a shorter sequence?
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The title and body seem to me to be asking different questions. It’s known that any Rubik’s cube position can be solved in at most $20$ moves (https://en.m.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube). But in the body it sounds like you are asking for a fixed sequence of moves which, when applied to any starting position, solves the cube at some point. Is that right? – Qiaochu Yuan Jan 04 '21 at 07:49
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Yes, you are correct. I could have worded it better. The question is what is the shortest fixed sequence that will result in ths cube being solved at some point. – Ethan Bottomley-Mason Jan 04 '21 at 07:52
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https://puzzling.stackexchange.com/questions/4820/is-it-possible-to-use-one-sequence-of-moves-to-solve-the-rubiks-cube-from-any-p Does this help? – Jan 04 '21 at 07:53
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Devil's Algorithm http://anttila.ca/michael/devilsalgorithm/ This should help. – Jan 04 '21 at 07:56
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Here is 1 from math stack exchange https://math.stackexchange.com/questions/184760/brute-force-method-of-solving-the-cube-how-many-moves-would-it-take?rq=1 – Jan 04 '21 at 07:57
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Does this answer your question? Brute force method of solving the cube: How many moves would it take? – Jan 04 '21 at 07:58
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You are looking for what is sometimes called the "devil's algorithm", which is a Hamiltonian cycle over cube states.
People have wondered about such a thing for some time, and there are explicit constructions; see here for example.
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