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Let's say you a Rubik's cube that can be solved using a sequences of the 12 standard moves. Can it be solved if you are required to alternate between clockwise and counter-clockwise moves? If not, which positions can you solve a Rubik's cube using these moves?

One way to solve this problem would be to show that regular clockwise quarter turn on a Rubick's cube can be simulated by an even number of these alternating moves, since then you could string these plays together for every clockwise quarter turn required to solve the rubik's cube the regular way (we only need to consider clockwise quarter turns since any normally solvable rubik's cube can be solved using only clockwise quarter turns). I'm not sure if this is the only way though.

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  • $\begingroup$ Hmm, interesting question. One option is to scramble by only alternating clockwise and counterclockwise moves. If you scramble it that way, you can also solve it that way. (Kinda like a 180 degrees scramble & solve.) But just normal scrambling and then solving it by alternating clockwise and counterclockwise I dunno, I leave that to the mathematicians among us. Favorited since I'm also interested to know the answer. $\endgroup$ – Kevin Cruijssen Feb 24 '17 at 21:01
  • $\begingroup$ @KevinCruijssen yeah, I was wondering if math.se would've been better. $\endgroup$ – PyRulez Feb 24 '17 at 21:26
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    $\begingroup$ No, I think this puzzling SE is more on topic for this question, considering it's about a Rubik's Cube. I've also linked to your question on the TwistyPuzzles Forum. There are quite a lot of people on the forum that would be able to answer these kind of questions, so any answer they'll post I will post here as well to keep you informed. $\endgroup$ – Kevin Cruijssen Feb 24 '17 at 22:01
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No it can't. The number of positions which the corners can reach with alternating moves is less than the total. An analysis is here. The short of it is that there are 8! possible positions of the corners in a 2x2x2 but 8!/120 reachable positions if you only allow alternating moves. The analysis is done using GAP via brute force. The reasons why this strange group turns up in this place are mysterious.

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