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Inspired by Rubik's cubes with multiple sets of colors or stickers. Suppose there's a Rubik's cube with multiple solutions (represented by multiple sets of colors) made with special paint that reflect specific wavelengths and polarization of light, such that one can only see one set of colors at a time using a special type of filtering glasses, and nobody can see multiple sets of colors at the same time. A group of cubers are allowed to enter the room one by one, each person must borrow a random pair of the special glasses, so they can see a random set of colors only(each set equally likely to be seen), and each person must turn the cube once and then return the glasses and leave the room. They can discuss before the first person enters the room, but cannot communicate in any way afterwards. How many cubers does it take on average to reach a solved state in any set of colors, and what's their best strategy?

You can assume all sets of colors are solvable, and when one set is solved, all other sets are scrambled. The cube starts scrambled in all sets of colors, and you can assume the number of sets is known. Turning the cube 180 degrees is allowed. Specific answers are fine if no general answer is found. There can be as many cubers as needed, but they can't repeat visits.

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  • $\begingroup$ Is a no-op allowed? ie. Enter the room, put on glasses, observe the cube without touching, return glasses, leave $\endgroup$ Commented Jul 1 at 1:28
  • $\begingroup$ @ApexPolenta I don't think so, they are supposed to turn the cube :) But I'm curious how no-op can help significantly, it seems to me it shouldn't help that much if at all. $\endgroup$ Commented Jul 1 at 2:25
  • $\begingroup$ I'd be very surprised if there were any non-bruteforce solution to this. $\endgroup$
    – Deusovi
    Commented Jul 1 at 2:35
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    $\begingroup$ @alices_and_bobs It helps because I had an idea of a possible strategy. You'd need a group of cubers who know CFOP and have practiced identifying mid-algorithm states of the cube. Then, the strategy would be to concentrate on one particular set of colors, and just no-op for any others. Without the option to no-op, this strategy can't get off the ground, and in fact I'm not sure if any strategy could. In which case the fallback would be to rely on the Hamiltonian circuit, requiring quintillions of moves. $\endgroup$ Commented Jul 1 at 4:57
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    $\begingroup$ @alices_and_bobs Do you know if there is a solution to this that does not visit (almost) every state of the cube? $\endgroup$ Commented Jul 1 at 11:18

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