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I was using the beginner method to deliberately solve a 3X3 cube with one edge piece flipped. But, by the time I got to the top I discovered that I couldn't solve it with only one edge flipped. I had to also flip an edge on the top layer. Why is that?

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  • $\begingroup$ That question specifically asks for a mathematical explanation. I honestly don't remember that much about group theory and I was wanting a more plain language answer. Is that a good enough reason to keep this question open? $\endgroup$
    – crownjewel82
    Jun 1, 2014 at 13:06
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    $\begingroup$ Not a dupe. Edge =/= corner. $\endgroup$
    – clickbait
    Jun 9, 2018 at 18:26

1 Answer 1

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12 edges on a cube. 24 positions an edge piece can be in once you consider flipping. If you consider one move to be a single quarter turn, then you can move an edge to exactly 12 of those 24 positions in an even number of moves, and to the other 12 only in an odd number of moves. You should be able to convince yourself of this by trial and error, but if you want proof you could come up with names for the 24 positions and check that each possible move only moves edges from a position in the even set to one in the odd set or vice versa.

Now consider what a single move does. It moves each of four edge pieces from one set of positions to the other. This means if you started with (say) 6 in one position-set and 6 in the other, you now have 2-10 or 4-8 or 6-6 or 8-4 or 10-2, but you can't have an odd number of pieces that changed position-sets.

In order to flip a single edge piece, you need to move a single piece to the other position set without moving any of the others. But you can only switch the position-sets of an even number of pieces at a time.

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  • $\begingroup$ While this answer is a good intuitive explanation, I'm not completely certain it's a sufficient one. For instance, on the tetraminx, three pieces change positions, but one edge still cannot be flipped. $\endgroup$
    – user20
    May 31, 2014 at 5:26
  • $\begingroup$ on the tetraminx you need to make an even number of moves to get the non-edge pieces back to where you want them $\endgroup$
    – greg m
    May 31, 2014 at 6:47
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    $\begingroup$ no that doesn't explain it, sorry. i've never seen a tetraminx before so bear with me. but it seems like you can make three moves and end up where you started. anyway, just because you need a different explanation for the tetraminx's property, doesn't mean my explanation doesn't hold for the cube. (though for sure i could be mistaken) $\endgroup$
    – greg m
    May 31, 2014 at 6:53
  • $\begingroup$ That's fair, but it isn't always the case. Your writeup of the intuitive proof is good, but it doesn't necessarily hold true as a line of reasoning. It is still possibly/probably true, though! $\endgroup$
    – user20
    May 31, 2014 at 7:05
  • $\begingroup$ sure, any proof can contain a mistake. but i presume you can't see anything wrong with this one as it applies to the cube? $\endgroup$
    – greg m
    May 31, 2014 at 7:20

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