Message #3574

Subject: Re: Newbie queries on Magic Tile - Klein Rubik
Date: Sat, 26 Nov 2016 13:38:59 -0800

I can’t help much with your questions #1 and #3, but as for #2, about parities - one of the interesting things about the Klein Bottle (and some other non-orientable surfaces) is that corners can be mirrored, so that one sticker will be in place but the other two will be swapped. For example, if reading clockwise you expect yellow-red-blue (YRB), on a normal cube you could have RBY or BYR, but on a Klein Bottle you may also see YBR, BRY, or RYB.

Now here’s the trick: mirroring the same corner twice in different ways will have the same effect as twisting it on a regular cube. That is, swapping stickers 1 and 2, then stickers 2 and 3, will do a 3-cycle of stickers. So one way to twist a single corner would be to mirror corners A and B, twist corner A (plus some other corner(s)), mirror corners A and B again, and then undo the twist. Now B is solved, but A will be twisted because there was a 3-cycle on its stickers in a way that is independent from the twisting steps above.