Rotating only one cubie

Filed under: math — jlm @ 22:00

I’ve entertained myself for countless hours with the Rubik’s Cube, but always as kind of a pointless diversion. It’s something to keep my mind occupied, but I’ve never seriously studied it. I’ve built up a lot of intuition, but when it comes to the group theory that the cubes embody, I have nothing to say. One of the bedrock concepts I’d developed from my intuition is that it’s impossible to rotate only one cubie. You can flip two edge cubies (any two), or four or all twelve, but not one or three. Similarly, you can rotate a corner cubie by 120°, but only if you cancel it out by rotating another corner cubie (any of them) by −120°, or rotate two others by 120° as well to add up to a full 360° (and so be equivalent to no rotation). Why is this? My intuitionistic argument is that a single 90° rotation of a face rotates each of the four edge cubies of the face by 90° and the same for the four corner cubies.

Quarter face-turn rotates 4 edge cubies     Quarter face-turn rotates 4 corner cubies

Since each turn rotates the edge cubies by 360° in total and every sequence is a composition of face turns, the “total rotation” of all the edge cubies together is a multiple of 360° and so equivalent to 0, meaning there can only be an even number of 180° edge cubie flips. The corner cubie rotations sum the same way, hence the corner cubie rotations that don’t cancel themselves out must sum to a multiple of 360°. The conclusions are correct: no sequence can do an odd number of edge flips or corner rotations that aren’t equivalent to zero. The argument is however unsound: adding rotations around different axes together willy-nilly like this can (in other contexts) produce complete nonsense — there’s some extra structure in the cube that makes this OK.

Anyway — I’ve “known” for 35 years or so that you cannot rotate a single cubie of the cube, you must also rotate other cubies of its type to cancel it out. But then, semi-recently, I encountered for the first time a “picture cube” where the orientations of all six center cubies had to be right. The first time I scrambled and solved it was fairly easy: first I solved it like the normal cube, and after that it was easy to work out how to rotate one center cubie by 90° and another by −90°. The second time I scrambled and “solved” it I was left with one center cubie upside down and everything else as it should be. And my brain kind of broke. The one center cubie had rotated 180° and everything else was in place. This had to be impossible, you can’t rotate just one cubie, you need to maintain a balance, there has to be another rotation to cancel it out, doesn’t there?!?! I felt like I had taken a number, multiplied and divided it by another number, and gotten a completely new number as a result. Gradually my faculties returned to me, and I thought back to why I thought you can’t have an unbalanced rotation, and it was blatantly obvious why the argument didn’t apply: a face turn rotates only one center cubie, so it does alter the total rotation of all the cubies after all. Each quarter turn preserves the total rotation of the edge and corner cubies, but alters the total rotation of the center cubies by 90°.

It took me about 15 minutes to completely shake off the feeling of unreality and get myself thinking in the paradigm where center cubies can all rotate independently. In another five minutes, it occurred to me that if I double-swapped the opposing edge cubies of one face and double-swapped the opposing corner ones as well, that was equivalent to a 180° face turn except the center cubie wouldn’t be rotated. That let me solve the configuration the cube was in. What about a 90° rotation? Impossible: the parity of the number of face quarter-turns was also the parity of the number of quarter-turns of all the center cubies summed together. So I had a full solution to the picture cube. Playing around with it further, I optimized the double-double-swap into a double-swap of an edge-and-corner-cubie complex and figured that was as far as I could take it. Yet, I’m sure that what’s going to stick with me the most is the eerie sense of irreality of staring at that lone center cubie, a thing that couldn’t be flipped, but was.

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