Cubies and Simple Groups

Even though I love to solve puzzles, it was only recently that I finally got around to learning how to solve the Rubik's Cube. I could only ever get one face at a time, and I never had the patience to learn how to keep that face from messing up while I tackled another part. I got a tip from a friend that the second phase involves working on the second “layer”, thinking of the completed face as being the top/bottom of the cube. But I still couldn't figure it out.

Then I got an email newsletter from the American Mathematical Society pointing me to an article in Scientific American about some new puzzles that are similar to the Rubik's Cube. (You can read about the puzzles here or play them here.) These puzzles are based on so-called simple groups, and require slightly different techniques to solve than the Rubik's Cube. It was at this point that I realized that I never learned the Cube, so I was probably at an even bigger disadvantage with these new puzzles.

At first I found a few web sites that teach you a series of moves to try depending on what state your cube is in. But this felt sort of hollow, as though I were simply memorizing sequences of moves with no real pattern behind them. Then I checked out a few books from the library that teach the “theory” behind the cube: Handbook of Cubik Math by Alexander H. Frey, Jr. and David Singmaster; and Rubik's Cube Compendium by Ernö Rubik, Tamás Varga, Gerzson Kéri, György Marx, and Tamás Vekerdy. These books explain various solution methods in terms of what you're actually trying to accomplish, which I found much easier to remember.

Now I can solve the Rubik's Cube, though I'm not very fast. It's still pretty satisfying, to mix up the cube, and then methodically solve the bottom layer, then the middle layer, and finally the top layer.

So what are these “groups”, and what do they have to do with puzzles? I don't have time to really explore group theory, the branch of mathematics that studies groups, but I can try to give a brief explanation. A group is a concept which involves some objects and a single operation on those objects. A really good example here is the integers (0, ±1, ±2, ±3, …) along with addition. For the Rubik's Cube, you can think of the objects as the rotations of the faces, and the operation is doing one rotation followed by another one. The goal of the Rubik's Cube is to “undo” all the rotations that put the cube in the scrambled state.

Simple groups are a little harder to define, but it suffices to say that you can build up any finite group out of finite simple groups. So they are like the building blocks of group theory. These groups really aren't “simple” in any usual sense of the word, and the puzzles in the Scientific American article above are quite challenging. I hope to tackle the first one soon, now that I've learned the basic concepts of the Rubik's Cube.