The 3x3x3 Rubik’s Cube, can you solve it?

Who among us has not lost at least one afternoon of their life to that most seductive of toys: The Rubik’s Cube? Originally invented by the Hungarian architect Erno Rubik in 1974, this cube – although apparently not its patents – have stood the test of time.

The beauty of the Rubik’s Cube, much like the beauty of mathematics, is that it seems totally impossible at first. But as soon as you learn the solution, it becomes totally trivial. The problem is to take this jumbled up cube, and perform a series of permutations (by twisting across various axes) to get each face to display a single color. For a 3x3x3 cube there are 4.3252×1019 possible permutations to chose from. That’s quite a lot. But even so, computations taking 35-CPU years by a bank of computers at Google show that the worst possible jumbling of the cube can always be solved in 20 or fewer moves. This maximum number of moves to solve a Rubik’s cube is known as God’s Number.

So this means that for any jumbling, you’re always only 20 moves away from a solved cube. Now you see where things start to get tantalizing. Of course you may not solve the cube *perfectly*, that is, you might use an algorithm that ends up taking more than God’s Number. But just knowing the solution is so close at hand is already fun. The difficulty then is in coming up with an algorithm to solve the cube, and most methods do this by breaking down the algorithm in to several sets of moves, or “macros.” And these can be best thought of as operations in group theory. We can think of permutations of the cube as elements of a group, *R*, whose binary operation is concatenation of moves. Then building the macros to solve the cube can be thought of in terms of commutators and conjugates, see this great explainer for the full story.

So, if you are looking for a holiday gift to ~~occupy~~ please your mathematical loved ones: look no further! Math’s Gear will meet all of your Rubik’s related needs with competition grade speed cubes of all dimensions. They even have the really fun looking — but I’ll admit, slightly intimidating — Skewb puzzle cube. And what better to accompany this gift of procrastination than a paper on group theory to go with it, or for the les mathematically inclined, a plain english explanation of the macros and algorithm.

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