Gleaning the Cube

I’ve become obsessed with remembering how to solve this damn Rubik’s Cube.

I use the white cross method, and I’m able to get the top and middle sections solved with relative ease. It’s the bottom that’s been kicking my butt.

The white cross method is a series of algorithms that start with solving for the edge cubies on the white side (forming a white plus sign or cross), then solving the white corner cubies, then solving the four middle edges.

This is where it breaks down for me. When flipped over, with the white center on the bottom and the yellow center on the top, there are three basic configurations you look for. From any of those three configurations, there are a series of algorithms that can be run to get to another interim step.

The goal here is to solve the yellow edges, then finish with the yellow corners. Voila, it’s solved!

However, it’s not so easy.

(This is about to get pretty numbers-heavy, so feel free to bail at this point. I won’t judge you.)

There are 26 physical cubies (as they’re called), but there are 54 individual cubie faces exposed – nine each of white, green, blue, orange, yellow and red.

Let me do the math for you. If you call the current state of the cube it’s configuration, then there are more than 43,250,000,000,000,000,000 possible configurations. That’s 43 quintillion 250 quadrillion. That’s a huge number. And only one of those 43 quintillion possible configurations is the “solved” configuration.

Someone much smarter than I am figured out that there is a one in 43 trillion chance – that’s 1/43,000,000,000,000 – that you’ll “accidentally” solve the puzzle.

In 1974, when the Hungarian professor of architecture Erno Rubik invented his Hungarian Magic Cube, the story goes that he developed the puzzle as a way to demonstrate to his students how to build a structure with multiple moving parts without the structure falling apart. He didn’t realize that he’d created a puzzle until the first time he scrambled it then tried to restore it to its original state.

One account I read stated that it took Rubik a full month of testing, developing nomenclature, and logging test algorithms before he discovered a solution. But to be sure, his original solution is just one of many.

The Rubik’s Cube – or the solution to it, anyway – falls into the category of Group Theory. When considered as a whole, or in subparts, the groups of algorithms that comprise any given solution allow the puzzle solver to successfully solve the puzzle.

With much more than 43 quintillion possible configurations, it came as a huge surprise to me that the God Number for the Rubik’s Cube is 20. The God Number, or the minimax value, is the least number of moves (algorithms) that an omniscient being would need to solve the given puzzle from any configuration. The upside of this is that it makes competitive speed cubing, as it’s called, much less mysterious. With the potential of being able to solve any given configuration in twenty moves or less, it’s just a matter of memorizing the various algorithms and practicing for hours on end. Much like learning tennis or guitar or cooking, it’s an acquired skill involving much practice and hours of determination to improve.

Just as interesting to me is the search by mathematicians for the Devil’s Number, or Satan’s Number. Whereas the God Number involves the minimum number it would take to solve any configuration just once, the Devil’s Number is concerned with determining the minimum number of algorithmic moves it would take to solve every one of the 43,250,000,000,000,000,000 different configurations. I know for certain that it’s a number between 43 quintillion and 865,000,000,000,000,000,000 (43 quintillion times 20), but anything beyond that I have to leave to someone with a more powerful computer than my Mac.

Whew, that’s a lot of numbers! I think I need a nap. But first I’m going to try to solve this last corner cubie…

(Credit for much of this goes to the SYSK podcast on Rubik’s Cubes and the geniuses that inhabit the math side of Wikipedia…)