Fri 5 Jan 2007
In a period of some confusion last month I considered returning to school and finishing my physics degree, and I undertook a review of abstract algebra, just to see if I was up to it. This incited a glance at Douglas Hofstadter’s articles on the Rubik’s Cube in his book Metamagical Themas, which happened to be on my shelf. I was fascinated by the thing. The story of its invention, the elegance of its construction, the sheer difficulty of solving the thing, and the untold number of people , who purposed to solve the damn thing, not to mention my own (lately latent) proclivity for puzzles (and the fact that I had never solved the Cube), virtually forced me to get my hands on one. I found an old Cube in an online auction, still in its original packaging from 1981. When it arrived, I did not immediately scramble it. Instead, I did some experiments – various combinations of turns repeated over and over, which I knew must lead back to the original position eventually. I discovered a few things during these explorations that would later be essential. On one such venture, though, I lost my place and could not recover. I tried intermittently for two weeks to solve the thing, but could only get to a certain point before I was stuck. Then I did something I had read how to do in Hofstadter’s book, something I had been meaning to do since I got the thing – I took it apart. If you have one lying around and have never done so, I encourage you to take it apart. It’s inner workings are surprising and beautiful. And don’t worry – you can do it without breaking it.
After doing that once, I resolved to actually solve it. After two weeks or so of occasional study, I was able, without severe pain, to set everything on the cube back to rights except six pieces – and I was completely lost as to what to do at that point. I would pick it up at least once a day, and, sometimes without moving it at all, set it back down in frustration. It was clear that if I was going to solve it, I would have to scrutinize the exact details of the cube’s changes under various moves, keeping meticulous notes, and try to calculate the solution. But I just didn’t care enough. Solving a Rubik’s Cube was not going to add anything to my life. Solving it would do nothing, I thought, besides cure me of the urge to solve it. It was like a crossword puzzle, or sudoku (I like to call it “sepuku”): in the end, it merely consumes time. Giving up will rid me of the urge in less time that it will take to solve the stupid thing. So one morning, instead of picking up the cube and playing with it as I had every morning for a month, I resigned from the pursuit, and left the cube alone.
But then, in that response, I noticed a pattern in my behavior – although puzzles are not my passion any more, and although solving them may in many ways be a waste of time, the real reason I was quitting was because I did not really think I could succeed. The problem seemed intractable, and I felt I would be wasting time and stabbing in the dark (as I had been, for the most part) endlessly. I knew the puzzle could not be solved by luck, because the number of possible positions was astronomical; you had to figure it out. And without realizing it, I was assuming that I was incapable of it. So about two minutes after giving up on it forever, I decided that I was going to solve the Rubik’s Cube – now. I spent hours – most of the day, in fact – transcribing the effect of a few simple operations in a notation I took from Hofstadter’s article, and calculating combinations. First I discovered a set of moves that “switched” 2 pairs of cubes at a time, but I still had to figure out how to get the cubes back to their home in the right orientation. After many more hours and pages of computation – much of it in a sort of “meta-notation” I was inventing on the spot – I thought I had found the solution. I went to my friend’s house to solve it in front of him. I needed a witness, in case anyone questioned the honesty of my solution method. I sat down, and executed all the turns, exactly as I had written them down; and as I completed the final turns, I saw all the little pieces fall exactly into place … except two corners, on opposite sides, that were in the right place, but turned the wrong way. You might imagine this was devastating, but I was invigorated by how much I had accomplished, and by how powerful a single, confident decision could be. So it was the next day when, after not only refiguring my notation, but also tracking down the errors I had made the previous day, I managed to put Humpty Dumpty back to square again.
The title of this article is no joke – I really did learn about life from solving the Rubik’s Cube. I admit that the following truths were not clear to me until now:
- If lots of people have accomplished something, then, no matter how difficult it might seem, it’s not that difficult. Otherwise all those people would not have succeeded.
- Self-doubt is sometimes the one and only barrier to success.
- Sometimes all you need to solve a problem are the right terms to think in.
- If something seems impossible, that only means it is difficult. It may not even be especially difficult.
- Most things have already been done. There’s no need to reinvent the wheel. (unless you really want to)
- It’s worth doing some things, even when you really don’t want to.
And Rubik’s Cube is a perfect example of the following fact: Sometimes you have to undo the progress you’ve made to progress further. That means that holding on to the partial solution is not always – or even usually – the best idea. This is of particular significance in political thought, where people are often unwilling to let go of the smallest gains in the interest of the bigger picture. The people of Michigan recently voted to forbid affirmative action in their state. This was seen as regressive by many, and it may indeed be, but it is clear that, if we are to have the kind of society we want, government-sanctioned discrimination must stop. It is also clear, though, that whenever it is stopped, we can expect more violations of the fairness we are trying to accomplish. So, how do we now “unbreak” what we have broken? Perhaps I should get cracking on group theory…