Metamagical Themas: Structure and Strangeness, Part 1
A diary of my journey with the Rubik's Cube
This is a brain dump about Section Four of Hofstadter’s collection of essays - Metamagical Themas. For an introduction to the book and hyperlinks to all the sections, see HERE. This section is broken up into two parts. This is Part 1. Part 2 will eventually be HERE… Also, I will soon return to psychiatry related topics as this blog is in fact, believe it or not, more than a Hofstadter commentary generator.
This section of Hofstadter’s book has essays themed around the following: Often things seem to be chaotic but in truth, there is structure and pattern behind the strangeness. The forces that silently drive the apparent chaos have mathematical structures. The Rubik’s Cube is a prime example. Order and chaos appear and disappear from its surface as it is solved. But it is truly never chaotic; it only appears so. Later we will explore fractals, strange attractor states, and other examples of there-is-order-in-the-chaos phenomenon. But for now, we will look at the 1980’s fad puzzle toy - the Rubik’s Cube.
Starting from scratch
Hofstader has 60 pages dedicated to the theory and algorithms behind the Rubik’s cube. As someone dedicated and strangely attracted to Rubik's Cubes, I still didn't think I could make it palatable. So instead, I tried to solve the mysterious puzzle on my own and document the experience. I had no experience with Rubik’s cube prior beyond that scene from Dude Where's My Car - which did not help me in any manner. Below, is an abbreviated look at that journey.
To start, I spent 30 minutes with a solved cube. Slowly taking it one, then two, then three twists away from the START position. I was hoping to pick up on some clues and patterns before scrambling it. Below are my findings.
Below is the cube twisted twice from START. My heart is already skipping beats.
I can’t explain why very well, but the cube at this stage feels like it has a pattern. I’m not sure when the pattern becomes non-pattern. Pattern recognition is subjective.
There is something to thinking about actions (i.e. twists) as consisting in a ‘2 and 4’ manner. A single twist will alter four faces all at once, but leave the other two faces untouched. Knowing this, we can choose which axis to initiate a twist on based on which face we want to preserve.
It's a strange feeling to 'solve' one side and all of a sudden everything else clicks together too.
It's a trap to solve for 'completion of a side'. Sometimes it looks really easy to almost get all the same color on a side, but that move is not a step in the right direction. It's better to aim for and set up patterns you recognize (such as in the photo above.)
It looks like there are nearly infinite positions, but there are not. Centerpieces never move. A corner piece will never be a middle-edge piece. Each corner piece is distinct in that it has three specific colors. Thus it belongs on a specific corner. Ultimately, every piece has one correct spot.
I wonder what is the ‘most scrambled’ this could be. After 20 turns is it maximally randomized? After that, further twisting doesn't make the number of steps to solving it any longer? Seems so. What is the longest number of turns you can make the cube away from its completed state? Can we mathematically determine the minimum number of turns to solve a cube if we know the starting position? How many unique positions are there? (I googled it - apparently 43 quintillion)
I imagine there are a handful of maneuvers used to move certain squares to certain slots while not impacting the 2-plane. They probably consist of 3-4 twists in which you scramble the 2-plane face temporarily, but it gets corrected in the end. Eventually, doing these 3-4 twists in a row becomes muscle memory and is treated by your brain as one 'move'. Kind of like how eating with a spoon involves a delicate act between your wrist, arm, mouth, neck, and tounge but your brain just lumps all these actions into one ‘move’ called ‘eat soup’. Stringing together these 'moves', solving a Rubik’s cube maybe only takes a dozen or two moves (moves =/= twist)?
Post scrambled.
Dear God, forgive me for my sins. I know not what I have done. - Someone in some religous text
Immediately I realize it is hard to get pieces into place without messing up others. It takes cognitive intention to follow algorithms not just turn. The problem is, I have no such algorithms. I can sit here for an hour thinking about what to move next and make no progress.
I can move a completed row of a square temporarily to a different axis so I can twist on an axis I need, then move the original row back into place. This feels very satisfying (and clever)
1 hour since the scramble
I got a completed face! But I think was a mistake. The edges don’t match. Ok… time to undo my work.
Imagine a corn maze. The maze runner is right next to the center of the maze- just one wall between her and the goal. But it is a dead end. And she has been on a dead end since the very start of the maze when she went left instead of right. That is what solving this feels like. I feel like I’m getting closer by completely solving a side, but I’m just being like our maze runner when I do that. Eventually, when I'm only one twist away, that last twist will solve all the faces simultaneously. With that in mind, no faces will be 'complete' at any point along the process until the very last twist. So I'm going to have to mess up my completed face.
2 hours since the scramble
I'm not looking for algorithms to solve faces, I'm looking for algorithms to effectively move particular pieces to particular places while minimizing screwing other faces up (via twisting them onto off-axises temporarily). In other words, there is no algorithm to follow such that you are always making the cube look closer to START with every step. Instead, you will be doing ‘moves’ (consisting of multiple twists) that move towards START, but within a move, chaos will temporarily seem to be appearing.
It is much harder to notice these algorithmic moves when it is scrambled than when it was in the START position. Like - a lot harder. 3
3 hours since the scramble
Ok - I was wrong earlier. The final twist will solve 4 faces, but the other 2 will have had to have been completed already. Maybe my one complete face wasn’t completely off base.
4 hours since the scramble
I give up.
Chess titan Paul Morphy once said “the ability to play chess is the sign of a gentleman. The ability to play chess well is the sign of a wasted life." I’m going to pretend I agree with him for now and apply his principle to a completely other domain because it is convenient for me and helps me burry my cognitive dissonance while I up a tutorial.
Using a tutorial1, I completed the cube in 30 minutes. In retrospect, most of my thoughts were on the right track, I just never was able to identify the important '4-twist move’ that the tutorial uses. I got Step 1 of 3 steps down on my own by completing a finished face with all the edges correct. I think I maybe would have accidentally gotten through step two by chance and brute force. Step three was a nightmare that really emphasized the whole ‘you need chaos in order to have order’ idea.
In short, solving a Rubik’s cube has plateaus of progress, that are temporarily lost in the progression toward completion. It looks like this:
In the graph above, I perhaps should have labeled the y-axis apparent order. Because when we are near the end of the Cube’s completion, Order really is right around the corner. It appears chaotic, but that’s just the appearance. Like a morning fog rolling away to reveal a landscape, it was right there the whole time you just couldn’t see it. That’s the thing about chaos - it’s an illusion. A pattern not yet understood. The patterns and mathematical structures that underly it may be difficult to grasp, but with experience and practice chaos dissipates.
This guy just makes Rubik’s cube videos. The tutorial I used has over 50 million views. The approach in the video I used was foolproof - it took significantly more twists, but it would guarantee to get you there. This video expands on that method and makes it clear that there depth of understanding well beyond ‘completing’ the cube. I am like the child holding up the fish they caught after their parent prepped the gear, brought the live bait, cast the line, and reeled it in while the child held onto the pole. Also like that child, I am still happy to have ‘caught a fish’.