A small parable of knowledge discovery follows. (Spoiler Warning! — if you have not yet independently figured out the Cube and wish to do so, skip this ^zhurnal entry!)
Back when Rubik's Cubes were a new invention, in early 1980, I got one and commenced messing around with it. (What's a Cube, you ask? The classic model, invented by Hungarian architect Erno Rubik, is a 3x3x3 block of cubelets; every face can rotate independently in 90 degree increments. The Cube begins with each side a single solid color. After a few quick twists the colors are scrambled, and it's highly nontrivial to undo the mess. Now you know enough to go on.) When they first encounter one, people react to the Cube in three disjoint ways:
- Some (most folks) are utterly disinterested.
- Some (engineering-personalities) see a physical artifact. They want to pry it apart and discover how the interlocking swivels and cubelets can possibly move about in three dimensions without coming unhinged.
- Some (scientist-personalities) see a mathematical artifact, a physical embodiment of a complicated set of transformations. They want to understand the idealized object and take control of it, so that they can "solve" the Cube: restore it to an initial state from any randomized configuration.
I was in the last category. For two solid weeks I played with my Cube (or rather, it played with me) — on the subway during morning and evening commutes, at the dinner table, in bed ... yes, I was obsessed ... I had a Cube Jones and couldn't shake it. There seemed to be no way to get a handle on this mysterious object. Everything I tried resulted in a horribly messy shuffling of cubelet corners, edges, and faces. Nothing stood still. How in the world could a human manage this kind of complexity?
Finally, late one night before falling asleep I had an epiphany, a tiny Aha! moment. I had so concentrated on the Cube that I started visualizing what would change if I twisted one face, then an adjacent face, then undid the twist of the first face, and finally undid the twist of the second face. Hmmm ... not much happens — three edge cubelets move, and so do four corners. OK, says I, what if I do that same pattern of twist-twist-untwist-untwist again? Well, the same edges and corners move another step. Do it again? Now the edges are all back to exactly where they started — but two pairs of corners are swapped. Zowie!
At that point I had to get up (sorry to disturb you, Paulette!), take my cube from the nightstand, go into another room, and turn on the light to make sure I wasn't dreaming. A brief experiment confirmed the happy hypothesis. I returned to bed confident that the Cube was now, for all practical purposes, understood. I slept well.
The fundamental insight that cracks the Cube is ridiculously obvious (in retrospect): do something, undo it, and repeat until a simple result emerges. There's a whole branch of mathematics (group theory) which deals with such things. To somebody fluent in that language, the Cube is rather trivial. It took a fortnight of near-total immersion, however, for a group-illiterate (such as I was) to discover the key.
I went on to figure out other transformations that manipulate edges and orientations of cubelets, and then to read about Cube Theory in various magazines (thanks again, Martin Gardner!), online Internet/USENET fora, and pamphlets. No books had yet been published on the subject. The best early write-up was by British mathematician David Singmaster. (I still remember the name of his colleague, knot theorist and professor Morwen Thistlethwaite.) On 20 September 1980 I took part in a Cube competition held by a local department store. The store had offered $100 gift certificates to anyone who could solve a Cube in under five minutes. I was the seventh to succeed that morning; there were a bunch of students from the University of Maryland there ahead of me. The store didn't keep its promise ... all I got was a T-shirt.
But nothing ever came close to that midnight flash of lightning when, after weeks of struggle, insight came. Eureka!
(see ^zhurnal RubikCubism2 (29 March 2001))
Friday, March 16, 2001 at 20:59:49 (EST) = 2001-03-16
(correlates: RubikCubism3, RubikCubism2, BlindLake, ...)