The Weekly: Collateral Mathematics


If I did my math right, there are 3^6 * 7! possible permutations.(3674160)

During my vacation, my mother in law gave me a Homer Simpson 2×2 Rubik’s Cube as a gag gift, which is the apex of absurd gift giving. So needless to say that she was surprised when I popped it out of its packaging and began clacking away. With a few minor inconveniences, Homer’s nose tends to catch and the top half is vastly larger than the bottom (really aurocorrect? Boron?), this is an awesome gift. Off and on I’ve always putzed around with the math behind Rubik’s Cubes, but I never got around to actually solving one. I’d examine it from the stance of group theory, then figuring out the number of various positions, then I’d start in on determining transformations that let you move various blocks around the cube. And that’s great, for a time.


I got everything but the penny in the heads up orientation.

But as I’ve alluded to previously, it’s a really slippery slope for me. It’s one thing to appreciate and understand the math behind a Rubik’s Cube, but when you start extending it to n-dimensions, or explore optimal block shuffling strategies, and the like it can get bad in a hurry. This isn’t to say that if you enjoy this sort of thing that you’re a bad person, it’s just that I don’t, not at that level of depth. And this has happened to me a lot. I mentioned my sordid World of Warcraft, Rawr-optimization experiences. On a previous vacation I played with a Find-It puzzle game for over two hours straight, muttering about packing fractions and plastic density and trying to characterize the effects of various motions on the column. When I did that board game write-up on the fourth of July, I cut a solid 3 paragraphs of mathematical analysis on the game Tsuro, which resulted in some moderately unclear reasoning as to why I didn’t care for it. Most of the time delving into topology and how many possible moves are available to you at any point of the game, does give you a strategic advantage, but certainly doesn’t make the game any more enjoyable.  Then there was the time my brother and I more or less broke the game Black Box, by coming up with a configuration of pieces that gave the seeking player a 1 in 5 chance of winning.  Aside from ruining that game for both of us, it also illustrates what this sort of analysis can do to game at its most extreme.

There are 135 separate ways of pairing 8 objects - not counting rotationally identical tiles.

There are 135 separate ways of pairing 8 objects – not counting rotationally identical tiles.

Some games have a probability space way too large for analysis, while others collapse down into more general rules as your understanding increases. Both of these are enjoyable, but the purely mathematical games just have that capability to warp my brain into this bizarre obsessive mode. This gives me advantages in analyzing and optimizing systems, but the real trick is providing the right analysis on the right system, and knowing when to stop. The real issue is that this extracts the fun out of things – sometimes it’s not the case, I enjoy zero-knowledge games where careful experimentation and observation are the only way to win.

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