Sunday, September 13, 2015

Doing Math: Archimedes' Stomachion

When I was younger, I had an obsession with puzzles. I would spend hours at a time putting puzzles together or playing with tangrams online. So when there was an option to try Archimedes' stomachion as part of work for this class I was stoked. I was fairly confident this wouldn't be too difficult, but soon found out how wrong this assumption was. If you don't know what the stomachion is, its basically a rectangle that has been cut into 14 distinct pieces and the goal is to put it back together.

I spent hours working on this but ultimately was never able to complete it. To start out, I just randomly started placing some of the bigger tiles into the corners and trying to put the smaller ones around them. Obviously, this strategy didn't really work too well. Here was my best attempt just randomly placing the tiles anywhere.
For a while, I thought the top left corner was right but I wasn't making any progress so decided to try a different approach. Instead of trying to fill in the rectangle, I started to focus on the individual pieces. The long light blue one in the bottom right corner was a real pain to find a decent spot for, so I literally spent about 20 minutes trying to find out the other piece that went along the top of it. As you can see the darker blue triangle worked pretty well.  I then spent the next hour or so doing the same thing for each of the other larger more annoying pieces like the big purple piece. Then I started to focus on filling out the sides of the entire rectangle. This was about the time my second big progress came when I filled out half of the rectangle.
At this point I was 100% positive I had gotten that half the rectangle correctly tiled and thought I would be done with the puzzle in no time. Not long after I came as close as finishing it as I possibly could without actually doing so.
Words cannot describe how disappointed I was when I found out that the last little piece didn't fit. At this point I finally caved and decided to look up the solution. As it turns out, Bill Cutler found out there are 17,152 different solution and 536 truly distinct ones. After finding out there are so many I'm a little upset I couldn't find one of them. However, it amazes me that Archimedes, or anyone, could have created something that seems so simple on the outside, but when actually analyzed is so much more complicated.

1 comment:

  1. I count this as a solution. First of all, the 17K allows flips, and I think, doesn't divide out for symmetry.

    Nice write up, great support with the pictures. But there's more to get at with your thinking. Maybe it can't be put into words. But how did you know what to try, how did you know when you were making progress? (complete)

    To extend your conclusion, I'd love to know how this compared to the puzzles of your youth or now. Sounds harder - what made it so? (consolidated)

    clear, coherent, content +