Sunday, September 13, 2015

Archimedes' Stomachion

Doing Math:

I love puzzles.  Love love love them.  I will spend an entire day putting together a giant jigsaw puzzles and be perfectly content.  So when I saw that one of our assignment options was trying to put together Archimedes' Stomachion I was very excited!  The Stomachion is basically a little 14 piece puzzle that you try to fit into a square.  I did this puzzle on Geogebra which you can check out here:  Stomachion on Geogebra.  Make sure your window is zoomed to show all the pieces and then you can rotate them and fit them in the square!

When I started to experiment with fitting the pieces in, I tried to be methodical and fit in the biggest pieces first.  With the big pieces in first, I could then see which smaller pieces could fit in between them.  One of the main goals was to find out which sides of the different pieces would fit together perfectly since this would make the most of the space.  At times I found it slightly difficult to spin and move the pieces exactly how I wanted them which made me wish I had physical pieces in front of me.

 This was one of my first attempts.  I tried to make bigger shapes out of the smaller pieces, like the yellow blue and green pieces at the top formed a triangle. This seemed like progress because it matched those ugly edges of the yellow piece up with others to make a shape that was pretty much the same as the triangle made from the long purple and green pieces.  Of course the two lighter blue triangles wouldn't fit, so I knew I wasn't maximizing the space as well as I needed to.

Next decent attempt.  Again I noticed certain pieces would make bigger shapes, like the rectangle in the bottom right which I kept from the first attempt.  Also the square in the top right corner.  However, I abandoned my first thoughts of putting in the biggest pieces first, which resulted in overlap from the red piece and vacant space around the small purple triangle.

This was my best and last attempt.  If there was a way to flip over that last purple triangle, it would have fit.  The right half of the square seems great.  The fact that it forms a rectangle and could be cut into multiple different triangles makes me think that it is correct.  However the left half doesn't have those same intracacies.  Unfortunately, I could not find another way to arrange it to make it fit.

Overall this was a very fun puzzle, despite being a little frustrating at times.  I always enjoy when math and puzzles connect.  Interestingly enough, after I worked on this puzzle (for a long time) I was browsing through the new book I got for class (The Math Book by Clifford Pickover) and it had a page talking about Archimedes and the Stomachion.  It even showed this solution, although it's not possible to make it this one in the Geogebra file without being able to reverse some pieces.  I like how every piece is a part of at least two bigger triangles. Plus there is a line splitting the square into two rectangles and another line on the diagonal forming two triangles.  I think it's so interesting how these strange pieces make up bigger shapes within the square.  So although I didn't find one of the "17,152 solutions" it was very fun, and interesting to examine the details of a final solution.


  1. Oh, what you found is definitely one of the solutions. Nice work! Well illustrated and thorough sharing of your thinking, too.
    5C's +, though I was wondering about your thoughts on 'so what' here.

  2. Pat Bellew - great math historian - saw this post (I tweeted it) and thought you might like this post of his: