The stomachion, the world’s earliest known puzzle

In my searches related to the aforementioned 3D puzzles, I came across some puzzle vendor and enthusiast sites, one of which contained the text of this New York Times article about the stomachion, a children’s game that seemed to be the subject of a manuscript Archimedes wrote 2200 years ago. From what I can tell, this is almost universally believed to be the oldest known example of a puzzle. This website points out that the stomachion is the same type of puzzle as the more familiar tangrams, which most of us did at some point in our youths.

The fascinating thing about Archimedes’ investigations of the stomachion was not that he wrote a treatise on how to play a children’s game, but rather that he founded the discipline of combinatorics by trying to figure out how many different ways the strips of paper in the stomachion could be arranged to form a square. As Gina Kolata wrote in her New York Times article, which focuses on the efforts at restoring and interpreting the manuscript as much as on its contents:

…a historian of mathematics at Stanford, sifting through ancient parchment overwritten by monks and nearly ruined by mold, appears to have solved the mystery of what the treatise was about. In the process, he has opened a surprising new window on the work of the genius best remembered (perhaps apocryphally) for his cry of “Eureka!”…

The Stomachion, concludes the historian, Dr. Reviel Netz, was far ahead of its time: a treatise on combinatorics, a field that did not come into its own until the rise of computer science.

The goal of combinatorics is to determine how many ways a given problem can be solved. And finding the number of ways that the problem posed in the Stomachion (pronounced sto-MOCK-yon) can be solved is so difficult that when Dr. Netz asked a team of four combinatorics experts to do it, it took them six weeks.
The diagram involved 14 pieces, and the word “multitude” seemed to be associated with it. Mr. Heiberg and those who followed him thought this meant that you could get many figures [of animals, plants, household objects, etc.] by rearranging the pieces.

“This is part of the reason people didn’t see what it was about,” Dr. Netz said. …[T]he old interpretation seemed trivial, hardly worth Archimedes’ time.

As he examined the manuscript pages, piecing together their text, he realized that what Archimedes was really asking seemed to be, “How many ways can you put the pieces together to make a square?” That question, Dr. Netz said, “has mathematical meaning.”

Archimedes was truly an amazing person. I use that word with the fullest extent of its meaning. It is difficult to understand, much less appreciate, how extraordinary and seemingly superhuman his mind was. I put him in a class with Da Vinci and Einstein and no one else who ever lived (that we know of). His founding of the field of combinatorics only adds to my already reverential and awestruck feelings towards him.

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