Archimedes‘ Stomachion

One of the mathematical papers by Archimedes, Ostomachion, also known as Archimeidus (the Latin square of Archimedes) or the synthetic ion, which survived fragmented versions in the Arabic version and the original Greek text of the Byzantine period, Archimedes Palimpsest.

The stomachion is a 14 piece dissection puzzle similar to tangrams. People can use these 14 flat pieces of various shapes blocks to create different objects, animals, plants and so on. However, it is not just a game which discovered by Archimedes. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. 

Arrange the Stomachion in the shape of a square, with the vertices of pieces occurring on a 12×12 grid. Taking the square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24. Simplifying by the common divisor 3, we obtain that them relative areas 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 7, and 8. Then it shows that the area of each piece in Stomachion is a multiple of 1/48 of the area of the whole square, therefore, simple fractions of areas can be constructed using the pieces of the Stomachion.

In November 2003, Bill Cutler argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square and he found that the pieces can be made into a square 17,152 ways. And there are 536 possible distinct arrangements of the pieces into a square, where solutions that are equivalent by rotation and reflection are considered identical.


Some people consider that this puzzle is the evidence of Greeks completely mastered the earliest science of combinatorics. In my view, it is no doubt that Stomachion represents a good example of an early problem in combinatorics. 

Reference List
GINA KOLATA NEW YORK, T. (n.d). Eureka: Ancient Archimedes puzzle solved. Toronto Star (Canada).

Kolata, G. (2003, December 14). In Archimedes' Puzzle, a New Eureka Moment. (cover story). New York Times. pp. 1-46.