The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word sigmatauomicronmualpha^'chiiotaomicronnu, meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.

Stomachion elephant

The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a 12×12 grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).

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, giving them relative areas 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 7, and 8. Interestingly, as noted by Coffin, it is always the case that all polygons formed by connecting points on a regular square grid must have areas in the ratios of whole numbers.


In November 2003, Bill Cutler found there to be 536 possible distinct arrangements of the pieces into a square, illustrated above, where solutions that are equivalent by rotation and reflection are considered identical (Pegg 2003).

See also

Dissection, Tangram

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Andrea. "Stomachion.", M. Liber XVII Cento Nuptalis.Chung, F. and Graham, R. "A Tour of Archimedes' Stomachion.", S. T. "Two-Dimensional Dissections Other Tangram-Like Puzzles." Ch. 1 in The Puzzling World of Polyhedral Dissections., E. Curiosités Géométriques. Paris, 1907.Ipsen, D. C. Archimedes: Greatest Scientist of the Ancient World. Enslow, 1988.Kadon Enterprises. "Archimedes' Square, 'Stomachion': The World's Oldest Known Puzzle.", G. "In Archimedes' Puzzle, a New Eureka Moment." The New York Times, pp. 1 and 32. December 14, 2003. McLeod, A. I. "Stomachion of Archimedes.", R. and Noel, W. The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity's Greatest Scientist. Philadelphia, PA: Da Capo Press, 2009.Pegg, E. Jr. "Math Games: The Loculus of Archimedes, Solved." Nov. 17, 2003. Pegg, E. Jr. "The Loculus of Archimedes, Solved.", C. "Stomachion Introduction.", C. "Stomachion Construction.", G. A. "Antique Puzzles.", J. The Tangram Book: The Story of the Chinese Puzzle with Over 2000 Puzzles to Solve. New York: Sterling, p. 11, 2003.Weisstein, E. W. "MathWorld Headline News: Playing in the Sandreckoner's Box." November 19, 2003.

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Cite this as:

Weisstein, Eric W. "Stomachion." From MathWorld--A Wolfram Web Resource.

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