Mathematical treatise

Ostomachion (after Suter; this version requires a lateral stretch by a factor of two to match that in the

Archimedes Palimpsest)

Ostomachion (after Suter): square reformed with some pieces turned over

Ostomachion figures mentioned by Ausonius and others (Bibliotheca Augustana)

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**Ostomachion**, also known as **loculus Archimedius** (Archimedes' box in Latin) and also as **syntomachion**, is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the *Archimedes Palimpsest*, of the original ancient Greek text made in Byzantine times.^{[1]} The word Ostomachion has as its roots in the Greek Ὀστομάχιον,^{[2]} which means "bone-fight", from ὀστέον (*osteon*), "bone"^{[3]} and μάχη (*mache*), "fight, battle, combat".^{[4]} Note that the manuscripts refer to the word as "**Stomachion**", an apparent corruption of the original Greek. Ausonius gives us the correct name "Ostomachion" (*quod Graeci ostomachion vocavere,* "which the Greeks called ostomachion"). The Ostomachion which he describes was a puzzle similar to tangrams and was played perhaps by several persons with pieces made of bone.^{[5]} It is not known which is older, Archimedes' geometrical investigation of the figure, or the game. Victorinus,^{[6]} Bassus^{[7]} Ennodius^{[8]} and Lucretius^{[9]} have talked about the game too.

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Game

The game is a 14-piece dissection puzzle forming a square. One form of play to which classical texts attest is the creation of different objects, animals, plants etc. by rearranging the pieces: an elephant, a tree, a barking dog, a ship, a sword, a tower etc. Another suggestion is that it exercised and developed memory skills in the young. James Gow, in his *Short History of Greek Mathematics* (1884), footnotes that the purpose was to put the pieces back in their box, and this was also a view expressed by W. W. Rouse Ball in some intermediate editions of *Mathematical Essays and Recreations*, but edited out from 1939.

The number of different ways to arrange the parts of the Stomachions within a square were determined to be 17,152 by Fan Chung, Persi Diaconis, Susan P. Holmes, and Ronald Graham, and confirmed by a computer search by William H. Cutler.^{[10]}
However, this count has been disputed because surviving images of the puzzle show it in a rectangle, not a square, and rotations or reflections of pieces may not have been allowed.^{[11]}