It is a 2200 year old 14-piece dissection of a square - a Tangram-style puzzle or game that is thought to have been played by several players using pieces made of bone or ivory. Various goals include re-forming the square, and forming various figures in the usual Tangram manner. One question was "How many ways can the pieces be arranged to form a square?" Other ancient authors, all later than Archimedes, mention the game as well, and it is not known whether the game itself or Archimedes' investigation of it is older.

The Dr. Wood version is made of colorful plastic and includes a tray and a booklet of 73 challenges. The color assignments (red, yellow, blue, black) to both sides of each piece allow for some interesting challenges.

A description of the Ostomachion was found in the Archimedes Palimpsest, a parchment codex dating from tenth-century Byzantium. The palimpsest contents were released publicly in digital form and can also be downloaded at archive.org.

As reported in a NY Times article by Gina Kolata, published Dec. 14 2003, according to Dr. Reviel Netz, a classics professor at Stanford, until the information in the palimpsest came to light, though it was known that Archimedes had investigated the Ostomachion, it was unlcear as to its significance. The palimpsest reveals that Archimedes was investigating the number of ways the pieces can form a square - an early foray into the science of combinatorics. A PDF of Netz' 2004 paper (with Fabio Acerbi and Nigel Wilson) Towards a Reconstruction of Archimedes' Stomachion can be found at matetam.com.

Archimedes demonstrated that the area of each piece is rationally commensurate with the square (in the ratio 1:48). The square is an integer multiple of the area of each triangular piece - the overall square is 144 units and the pieces are either 3,6,9,12, or 24 units in area (the pentagonal "house" shaped piece is 21 units). (I have read that the palimpsest refers to a version stretched to twice the width of the square version shown here, which is a bit confusing if the investigation is into a square, not a rectangle.) Archimedes himself might not have known the number of solutions - he refers only to "multitudes."

Joe Marasco gives a good chronology and account of the re-popularization of the puzzle at his website, barbecuejoe.com. Marasco says that he commissioned the work to determine the number of solutions, and that Bill Cutler answered the challenge, computing that excluding rotations, reflections, and permutations of identical pieces, there are 536 distinct solutions. Without exclusions, there are 17,152 solutions, independently confirmed by graph theorists Fan Chun and Ron Graham (the Kolata article incorrectly attributes the solution to Chun and Graham with confirmation by Cutler, but it is the other way around). Marasco also provides a diagram of the 536 solutions.

Kadon also has a great article on the Ostomachion. There is some information at George Miller's website. See also an article by Ed Pegg Jr.

According to 4umi.com, "The remaining Greek fragments of Stomachion appear on the very last folio in the Archimedes Palimpsest. The evidence suggests the palimpsest originally contained more pages, but several are missing at the end. Following the prayer book page numbering, the essay begins on folio 177 recto - 172 verso and continues on the reverse side 177 verso - 172 recto. It is preceded by The Method of Mechanical Theorems. The text, which remains quite fragmentary, sets out to compare the various angles of the puzzle pieces. A diagram of a Stomachion square is visible in the righthand column on 177 verso."

The same website tells us that "Roman poet Decimus Magnus Ausonius (ca. 310 – ca. 393) mentions the Stomachion in his description of a type of poetry in which various meters are put together, like the pieces of the puzzle, in the introduction to his book Cento Nuptialis (A Wedding Cento)."

Ausonius writes "... [the Cento] is like the puzzle which the Greeks have called ostomachia. There you have little pieces of bone, fourteen in number and representing geometrical figures. For some are equilateral triangles, some with sides of various lengths, some symmetrical, some with right angles, some with oblique: the same people call them isosceles or equal-sided triangles, and also right-angled and scalene. By fitting these pieces together in various ways, pictures of countless objects are produced: a monstrous elephant, a brutal boar, a goose in flight, and a gladiator in armour, a huntsman crouching down, and a dog barking — even a tower and a tankard and numberless other things of this sort, whose variety depends upon the skill of the player. But while the harmonious arrangement of the skilful is marvellous, the jumble made by the unskilled is grotesque."