Editing Archimedes Palimpsest (section)

===Stomachion===
{{Main|Ostomachion}}
[[Image:Stomachion.JPG|thumb|''[[Ostomachion]]'' is a [[dissection puzzle]] in the Archimedes Palimpsest (shown after Suter from a different source; this version must be stretched to twice the width to conform to the Palimpsest)]]

In Heiberg's time, much attention was paid to Archimedes' brilliant use of [[indivisibles]] to solve problems about areas, volumes, and centers of gravity. Less attention was given to the ''[[Ostomachion]]'', a problem treated in the palimpsest that appears to deal with a children's puzzle. [[Reviel Netz]] of [[Stanford University]] has argued that Archimedes discussed the ''number of ways'' to solve the puzzle, that is, to put the pieces back into their box. No pieces have been identified as such; the rules for placement, such as whether pieces are allowed to be turned over, are not known; and there is doubt about the board.

The board illustrated here, as also by Netz, is one proposed by [[Heinrich Suter]] in translating an unpointed Arabic text in which twice and equals are easily confused; Suter makes at least a typographical error at the crucial point, equating the lengths of a side and diagonal, in which case the board cannot be a rectangle. But, as the diagonals of a square intersect at right angles, the presence of right triangles makes the first proposition of Archimedes' ''Ostomachion'' immediate. Rather, the first proposition sets up a board consisting of two squares side by side (as in [[Tangram]]). A reconciliation of the Suter board with this Codex board was published by [[Richard Dixon Oldham]], FRS, in ''Nature'' in March, 1926, sparking an ''Ostomachion'' craze that year.

Modern [[combinatorics]] reveals that the number of ways to place the pieces of the Suter board to reform their square, allowing them to be turned over, is 17,152; the number is considerably smaller – 64 – if pieces are not allowed to be turned over. The sharpness of some angles in the Suter board makes fabrication difficult, while play could be awkward if pieces with sharp points are turned over. For the Codex board (again as with Tangram) there are three ways to pack the pieces: as two unit squares side by side; as two unit squares one on top of the other; and as a single square of side the square root of two. But the key to these packings is forming isosceles right triangles, just as [[Socrates]] gets the slave boy to consider in [[Plato]]'s ''Meno'' – Socrates was arguing for knowledge by recollection, and here pattern recognition and memory seem more pertinent than a count of solutions. The Codex board can be found as an extension of Socrates' argument in a seven-by-seven-square grid, suggesting an iterative construction of the side-diameter numbers that give rational approximations to the square root of two.

The fragmentary state of the palimpsest leaves much in doubt. But it would certainly add to the mystery had Archimedes used the Suter board in preference to the Codex board. However, if Netz is right, this may have been the most sophisticated work in the field of combinatorics in Greek antiquity. Either Archimedes used the Suter board, the pieces of which were allowed to be turned over, or the statistics of the Suter board are irrelevant.