The Archimedes Palimpsest

How many ways can you have a stomach ache?
By Reviel Netz of Stanford University

A scholarly article published in early 2004 notes that "The Stomachion is something of a poor relation". This has not been the case recently. Indeed, besides the word "Palimpsest" itself, study of the Archimedes Palimpsest has now made this funny Greek word, too - "Stomachion" - into something of a household name. It probably never was so even in antiquity itself. The evidence suggests that antiquity knew a game called "Stomachion", always considered a tough challenge. The very name is not very well understood today, but apparently it refers to the difficulty of the game: our word "stomach" comes from the Greek, so that "Stomachion" means, literally, "that which relates to the stomach". All occurrences of the word in classical texts are either references to the game or (much more often) medical references to a belly-ache. Most likely, then, the game literally meant "belly-ache". It appears that while modern gamers feel a "headache" with a difficult problem, their ancient counterparts felt a "bellyache". Anatomy is a matter of history.

The headache was about this: take a square, divided into 14 pieces according to a pre-defined pattern. Now shuffle well the 14 pieces. And then try and put them together into a square. This is surprisingly difficult.

What turned out to have been even more difficult – what has caused, as it were, even more headache – was making sense of Archimedes' foray into the game. We always knew he has something to do with this game, because some ancient authorities refer to the game simply as "Archimedes' Box". In 1899, an Arabic text was published based on a manuscript in Berlin, containing a brief passage claiming to derive from a work by Archimedes on the "Stomachion". And then, in 1906, the Palimpsest was discovered.

Here is where the Stomachion is in really bad luck. Archimedes' treatise on the subject was apparently the last treatise in the original Archimedes manuscript, from which the Palimpsest was made. Most of its pages were in bad shape already in the 13th century, and so were discarded by the makers of the Palimpsest. Only a single page survived - the first one. Thus, we have the introduction to the Stomachion, together with a brief introductory theorem. This is in principle very useful – the introduction, after all, should state what the treatise is about and so should furnish us with crucial information – but worse is to come: the maker of the Palimpsest inserted this page right towards the end of the book. At around the 16th century, apparently, a few pages were lost from the end of the book (perhaps to a fire) and the page containing the introduction to the Stomachion now became the very last page of the book. The result is that this is now among the most difficult pages in the book, indeed was very difficult to read already when Heiberg came to look at it. Since the Arabic fragment is also very tantalizing (and is, anyway, no more than a very late compilation), the end result is that, up until recently, we could not put together this particular jigsaw puzzle. No one knew what Archimedes' treatise was about. The standard view was, we shall never know.

The interpretation put forward in 2004, based on the new readings made possible thanks to digital technologies, is still tentative. Nothing better is possible, seeing that our evidence is so fragmentary. But this interpretation is widely considered to be very likely. This is quite a result, too: it makes Archimedes into the first author of an entire field of mathematics, one that until recently we thought no ancient author has ever worked in. This is the field of combinatorics.

Combinatorics, as its name suggests, is the study of the number of possible combinations.

Suppose you have three blue socks: how many pairs can you form?
Answer: three. Each choice of pair leaves exactly one sock "unmatched", so that the number of pairs is the same as the number of individual socks.
Suppose you have three green socks, too, and you wish to form bicolored pairs, a green and a blue in each. How many combinations?
Answer: nine. Each choice of blue is allowed, and so is each choice of green, allowing the entire "multiplication table" of three-by-three, i.e. nine.

This kind of study, we can immediately see, is a far cry from the world of geometrical curves studied so extensively by Archimedes and by his contemporaries. For this reason, no one thought of combinatorics in connection with the Stomachion, looking instead, in vain, for some interesting, strictly geometrical problem associated with the game.

More recently, Fabio Acerbi, an Italian historian of mathematics, has definitely proved the existence of an interest in combinatorics in antiquity (even if later than Archimedes himself). This has allowed us finally to interpret the treatise. A fuller reading of the introduction has shown that Archimedes was addressing the fact that there are many ways of solving the puzzle, given the possible ways of interchanging one solution for another. The evidence suggests, therefore, that the treatise was a study in the possible substitutions of one solution by another, aimed at calculating the total number of such solutions.

Prodded by the Palimpsest team, modern combinatorists (Profs. Diaconis and Holmes of Stanford, together with Profs. Chung and Graham of UCSD) have tackled this question – how many ways to solve the Stomachion puzzle – and have come up with the number 17,152 (or 536 geometrically distinct solutions, each multiplied by 32 symmetries). We cannot tell, of course, whether Archimedes came up with the right answer. But from all we know of him, more likely than not, he did. An ancient game came back to life – and with it, an entire ancient field of science.