Archimedes' Stomachion

In early 2003, a retired businessman named Joe Marasco inquired of Kadon whether we knew of anyone who made this puzzle, attributed to Archimedes (287 to 212 B.C.) and called variously Stomachion or the Loculus of Archimedes. It consists of 14 tiles forming a 12x12 unit square, where each tile's area is a whole number (diagram courtesy of Prof. Chris Rorres of the University of Pennsylvania):

We said we knew of no sources but would be glad to make it for him. And Joe commissioned a number of them for himself and his friends. Turns out he had been in touch with scholars studying a rare historical find: a 174-page manuscript, or "palimpsest," of Archimedes' mathematical method and incidentally describing this very puzzle.

Joe then urged us to make the puzzle a regular part of Kadon's product line and plied us with Web references of the most fascinating information. Turns out it was a really hot topic among scholars and mathematicians. It was irresistible, and in August 2003 we premiered our beautiful Lucite version of Archimedes' Square, the world's oldest tiling puzzle, at the annual International Puzzle Party.
 

Hot on the trail, Joe then challenged programmers to identify, with proven accuracy, exactly how many solutions this puzzle has. Archimedes himself may not have known the exact number, referring only to "multitudes." Well, we know some savvy solvers with computer programs. We asked around a bit, and Ed Pegg Jr. steered us to our friend Bill Cutler, a puzzle designer in his own right, who had the right program and was intrigued enough by the question to adapt it for the Stomachion.

To speed the task among all his researchers, Joe offered a $100 bounty for the first correct solution. Bill Cutler won handily. In only 5 days, just before midnight on October 31, 2003, after double-checking, Bill supplied the answer: there are 536 all-different ways to assemble the pieces of the puzzle into a square. For purposes of counting only the distinct solutions, he did not include mere rotations or reflections of a single solution, nor did he count as different the exchanging of two identical tiles. You can also see all 536 solutions on the Mathpuzzle website, and an article by Ed Pegg with additional historical notes and references.

A little later, another group independently came up with the number 17,152, and that is the figure reported in a New York Times front page article on December 14, 2003, about this historic discovery. The article mentioned Dr. William Cutler as "confirming this result." We, of course, had already included Bill's 536 in our product descriptions, and were receiving "corrections" from folks who had read the Times. It took some explaining to point out that 17,152 was the full total, including all those rotations and reflections, etc., and dividing it by 32 would, in fact, yield the essential 536. By the way, it was very sweet to see in the article a picture of Prof. Reviel Netz, who is on the manuscript deciphering team, holding one of the puzzles we had made for Joe Marasco. The Times' copyright prohibits our showing the picture here.

This team's work with Archimedes' manuscript was also the subject of a NOVA special aired on PBS on September 30, 2003. The program did not mention the puzzle, only the amazing way the manuscript had come to light and the techniques being used to decipher the original writing underneath a second layer of religious texts. Such overwritten documents are called a "palimpsest." Read more about the Archimedes palimpsest on the NOVA website.

The manuscript had been discovered accidentally in 1846, but it lay in obscurity for over a century. In October 1998 it resurfaced and was auctioned in New York for two million dollars. The anonymous owner has graciously made it available for scholars to study. This exciting artifact is a unique source of evidence for Archimedes' thought and contains the oldest, by far, surviving description of his work in the original Greek. Among its many treasures is the only evidence we have for the treatise known as the Method, in which Archimedes combined physics and mathematics.

His description of the world's oldest documented puzzle supports the belief that Archimedes was also exploring combinatorics, a branch of mathematics dealing with combinations and permutations. Over 2200 years later, most of Kadon's puzzles fit into this category. Accordingly we've been exploring other shapes to make with Archimedes' 14 tiles, such as convex figures and tetratans solved by Bernd Rennhak of Germany.

Footnote:  In 2011, we added colors into our Stomachion. Kate Jones calculated the division so each color has exactly the same total area. By happy coincidence, no two pieces of the same color share sides in this basic solution:

Update:   Joe Marasco has proposed making June 5 (May 36th, or 536) "Archimedes Day" as an annual celebration of this greatest of mathematicians of antiquity. We heartily endorse this idea.

Update, December 2, 2011:   The palimpsest had been entrusted to the conservators and scholars at the Walters Art Museum in Baltimore, MD, in January 1999. They assembled an international team of experts, and by October 2011, their work of dismantling, reconstructing, and bringing the original writings into view was mostly completed, and the Walters offered a magnificent multi-room exhibit, "Lost and Found: The Secrets of Archimedes", of the entire story for visitors to appreciate. The exhibit ran from October 16, 2011, to January 1, 2012.

This is the rarest of manuscripts, a treasure of the human race, and a tribute to the skills, intelligence, patience and technological prowess of today's scholars to have retrieved the secrets buried in the badly damaged and time-worn parchment. One of the surprises was the discovery that the palimpsest contained works not only by Archimedes but six other writers from antiquity. The palimpsest was "recycled" from 7 different volumes.

The entire project is beautifully documented in two huge volumes, available from the Walters gift shop for $140 each. They also have a DVD about the exhibit. A giant model of the Stomachion in a flexible, carpet-like material was in the center of one of the exhibit rooms for visitors to attempt to solve. A note in the souvenir book reflected the complexity of this puzzle. Wrote one visitor: "This puzzle is much too hard for 5 doctors to solve without cheating." The gift shop had also carried small 4" versions of the puzzle made in colorful wood but were sold out by the time we visited on December 1.

We were able to chat with Will Noel and Abigail Quandt, two of the major team leaders of the palimpsest project, and learned that only 6 leaves remained to be processed and decoded. The rest of the book had been returned to the anonymous owner, whose goal in lending the book had been to get it deciphered and to share its treasures with the world, free on the Internet. Abigail laughed when she remarked that she had spent most of these years of study behind a microscope.

For more of the story, see these source pages at The Archimedes Palimpsest official website:

See also a marvelous presentation about the Archimedes palimpsest by William Noel on TED.com.

Update, May 2, 2013:   View a superb BBC Documentary on the story of Archimedes and the palimpsest on YouTube, Archimedes' Secret (opens in new window).

Update, July 1, 2017:   Kadon has produced a monograph reporting the full research results of the Tricolor Stomachion and its 1072 tricolor solutions, obtained during an intense 6-month project by the team of Joe Marasco, Kate Jones and Alex Streif.

Update, November 20, 2018:   Joe Marasco has composed a fascinating PowerPoint presentation with commentary as a YouTube video about the Tricolor Stomachion. See also Joe's other classy videos on his YouTube playlist.
 

Archimedes' Square is part of Kadon's collection of Treasured Oldies