Math Games

The Loculus of Archimedes, Solved

Ed Pegg Jr., November 17, 2003

The world's oldest puzzle has been solved. 2200 years ago, Archimedes invented a puzzle variously called the Loculus, the Stomachion, the Ostomachion, the Syntemachion, or Archimedes' Box. In November of 2003, Bill Cutler used a computer program to enumerate all solutions. Barring rotations and reflections, there are 536 distinct solutions.

The seventeenth Cutler solution of the Loculus
Figure 2. Bill Cutler's 17th solution of Archimedes' Box

The 14 pieces of the Loculus are displayed above. Note that pieces 6 and 7 are both duplicated, and that piece pairs 1&2, 9&10, and 11&12 will always be found together. One of the better pages about the Loculus is by Tangram Fan.

The poet Magnus Ausonius (310-395 A.D.) writes about this puzzle in his book Liber XVII Cento Nuptalis. He says: "Diffinduntur autem per caesuras omnes, quas recipit versus heroicus, convenire ut possit aut penthemimeris cum reliquo anapaestico aut trochaice cum posteriore segmento aut septem semipedes cum anapaestico chorico aut post dactylum atque semipedem quidquid restat hexametro, simile ut dicas ludicro quod Graeci ostomachion vocavere. Ossicula ea sunt: ad summam XIV figuras geometricas habent. Sunt enim aequaliter triquetra, vel extentis lineis, vel ejusdem frontis, vel rectis angulis, vel obliqui: isoskele ipsi, vel isopleura vocant, orthogonia quoque et skalena. Harum verticularum varis coagmentis simulantur species mille formarum: elephantus belua aut aper bestia, anser volans et mirmillo in armis, subsidens venator et latrans canis, quin et turris et cantharus et alia hujusmodi innumerabilium figurarum, quae alio alius scientius variegant. Sed peritorum concinnatio miraculum est, imperitorum junctura ridiculum."

Kadon Enterprises sells the puzzle in their catalog, as Archimedes' Square. Their catalog description:
   What marvel of antiquity be this,
   This fabled square of 14 parts comprised?
   The legends credit Archimedes' wit
   With clever cuts that render every tile
   An integer.  All sum to twelve by twelve.
   Solve 18 figures lore has handed down,
   Like unto tangrams of a later time,
   And many new designs discovered since.
   Behold the oldest puzzle ever told,
   Our heritage of mind, millennia old.
   Now scholars scramble to decode, with zest,
   Archimedes much-prized Palimpsest,
   A scroll long lost, inscribed by his own hands,
   A rarest find from Greek and Latin lands.

The Palimpsest has seen mold, fire, knives, theft, lawsuits, and other abuse over the past few thousand years. His treatise On Floating Bodies is contained within its pages, along with such things as the Archimedean Spiral. Originally, Archimedes wrote his works on scrolls, which were continually copied onto fresher scrolls by others as the older scrolls aged. In the meantime, various empires grew and collapsed...

In the 9th and 10th centuries, Constantinople became a center of learning. The Emperor-scholar Constantine VII Porphyrogennitos worked to salvage the available ancient works, and made a fresh copy of Archimedes' Palimpsest. In 1204, the Fourth Crusade sacked the city, and destroyed many of the texts. Shortly afterwards, the book was cut up, the ink was scraped off, and a Christian prayerbook (the Euchologion) was written over the pages. Fittingly, the word Palimpsest comes from a Greek term meaning "scraped again".

By the sixteenth century, the marred book resided at the Greek Orthodox Monastery of Mar Saba. The reverend George Croly described the book being there in 1839. In 1906, the Danish philologist Johan Ludvig Heiberg studied the book at the Constantinople library. His discovery of lost Archimedean works made front page news on July 16th, 1907. The book was lost again a few years later. In 1998, the book was discovered again. After a brief lawsuit, the Palimpsest was sold at auction for two million dollars. More details on the Palimpsest can be seen at walters.org, the NOVA special, and the Math Archives.

Reviel Netz is one of the primary researchers of the Palimpsest. For the Loculus, he turned to Joe Marasco to possibly get a definitive answer for the number of solutions. A longtime collector of Kadon's puzzles, Joe convinced Kate Jones to make a version of the Loculus for sale. Kate asked me (Ed) if I could solve it. I couldn't. But I knew who could.

That person was Bill Cutler. The name should be familiar to recreational mathematicians. In 1977, Bill did a computer analysis of Burr Puzzles, and found 369 pieces that could be assembled in 119979 different ways. Martin Gardner devoted his January 1978 column to Bill's discoveries. IBM Research has a nice burr puzzle site of a similar structure, though the program there is by Juerg von Kaenel. Modern burr puzzle designs can be found in Cubism For Fun.

In 2000, after getting a fascinating answer to a problem based on the Pythagoras figure, I started looking at 1-2 and 1-3 right triangles. ArctTan[1] == ArcTan[1/2] + Arctan[1/3]. If these triangles are reflected along the hypotenuse, a kite shape results. I cut twenty of these kites out of a piece of paper, along with some dominoes, and played with them. To my surprise, I was able to make a 7×7 square in a bizarre way, and this became my Kites and Bricks puzzle.

Kites
Figure 3. Kite pairs

The solutions of Kites and Bricks don't follow grid lines. In fact, one of the angles is the same as the hypotenuse in a 7-24-25 right triangle. When Bill Cutler saw the puzzle, he mentioned that he had long planned to write a vector-based solving program, but hadn't yet seen a puzzle that would be interesting to analyze. Kites and Bricks fit the bill, so he wrote the program, and verified that the 5 human-found solutions were complete.

When Kate Jones contacted me, I thought of Bill's program. Within a few days, he sent us a complete list of solutions. A few days after that, I'm writing up this synopsis.

The history of the Loculus may seem complicated, but this is nothing compared to the Tangram. For that, I highly recommend Jerry Slocum's 2003 work, The Tangram Book: The Story of the Chinese Puzzle. Martin Gardner: "The little-known facts are all here, for the first time, in a wonderfully readable and richly illustrated volume." The history of the Loculus comes near the start of Jerry's book.

The world's oldest puzzle finally has a complete answer. Bizarrely, it really wasn't that hard. None of these solutions would be particularly hard to find. Most of them are easily derived from other solutions, by swapping, reflecting, and rotating various sections. With a systematic approach, I'm sure that Archimedes, or anyone following him, could have listed all the distinct solutions within a few weeks of work.

References:

Gardner, Martin. Mathematical Games. Scientific American. Jan 1978.

Kadon Enterprises. Archimedes Square: The world's oldest known puzzle. http://www.gamepuzzles.com/archsqu.htm.

Lambrou, Michael. Archimedes Palimpsest. http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jul99/0034.html.

MacTutor History of Mathematics. Archimedes. http://turnbull.mcs.st-and.ac.uk/history/.

Slocum, Jerry. The Tangram Book: The Story of the Chinese Puzzle. New York: Sterling, p. 11, 2003.

Weisstein, Eric W. Ostomachion. Eric Weisstein's World of Mathematics. http://mathworld.wolfram.com/.

Mathematica Code:

A notebook for this column is available at the Mathematica Information Center.

Loculus17 =
{{{0, 0}, {0, 12}, {2, 10}, {0, 0}},
{{0, 0}, {2, 4}, {6, 0}, {0, 0}},
{{0, 0}, {2, 10}, {4, 8}, {0, 0}},
{{0, 12}, {6, 12}, {2, 10}, {0, 12}},
{{4, 8}, {4, 2}, {2, 4}, {4, 8}},
{{4, 11}, {6, 12}, {10, 8}, {4, 11}},
{{6, 12}, {12, 12}, {10, 8}, {6, 12}},
{{7, 8}, {8, 4}, {7, 2}, {7, 8}},
{{10, 8}, {8, 4}, {7, 8}, {10, 8}},
{{12, 6}, {12, 4}, {9, 6}, {12, 6}},
{{12, 12}, {12, 6}, {9, 6}, {12, 12}},
{{2, 10}, {4, 11}, {10, 8}, {4, 8}, {2, 10}},
{{9, 6}, {12, 4}, {12, 0}, {6, 0}, {9, 6}},
{{6, 0}, {4, 2}, {4, 8}, {7, 8}, {7, 2}, {6, 0}}};

(*Part of Figure 2: *) Show[Graphics[Map[Line[Join[#, {First[#]}]] &, Loculus17, AspectRatio -> Automatic]];

(*For the full Figure 1 and 2, see http://library.wolfram.com/infocenter/MathSource/5108/*)


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Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com. He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.

Solutions of the Loculus

Figure 1. The 536 solutions of the Loculus, as found by Bill Cutler