**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 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 Kadon Enterprises
sells the puzzle in their catalog, as Archimedes' Square.
Their catalog description: The 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' 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 Reviel Netz is one of the primary researchers of the
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 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.
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 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.
Gardner, Martin. Mathematical Games. Kadon Enterprises. Lambrou, Michael. Archimedes MacTutor History of Mathematics. Archimedes. http://turnbull.mcs.st-and.ac.uk/history/. Slocum, Jerry. Weisstein, Eric W. Ostomachion.
A notebook for this column is available at the Loculus17 = (*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/*) 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 |
Figure 1. The 536 solutions of the Loculus, as found by Bill Cutler |