Think inside the box. A few of the 536 ways to arrange the shapes in Archimedes' puzzle.

Stay in the Tub, Archimedes

Perhaps it wouldn't make him run naked through the streets shouting "eureka," but Archimedes would no doubt be pleased that one of his puzzles has been completely solved more than 2000 years later.

In the third century B.C.E., Archimedes posed a geometric mindbender: How can one get a particular set of 14 irregular triangles and quadrilaterals to fit together into one big square? Finding one solution isn't that hard, but nobody knew how many solutions there are. "When you first start looking at it, it seems like it might have thousands and thousands of solutions," says mathematician Ed Pegg, who works at Wolfram Research, a mathematics software company in Champaign, Illinois.

But just this month, puzzlemaker Bill Cutler of Palatine, Illinois, put the 2-millennium-old poser to rest. Using a computer's brute force, Cutler figured out by trial and error that there were only 536 solutions to the puzzle, excluding rotating and reflecting the final assembled square. One element that made the problem tractable was that there are three pairs of pieces that need always be together--one side of each of those pieces is a length that only fits together with its partner.

That constraint as well as the fact that there were obviously lots of solutions limited Archimedes' puzzle's appeal, says Pegg. "It really wasn't all that good of a puzzle," he says. "So it went by the wayside." Nevertheless, he thinks that the solution to the ancient problem is a victory.

Related sites
More information about the puzzle
Bill Cutler's puzzle company
A biography of Archimedes