The Stomachion: Archimedes' Lost Puzzle

Quick answer: the Stomachion is a square cut into 14 pieces, attributed to Archimedes in the third century BC, and the oldest mechanical puzzle we know of. It has 17,152 solutions, a number nobody established until 2003.

Best for: anyone who thinks puzzle history starts with Victorian parlour games.

Most old puzzles are old in the sense that somebody's grandfather had one. The Stomachion is old in the sense that Archimedes appears to have been thinking about it, and then it vanished for roughly a thousand years.

What Is the Stomachion?

A square divided into 14 pieces: a mixture of triangles, quadrilaterals and five-sided figures, all with vertices on a regular grid.

Mechanically it behaves like a tangram, with the same two rules: use every piece, overlap nothing. It is harder than a tangram for the obvious reason that fourteen pieces give you far more ways to be wrong, and for a less obvious one, which is that several of the pieces are similar enough to be confused with each other.

It is also called the Ostomachion, and the name is unglamorous: it comes from the Greek for stomach, and is usually read as something like "bellyache", presumably describing the effect of attempting it.

Did Archimedes Really Invent It?

He wrote about it, which is the strongest claim the evidence supports.

What survives is a treatise attributed to Archimedes inside the Archimedes Palimpsest, and the story of that document is the reason this puzzle is famous. A palimpsest is a manuscript that has been scraped clean and written over because parchment was expensive. This one was first written in Constantinople, probably in the tenth century, and later scraped so the parchment could be reused for a prayer book.

So the oldest known mechanical puzzle survived by being erased badly. The original ink left faint traces under the newer text, and modern imaging recovered what was underneath.

Whether Archimedes devised the puzzle or was analysing an existing one is unknown. What the treatise makes clear is the question he was interested in, and it is not the one you would expect.

How Many Solutions Does the Stomachion Have?

17,152. And the striking thing is how recently anybody knew that.

The number breaks down as 536 geometrically distinct solutions, each appearing in 32 variants once you count rotations and reflections as different.

Getting there took modern computing and modern mathematics. In October 2003, Bill Cutler wrote a program that performed an exhaustive search and produced all 536 distinct arrangements. Shortly afterwards a group including Fan Chung, Persi Diaconis, Susan Holmes and Ron Graham supplied the theoretical framework explaining why there are that many rather than some other number.

That is a gap of more than two thousand years between the question being asked and the answer being known.

Why Does the Solution Count Matter?

Because of what it suggests Archimedes was doing.

If you are designing a puzzle, you want one solution, or few. If you are asking how many ways can these pieces form a square, you are not posing a puzzle at all. You are doing combinatorics, and combinatorics is not supposed to have existed as a field for another fifteen hundred years or so.

The reading that the surviving text supports is that the counting was the point. That would make the oldest known mechanical puzzle not really a toy but a piece of mathematics wearing a toy's clothes, which is a description that fits a surprising number of good puzzles. The mathematics of the modern equivalents is covered in the guide to mechanical puzzles.

How Does It Compare to a Tangram?

Same idea, twice the pieces, and a completely different experience.

A tangram has seven pieces and usually one intended solution per silhouette, so when you place a piece wrongly the puzzle tells you fairly quickly. The Stomachion has fourteen pieces and hundreds of valid squares, which sounds easier and is not: with that many ways to be right, you get almost no feedback that you are on a wrong path until the last few pieces refuse to fit.

The pieces are also cut on a grid, so edges match in many combinations that look promising and lead nowhere. A tangram's five triangles are obviously different sizes. Several Stomachion pieces are nearly interchangeable until they are not.

It is the better puzzle for a mathematician and the worse one for an afternoon.

How Do You Solve a Stomachion?

Much the way you solve any dissection, with one addition.

Place the largest pieces first. They have the fewest positions available and they decide most of what follows, exactly as the two big triangles do in a tangram.

Work the corners. A right-angled corner of the square has to be filled by a piece with a matching corner, which immediately eliminates most of the set.

Keep the near-identical pieces apart. This is the Stomachion-specific trap. Several pieces are similar without being the same, and a solution that refuses to close is often correct except that two lookalikes have been swapped.

Accept that you are finding one of thousands. With 536 distinct solutions, your arrangement is almost certainly not the one in the book, and that is fine. Unlike most dissections there is no single correct answer to find.

Can You Still Get One?

Not from us, and it is worth being straight about that. The Stomachion is public domain and widely reproduced, mostly in wood or card.

If what appeals is the dissection problem rather than the history, the modern equivalents are better puzzles, because they are designed to have one solution instead of thousands. Three-Piece Block and Multi-Grain Burr are Stewart Coffin dissections in three dimensions, the flat version is the tray puzzle family like Trivial and Galette, and the whole category is in assembly puzzles explained.

The wider range is in mechanical puzzles and packing puzzles.

Frequently Asked Questions

What is the Stomachion puzzle?

A square cut into 14 pieces, attributed to Archimedes in the third century BC and generally considered the oldest known mechanical puzzle. The goal is to rearrange all 14 back into the square, using every piece without overlaps.

How many solutions does the Stomachion have?

17,152 in total, which is 536 geometrically distinct solutions each counted in 32 orientations. Bill Cutler established the 536 by exhaustive computer search in October 2003, and a group including Fan Chung and Ron Graham then explained the result theoretically.

Is the Stomachion older than the tangram?

By roughly two thousand years. The Stomachion dates to the third century BC; the earliest record of the tangram is from around 1796, despite frequent claims that tangrams are ancient.

What is the Archimedes Palimpsest?

A medieval manuscript containing copies of Archimedes' work that was later scraped clean so the parchment could be reused for a prayer book. The original writing left faint traces, recovered by modern imaging, and the Stomachion treatise is among the texts it preserved.

Why is it called the Stomachion?

The name derives from the Greek for stomach and is usually translated as something like bellyache, which is generally taken as a comment on the experience of trying to solve it.