Tangram: History, Maths and How to Solve It

Quick answer: a tangram is a square cut into seven flat pieces, called tans, which are rearranged to form silhouettes. Every piece must be used, none may overlap, and the difficulty is that the target shape shows you its outline and nothing about what is inside it.

Best for: anyone who wants the cleanest possible example of why a puzzle with no mechanism can still be hard.

The tangram is the most widely taught mechanical puzzle in the world. It is also, almost always, taught with a history that is not true.

What Is a Tangram?

Seven pieces cut from a single square: five triangles of three sizes, one square and one parallelogram. Together they tile the original square exactly, which is the first thing worth noticing, because it means every solution has precisely the same total area and you can never be short.

The puzzle is to rearrange all seven into a given outline. Two rules, and they are the whole game: use every piece and do not overlap.

That makes it a dissection puzzle, a sub-type of the assembly family set out in the guide to mechanical puzzles. There is no mechanism, nothing is hidden, and you can see the entire problem from the first second, which is exactly what makes it a good puzzle rather than a trivial one.

Where Did the Tangram Actually Come From?

China, and far more recently than you have been told.

Tangrams are routinely described as ancient, thousands of years old, Tang dynasty. There is no evidence for any of it. The earliest known record dates to around 1796, in a book that is referenced in the historical record but which nobody has ever found. The oldest surviving physical sets date to 1802, and a Chinese book of tangram problems from 1813 exists.

So the honest answer is late eighteenth century, with the usual caveat that an absence of earlier records is not proof of absence.

There is a good reason the records are thin, and it is the most interesting thing in the puzzle's history. In China at the time it was considered a game for women and children, and things considered unserious do not get written about by the people doing the writing. The tangram was not documented because nobody thought it mattered.

It reached Europe and America early in the nineteenth century and caused a genuine craze, which is when most of the invented ancient history was attached to it, because an exotic origin sold better than a recent one.

How Many Shapes Can You Make With a Tangram?

Infinitely many in general, and exactly thirteen if you want them convex.

A convex shape is one with no indentations: every straight line between two points inside it stays inside. In 1942, Fu Tsiang Wang and Chuan-Chih Hsiung proved that the seven tans can form exactly thirteen convex shapes and no more. Not approximately thirteen, not thirteen that anyone has found. Thirteen, proven, finished.

That is a remarkable result for a children's toy, and it is the reason tangrams keep appearing in mathematics departments. The classic puzzle books contain thousands of target silhouettes, nearly all of which are non-convex, and that set is unbounded.

What Is a Tangram Paradox?

Two figures, apparently identical, except one of them is missing a piece of itself. Both are built from all seven tans.

The best-known examples show a pair of nearly matching outlines where one has, say, a foot or a corner the other lacks. Since both use every tan and no tan changes size, the shapes must have equal area, and they plainly do not look like they do.

The resolution is the same every time: the two outlines are not actually the same shape. One is subtly fatter elsewhere by exactly the area that appears to be missing, and the eye cannot see a difference spread thinly across a long edge. It is the flat cousin of the dissection paradoxes discussed in impossible objects, and we sell one that works the same way, Infinity Chocolate, where rearranging the pieces appears to create a square from nothing.

How Do You Solve a Tangram?

By placing the pieces that have the fewest options first, which is the opposite of what most people do.

Start with the two large triangles. They account for half the total area between them, so in any silhouette they can only sit in a handful of places. Most outlines are decided by where those two go, and everything after is tidying up.

Work the corners and the extremities. A sharp point in the outline must be made by a sharp piece, and a long straight edge usually comes from a long piece edge rather than several short ones meeting.

Remember the parallelogram is the awkward one. It is the only piece that cannot be matched by its own mirror image without being flipped over. If a solution refuses to close, the parallelogram being the wrong way up is the most common reason.

Do not trust the picture's proportions. Printed silhouettes are often drawn slightly wrong, and people spend a long time trying to match a shape that was never accurate.

Why Are Tangrams Used in Schools?

Because they make spatial reasoning visible with no reading involved.

A child who cannot place the pieces has not misunderstood an instruction. The goal is obvious and the obstacle is purely spatial, which is rare in a classroom task. Tangrams also teach area conservation without saying so: the pieces always total the same amount however they are arranged, which is a surprisingly hard idea to convey any other way.

The broader evidence on spatial training, including what it does and does not show, is set out in 3D spatial reasoning puzzles.

What Should You Try After a Tangram?

We do not sell tangrams, so this is not a sales pitch. The honest next steps depend on what you enjoyed.

Nothing we make is really a tangram, and it would be a stretch to claim otherwise. The honest nearest relatives are the flat tray puzzles, which share the important properties: flat pieces, a defined boundary, every piece used, nothing overlapping. Trivial and Fish Tank by Goh Pit Khiam are three pieces each, and Galette is five tetrominoes into a frame. What they add is a container, which the tangram deliberately does without.

If you liked that a simple-looking arrangement defeated you, the step up is three dimensions. Three-Piece Block and Multi-Grain Burr are both Stewart Coffin dissections, and assembly puzzles explained covers the family. The classic three-dimensional relative of the tangram is the Soma Cube.

The full range is in mechanical puzzles, with packing puzzles the nearest shelf.

Frequently Asked Questions

What is a tangram?

A square cut into seven flat pieces called tans: five triangles, one square and one parallelogram. The puzzle is to rearrange all seven into a given silhouette, using every piece and overlapping none.

How old is the tangram?

Much younger than usually claimed. The earliest known record is from around 1796, surviving sets date to 1802, and a Chinese problem book from 1813 exists. There is no evidence for the common claim that it is thousands of years old.

How many tangram shapes are possible?

Unlimited in general, but exactly thirteen convex shapes. Fu Tsiang Wang and Chuan-Chih Hsiung proved that result in 1942, and it is a proof rather than a count of what anyone happened to find.

What is the trick to solving tangrams?

Place the two large triangles first. They are half the total area between them, so they can only go in a few places in any outline, and most silhouettes are decided by that choice. Watch the parallelogram too, since it is the only piece that must sometimes be flipped over.

Why do two tangram figures look different if they use the same pieces?

Because the outlines are not actually identical. Since all seven tans are used in both, the areas must match, so one figure is slightly wider somewhere by exactly the amount that appears missing elsewhere. The eye cannot detect a small difference spread along a long edge.

Is a tangram a mechanical puzzle?

Yes, of the dissection kind, which sits in the put-together or assembly family. It has no mechanism and nothing is hidden, which is what makes it a clean example of a puzzle that is hard purely because of geometry.