The story of the tangram

The tangram (Chinese qīqiǎobǎn) is a square cut into seven flat pieces. You rearrange them, with no gaps and no overlaps, to make all kinds of shapes. The seven pieces are two large triangles, one medium triangle, two small triangles, a square and a parallelogram. With all seven you can make people, animals, houses, boats and endless more.

Seven areas that add up

If the original square has area 16, the pieces measure 4 and 4 (large triangles), 2 (medium triangle), 1 and 1 (small triangles), 2 (square) and 2 (parallelogram): 16 in total. This site's tests check those areas, and that every shape study on the puzzle page also has area 16. Since you always use all seven pieces, every tangram figure, however different it looks, covers exactly the same area.

Only 13 convex shapes

A convex polygon has no dents: every corner points outward. Using all seven pieces, you can make exactly 13 different convex polygons (counting congruent ones as the same): 1 triangle, 6 quadrilaterals, 2 pentagons and 4 hexagons. There is only one triangle, and the square is one of the thirteen.

This was proved in 1942 by two Chinese mathematicians, Fu Traing Wang and Chuan-Chih Hsiung, in a paper called “A Theorem on the Tangram”. Note that this site's program does not check the 13 itself; we are reporting a result proved in the literature.

From China, on trade ships

The tangram is said to have come from China, but nobody knows who invented it or when. Some suggest it was inspired by an older Chinese table game, the Yanjitu, though that is only a theory. Among the oldest surviving Chinese books are one from 1813 and another from 1858. The puzzle was clearly being played by the turn of the 19th century. The Chinese name means “seven boards of skill”.

In the early 1800s, ships trading with China carried the tangram to Europe and America. A record says an American captain brought a Chinese book from Canton in 1815. In 1817 the first Western book on it appeared in England, and the craze spread through France, Denmark, Italy and the United States, peaking some time between the 1810s and about 1820. Sources differ on the exact years.

The word “tangram” is first found in an American book of 1848. Where it comes from is unknown.

The god from 4,000 years ago never existed

The American puzzle maker Sam Loyd published The Eighth Book of Tan in the early 1900s, with some 700 designs. It told of a god named Tan who supposedly created the puzzle 4,000 years ago. That history was invented. In 1910, Sir James Murray, editor of the Oxford English Dictionary, wrote to the puzzle writer Henry Dudeney that Chinese scholars could find neither Tan nor his book in any Chinese source. So set aside any “4,000 years” or “Tan” story you may come across.

A similar puzzle in Japan

In 1742 a Japanese book called Seishō Nagon Chie no Ita appeared. It is also a silhouette puzzle of seven boards, but the boards are cut differently from a tangram. It is neither a parent nor a Japanese version of the tangram, but a separate idea. Its preface names the poet Sei Shōnagon, which nobody believes; the real designer is unknown.

How to think about it

• Place the large triangles first. They have the fewest places to go, so deciding them early makes the rest clearer.
• Watch the slopes and corners. Diagonal edges call for triangles or the parallelogram. Look for the piece with the same corner as a sharp point of the shadow.
• The parallelogram can flip. Every other piece looks the same flipped over, but the parallelogram changes. Try it when you are one step short.
• Other cuts count too. On this site, covering the shadow completely is a win, so an arrangement different from the designer's is also correct.

Stuck? The Hint button shows where one piece goes, as a dotted outline. It is one possible placement, not the only one.