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The tangram () is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pieces without overlap. Alternatively the tans can be used to create original minimalist designs that are either appreciated for their inherent aesthetic merits or as the basis for challenging others to replicate its outline. It is reputed to have been invented in China sometime around the late 18th century and then carried over to Americas and Europe by trading ships shortly after. It became very popular in Europe for a time, and then again during World War I. It is one of the most widely recognized dissection puzzles in the world and has been used for various purposes including amusement, art, and education.
In either case, the first known use of the word is believed to be found in the 1848 book Geometrical Puzzle for the Young by mathematician and future Harvard University president Thomas Hill. Hill likely coined the term in the same work, and vigorously promoted the word in numerous articles advocating for the puzzle's use in education, and in 1864 the word received official recognition in the English language when it was included in Noah Webster's American Dictionary.
Several Chinese sources broadly report a well-known Song dynasty polymath Huang Bosi 黄伯思 who developed a form of entertainment for his dinner guests based on creative arrangements of six small tables called 宴几 or 燕几( feast tables or swallow tables respectively). One diagram shows these as oblong rectangles, and other reports suggest a seventh table was added later, perhaps by a later inventor.
According to Western sources, however, the tangram's historical Chinese inventor is unknown except through the pen name Yang-cho-chu-shih (Dim-witted (?) recluse, recluse = 处士). It is believed that the puzzle was originally introduced in a book titled Ch'i chi'iao t'u, which was already reported as lost in 1815 by Shan-chiao in his book New Figures of the Tangram. Nevertheless, it is generally believed that the puzzle was invented about 20 years earlier.
The prominent third-century mathematician Liu Hui made use of construction proofs in his works and some bear a striking resemblance to the subsequently developed banquet tables which in turn seem to anticipate the tangram. While there is no reason to suspect that tangrams were used in the proof of the Pythagorean theorem, as is sometimes reported, it is likely that this style of geometric reasoning went on to exert an influence on Chinese cultural life that lead directly to the puzzle.
The early years of attempting to date the Tangram were confused by the popular but fraudulently written history by famed puzzle maker Sam Loyd in his 1908 The Eighth Book Of Tan. This work contains many whimsical features that aroused both interest and suspicion amongst contemporary scholars who attempted to verify the account. By 1910 it was clear that it was a hoax. A letter dated from this year from the Oxford Dictionary editor Sir James Murray on behalf of a number of Chinese scholars to the prominent puzzlist Henry Dudeney reads "The result has been to show that the man Tan, the god Tan, and the Book of Tan are entirely unknown to Chinese literature, history or tradition." Along with its many strange details The Eighth Book of Tan's date of creation for the puzzle of 4000 years in antiquity had to be regarded as entirely baseless and false.
The puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries. This was mostly due to a pair of British tangram books, The Fashionable Chinese Puzzle, and the accompanying solution book, Key. Soon, tangram sets were being exported in great number from China, made of various materials, from glass, to wood, to tortoise shell.
Many of these unusual and exquisite tangram sets made their way to Denmark. Danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm. The first of these was Mandarinen (About the Chinese Game). This was written by a student at Copenhagen University, which was a non-fictional work about the history and popularity of tangrams. The second, Det nye chinesiske Gaadespil (The new Chinese Puzzle Game), consisted of 339 puzzles copied from The Eighth Book of Tan, as well as one original.
One contributing factor in the popularity of the game in Europe was that although the Catholic Church forbade many forms of recreation on the sabbath, they made no objection to puzzle games such as the tangram.
More internationally, the First World War saw a great resurgence of interest in tangrams, on the homefront and trenches of both sides. During this time, it occasionally went under the name of "The Sphinx" an alternative title for the "Anchor Puzzle" sets.
The two-monks paradox – two similar shapes but one missing a foot:
The Magic Dice Cup tangram paradox – from Sam Loyd's book The 8th Book of Tan (1903). Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes. (Notice that the one on the left is slightly shorter than the other two. The one in the middle is ever-so-slightly wider than the one on the right, and the one on the left is narrower still.)
Clipped square tangram paradox – from Loyd's book The Eighth Book of Tan (1903):
Of these seven pieces, the parallelogram is unique in that it has no reflection symmetry but only rotational symmetry, and so its mirror image can be obtained only by flipping it over. Thus, it is the only piece that may need to be flipped when forming certain shapes.
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