Monday 31 December 2012

Dissection Puzzles, a Little History

It is known that Archimedes created a game/puzzle with the dissection of a square into 14 pieces.  The object of the puzzle is to put the pieces back together to form a square. A more difficult question, unknown for over 2000 years, is how many unique ways are there of putting the pieces together to form a square. Bill Cutler used a computer program to show that there are 536 unique ways to assemble the pieces not counting similar rotations and reflections. All 536 solutions are visible in this article from Ed Pegg's web site.

The Archimedes Palimpsest is a parchment codex palimpsest, which originally was a 10th-century Byzantine Greek copy of an otherwise unknown work of Archimedes of Syracuse and other authors. It was overwritten with a Christian religious text by 13th-century monks. The erasure was incomplete, and Archimedes' work is now readable after scientific and scholarly work from 1998 to 2008 using digital processing of images produced by ultraviolet, infrared, visible and raking light, and X-ray.

The Palimpsest is the only known copy of "Stomachion". The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the ancient Greek word for throat or gullet, stomachos (στόμαχος). Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots ofὀστέον (osteon, bone) and μάχη (machē – fight). The puzzle is also known as the Loculus of Archimedes or Archimedes'Box. Loculus seems to be a word related to the division of a tomb area into small chambers for different bodies and is related to the diminutive of locus for a point or place, thus "a little place". (I don't yet get the connection between the puzzle and stomach. Bob Mrotek pointed out to me in a comment that the Stomahion may be a later condesation of the Greek Ostomachion, Ὀστομάχιον, "a word meaning a fight (μάχη, mákhion) with bones (ὀστέον, ostéon) in reference to the pieces which were often made out of ivory." Which now appears as well on Wikipedia.)

The puzzle is sold by Kadon as Archimedes Square:

Amazingly, the next "put-together" puzzle of geometric shapes didn't appear until 1742.  The wisdom plates of Sei Shonagon is a seven piece puzzle.  They appeared in China 71 years before the more famous Tangram puzzle.  The second edition of this puzzle, published in 1743, is the oldest known surviving puzzle of this type.


Tangram is a name of a Chinese puzzle of seven pieces that became popular in Europe around the middle of the 19th century. It seems to have been brought back to England by Sailors returning from Hong Kong. The origin of the name is not definite. One theory is that it comes from the Cantonese word for chin. A second is that it is related to a mispronunciation of a Chinese term that the sailors used for the ladies of the evening from whom they learned the game. [Concubines on the floating brothels of Canton, Hong Kong, and many other ports belonged to an ethnic group called the Tanka whose ancestors came from the interior of the country to become fishermen and pearl divers. They were considered as non-chinese by the govenments of China until 1731. They were unique among Chinese women in refusing to have their feet bound. ] A third suggestion is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh in Japan which celebrates the reunion of the weaver (vega) and the herdsman (altair). David Singmaster, below, suggests the name was made up by puzzle master Sam Loyd, but I favor Harvard President Thomas Hill (below) . Whatever the origin of the name, the use of the seven shapes as a game in China were supposed to date back to the origin of the Chou dynasty over one thousand years before the common era. The Chinese name is Ch'i ch'iao t'u which translates, so I am told, as "ingenious plan of seven".

It appears however, that the game and the name are both much more modern than believed. From the website, I found that " The Tangram was invented between 1796 and 1802 in China by Yang-cho-chu-shih. He published the book Ch'i ch'iao t'u (Pictures using seven clever pieces). The first European publication of Tangrams was in 1817. The word Tangram itself was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young. He became the president of Harvard in 1862, and also invented the game Halma.
When Tangrams hit Europe they were an immediate success.  A puzzle museum on-line boasts a collection of a dozen books which were all written within a year of 1817 when tangrams were supposedly introduced into Europe:

Here is some history of the game by David Singmaster, one of the world's foremost authorities on recreational mathematics,

TANGRAMS. These are traditionally associated with China of several thousand years ago, but the earliest books are from the early 19C and appear in the west and in China at about the same time.
(although the image below with tangram problems to create was printed in Japan in 1795, and  Utamaro’s “Tagasode” is a famous 1804 Japanese blockprint that shows Tagasode and her servent trying to solve Tangrams), Indeed the word 'tangram' appears to be a 19C American invention (probably by Sam Loyd). A slightly different form of the game appears in Japan in a booklet by Ganreiken in 1742. Takagi says the author's real name is unknown, but Slocum & Botermans say it was probably Fan Chu Sen. There is an Utamaro woodcut of 1780 showing some form of the game (not yet seen by me). I have seen a 1786 print - Interior of an Edo House, from The Edo Sparrows or Chattering Guide - that may show the game. Needham says there are some early Chinese books, and van der Waals' historical chapter in Elffers' book Tangram cites a number with the following titles.
Ch'i Ch'iao ch'u pien ho-pi. >1820.
Ch'i Ch'iao hsin p'u. 1815 and later.
Ch'i Ch'iao pan. c1820.
Ch'i Ch'iao t'u ho-pi. Introduction by Sang-Hsi Ko. 1813 and later. remarks inserted from a description of the book in Tangram by Joost Elffers, {Located in the Leiden Library #6891: This book, with an introduction by Sang-Hsi Ko, is, as far as is known, the oldest example of a Chinese game-book. ] I would like to see some of these or photocopies of them. I would also be interested in seeing antique versions of the game itself. The only historical antecedent is the 'Loculus of Archimedes', a 14 piece puzzle known from about -3C to 6C in the Greek world. Could it have travelled to China? I found a plastic version of the Loculus on sale in Xian, made in Liaoning province. I wrote to the manufacturer to get more, but have had no reply.
For the 10th International Puzzle Party, Naoki Takashima sent a reproduction of a 1881 Japanese edition of an 1803 Chinese book on Tangrams which he says is the earliest known Tangram book.
Jean-Claude Martzloff found some some drawings of tangram-like puzzles from a 1727 booklet Wakoku Chie-kurabe, reproduced in Akira Hirayama's T“zai S–gaku Monogatari Heibonsha of 1973. Takagi has kindly sent his reprint of this booklet, but I am unsure as to the author, etc.

*Utamaro’s “Tagasode”
The "Ganriken" mentioned in Dr. Singmaster's post was a pseudonym, and it seems unclear who the actual author was.  The book was titled "Sei-Shonagon Chie-no-Ita", which translates to "the ingenious pieces of Sei Shonagon". From Wikipedia, "Sei Shonagon was a lady-in-waiting at the Japanese Imperial Court in the beginning of the 11th Century. She kept a personal diary of sorts in which she wrote down her experiences but mainly her feelings. Such diaries were common at the time and were called pillow books because these books were often kept next to people's pillows in which they would write their experiences and observations. The Pillow Book of Sei Shonagon gives an invaluable insight into the world of the Imperial Court of Kyoto a thousand years ago. Sei Shonagon's observations are witty, wry, poignant, and at times condescending."

Tangrams received another boost in popularity when Charles Dodgson, writing as Lewis Carroll, used them to create illustrations of the Characters in the "Alice" books. In the Penguin Books translation of Tangram by Joost Elffers he states that English Puzzle writer H. E. Dudney purchased a copy of a play-book called The Fashionable Chinese Puzzle from Dodgeson's estate. This book seems to be the most common source of the assertion that Napoleon was an avid Tangram player,

And recently I found this beautiful set of 19th Century Tangram dishes made in China,which are in the Hikimi Town Puzzle Museum in Japan.


Bob Mrotek said...

Posibility - the Roman poet Ausonius wrote in his Cento Nuptialis that the Greek called it Ostomachion, a word meaning a fight (μάχη, mákhion) with bones (ὀστέον, ostéon) in reference to the pieces which were often made out of ivory.

Pat's Blog said...

Bob, Thanks loads, I haven't seen that. Will vvresearch

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