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QUERIES ON ORIENTAL SOURCES IN RECREATIONAL MATHEMATICS
by Prof. David Singmaster
Last amended on 4 agosto 1996.
I am working on a history of recreational mathematics. I have found a
number of topics which have Chinese and Japanese connections which I am
gathering here for convenience in correspondence. Separate letters deal
with Middle Eastern questions, i.e. Egyptian, Babylonian, Indian, Arabic,
Persian and Turkish, and with Russian questions.
Most of the questions here relate to China, but there may be Japanese
connections unsuspected by me and any oriental information on the topics
would be of interest. Some of these concern situations where problems are
known from China and Europe, but there are no Indian and Arabic sources
known to me; that is, the apparent transmission from the Orient has a gap in
it. For some topics, the usual transmission may be inadequate to explain
the early history.
For example, the cistern problem appears almost simultaneously in China
and Alexandria. Hero(n)'s work gives two problems, both incorrectly solved,
while the Chiu Chang Suan Ching (Jiu Zhang Suan Shu) gives a clear example
with 5 pipes and several related problems.
As another example, after 5C to 7C China, the Hundred Fowls problem is
first known to appear in Europe, Egypt and India almost simultaneously in
the late 9C. This is faster than any other example of transmission that we
know of. Further, the problem is well developed in all three places,
especially in Egypt where Abu Kamil gives a problem with five varieties of
bird and says there are 2676 solutions.
Tait's Counter Puzzle and the Chinese Rings are further examples where
there is no sign of the usual transmission through India and the Arabs.
There are also a few cases where transmission seems to have gone from
Europe to the Orient - Tangrams and the Josephus Problem are examples where
there is no sign of the usual transmission, and the transmission may well
have gone to the East.
I wonder if there was some transmission over the Silk Road or other
central Asian trade routes which could have brought some information
directly to Europe, bypassing the Indians and Arabs. If so, there may be
some evidence for this in the folk cultures of the central Asians and
Russians, but I can find nothing about this and would be delighted to hear
from anyone who does know about this.
The recreational questions are discussed more fully in my Sources or
the Queries thereto. I am preparing the seventh preliminary edition of
I have recently obtained: Marguerite Fawdry; Chinese Childhood;
Pollock's Toy Theatres, London, 1977. This describes many toys and puzzles,
but fairly sketchily and not always accurately. Pp. 182-183 give sources
in Chinese (or Japanese) which I have not seen and I would appreciate any
help in obtaining and translating these. In particular, she cites Liu
Hsieh's Wen-hsin Tiao-lung for 3C puzzles and Dream of a Red Chamber, in
which the hero solves metal puzzles.
Stewart Culin; Korean Games, with Notes on the Corresponding Games of
China and Japan; University of Pennsylvania, Philadelphia, 1895, (Reprinted
as: Games of the Orient; Tuttle, Rutland, Vermont, 1958. Reprinted under
the original title, Dover and The Brooklyn Museum, 1991.) cites a 1714 (or
1712) Japanese book: Wa Kan san sai dzu e ("Japanese, Chinese, Three Powers
picture collection"), published in Osaka.
NIM GAMES. Nim is first described by C. L. Bouton in 1902. He claimed
that it was widely played in America and was called Fan-Tan by the Chinese.
He also later admitted that the identification with Fan-Tan was wrong. He
later admitted that he coined the word Nim from the German word 'nimm', the
imperative of take. Interestingly, Luo Jianjin and Siu Man-Keung tell me
there is a Chinese character, nian, pronounced 'nim' in Cantonese, which
means to pick up or take. However, there seems to be no historic connection
between these words.
Wythoff's Nim, described by Wythoff in 1907, has two piles and one can
take any amount from one pile or the same amount from both piles. The
English translation of A. P. Domoryad's Russian book on mathematical games
says this 'is the Chinese national game of TSYANSHIDZI ("picking stones")'.
I have only seen this in English translation, so the original Chinese word
is hard to determine. Prof. Siu could not work out what the Chinese was.
Winning Ways says it is called Chinese Nim or Tsyan-shizi. Is there any
evidence for any games of this kind in China?
MORRIS GAMES. On p. 12 of Fawdry is a scene, apparently from the
Hundred Sons scroll of the Ming period, in which some children appear to be
playing on a Twelve Men's Morris board. In Culin's Korean Games, p. 102,
section 80: Kon-tjil - merrells is a description of the usual Nine Men's
Morris. He states the Chinese name is Sám-k'i (Three Chess) and
continues: "I am told by a Chinese merchant that this game was invented by
Chao Kw'ang-yin (917-975), founder of the Sung dynasty." This is the only
indication of an oriental source of these games that I have seen.
SNAKES AND LADDERS. The only paper that I have seen on this traces it
back to medieval India. However, Fawdry, p. 183, cites Nagao Tatsuzo's
Shina Minzoku-shi [Manners and Customs of the Chinese], Tokyo, 1940-1942,
for a 7C Chinese version of the game called Sheng-kuan T'u [The Game of
Promotion] and this game is also described by Bell and Cornelius.
SQUARING THE SQUARE. There is a paper of relevance by Michio Abe; On
the problem to cover simply and without gap the inside of a square with a
finite number of squares which are all different from one another [in
Japanese]; Proc. Phys.-Math. Soc. Japan 4 (1932) 359-366. I have not been
able to locate a copy of this. Can anyone supply a photocopy?
TAIT'S COUNTER PUZZLE. The problem is to shift a row of black and
white markers from BBBBWWWW to BWBWBWBW, moving two counters as a pair.
P. G. Tait describes it in 1884, but T. Hayashi reports that it appears in
a book of Genjun Nakane in 1743. Does the problem appear in China? Can
anyone supply a copy of Nakane or similar early Japanese occurrences?
FOX AND GEESE, ETC. These are board games with asymmetric forces. Fox
and Geese is supposed to be medieval, even 1st millennium, but Murray's
History of Chess cites a North Asian version of Bouge-Skodra (Boar's
Chess). Are there any early Chinese forms? Culin cites a Japanese version
called Yasasukari Musashi from Wa Kan san sai dzu e of 1712.
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. 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.
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
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 Tôzai
Sľgaku Monogatari Heibonsha of 1973. Takagi has kindly sent his reprint of
this booklet, but I am unsure as to the author, etc. Can anyone provide
more details? Hirayama's book might be interesting to see.
SIX-PIECE BURR = CHINESE CROSS. This is a puzzle comprising six square
rods with notches that assemble into a kind of star with two rods along each
of the three axes in space. The first known reference is in a Berlin
catalogue of 1790 and then in a Nuremberg toy catalogue of 1801 where it is
called 'The small Devil's claw'. One of the next references is 1860 where
it is called 'The Chinese Cross'. This is one of the puzzles that appears
in 19C 'Sunday Chests' which were collections of ivory puzzles from China.
Is there any Oriental history of this or similar assembly puzzles, or indeed
of the Sunday Chests?
DEAD DOGS AND TRICK PONIES. There is a pattern of overlapping bodies
and heads so that the same head can be viewed as part of several bodies.
Examples are known from medieval Persia (Rza Abbasi, (1587-1628)) and Edo
period Japan (17C - 19C). There are said to be other examples from India
and/or China. I would like to know of early examples.
MOIRÉ PATTERNS. This technique derives from China, and was introduced
into France in 1754. I'd like to know more about its Chinese history.
ROUND PEG IN SQUARE HOLE OR VICE VERSA. The late 16C poet Wang
Tao-K'un mentions 'square handles ... into the round sockets'. Are there
other early references to this idea, in China or elsewhere?
THE JOSEPHUS OR SURVIVOR PROBLEM. This is the problem of counting out
every k-th person from a circle. It was a common medieval European problem
in the form where half of a group is to be eliminated. The usual form
involved 15 Christians and 15 Turks on a ship in a storm. The captain
announces that half of the passengers must go overboard and one of them says
that everyone should get in a circle and be counted off by 9s. Cardan
suggested that this might be the way in which Josephus survived. It appears
in the Japanese literature as early as 1627, with 15 children and 15
stepchildren counted by 10s, but with one child (the 15th) skipped, until
only one is left. Ahrens cites some indications that it may go back to the
11C in Japan and believes that the problem arose independently in Japan.
Takagi has sent me an article by Shimodaira (source and date not given) on
the recreational problems in Jing_ki, but this doesn't indicate the date of
the second edition which first contained these problems. Shimodaira states
the Japanese name of the problem, Mamakodate, first occurs in the essay
Tsurezuregusa by Kenk_ Yoshida (1283-1350), but it's not clear if the
problem itself is given there. Shimodaira gives the problem in the form
where the 15th stepchild protests that he is about to be eliminated and the
stepmother agrees to restart from him. Ahrens says that Jing_ki has the
stepmother doing this by accident and the bit about the 15th stepchild
first occurs in Miyake Kenry_, 1795. Is there any Chinese material on this
problem? I have recently seen an article which claims an Irish origin of
the problem, c800, and which gives early medieval forms called the Ludus
Sancti Petri. It is often thought to derive from the Roman practice of
decimation. Murray's History of Chess mentions 10 diagrams of this in an
Arabic chess MS of c1370, possibly referring to a c1350 work. Murray
asserts the problem is of Arabic origin. Wakoku Chie-kurabe shows a version
with 8 and 8 counted by 8s, such that either group can be the group
counted out first!, depending on where one starts.
A TRAVELLING MERCHANT PROBLEM or THE MONKEY AND THE COCONUTS. A
merchant travels to three fairs. At each, he doubles his money and spends
1000. At the end he has no money left. How much did he have? An
alternative formulation has a man carrying apples who has to pass three
porters, each of whom takes half the apples and half an apple more. If the
final amount is given, the problem is determinate and readily solved. Such
problems already appear in the Chiu Chang Suan Ching (Jiu Zhang Suan Shu)
and later in Mahavira, Sridhara, Bhaskara and Zhu Shijie's Siyuan Yujian.
If the final amount is not given, the problem is indeterminate and much more
interesting and is known as The Monkey and the Coconuts Problem from a 1927
version. The only indeterminate examples known prior to 1725 are in
Mahavira. Are there other Chinese examples, either determinate or
CISTERN PROBLEM. One pipe fills a cistern in 2 days, another in 3
days, how long for both together? The earliest examples known are in the
Chiu Chang Suan Ching (Jiu Zhang Suan Shu), Hero(n)'s Peri Metron, the
Bakhshali MS and The Greek Anthology. It is popular in India from about
850. Are there other early Chinese examples? I have recently found that
the equivalent assembly problems which occur in the Chiu Chang Suan Ching
(Jiu Zhang Suan Shu) also occur in Old Babylonian.
THE CHINESE RINGS and other WIRE PUZZLES. Ch'ung-En Yü's Ingenious
Ring Puzzle Book calls this 'Nine Interlocked-Rings Puzzle' and says it was
well known during the Sung (Song), but I have not seen any such material and
Needham does not cite any. On pp. 70-72, Fawdry cites The Stratagem of
Interlocking Rings, a Chinese musical drama of c1300 - can anyone provide
more information about this? The first European mention is in Cardan
(1550). Gardner reports that there are 17C Japanese haiku about it and that
it is used in Japanese heraldry - can anyone supply examples? Are there
other examples of wire puzzles in China before about 1890 when they become
common in Europe? Many of the common string and ring puzzles appeared in
Sunday Chests in the 19C. Fawdry, pp. 70-71 illustrates one such, but
claims it is 18C, which seems unlikely since it contains a tangram, and p.
74 shows a French chest and she identifies several string and wire puzzles
as being Chinese. String and wire puzzles are popular in modern India and
Pakistan. Stewart Culin; Korean Games; pp. 31-32, says there are many
Japanese ring puzzles, called Chiye No Wa, and shows one which seems to be
10 rings linked in a chain - possibly the simple type of puzzle ring??
MAGIC SQUARES. These are certainly Chinese in origin. Needham, Ho
Peng Yoke, Camman and Lam cite a number of books which are not available in
translations. These are the following.
Hsu Yo (Xu Yue). Shushu Jiyi.
I Wei Ch'ien Tso Tu. c1C.
Ta Tai Li Chi. c80. Chap 67: Ming Thang. Or: Chap. 8, p. 43 of
Szu-pu ts'ung-k'an edition, Shanghai, 1919-1922.
Ts'ai Yuan-Ting. c1160. (He is cited by an author, quoting Needham,
but I can't find him in Needham.)
I would be interested in seeing these and any similar early Chinese
Li Yan & Du Shiran's Chinese Mathematics - A Concise History, section
5.6 (and other books) describes 6 x 6 magic squares on iron tablets, using
Arabic numerals, found at Xian in 1956. Can anyone give more details on
these? These are apparently in the new museum in Xian.
THE HUNDRED FOWLS PROBLEM. A man buys 100 fowls for 100 cash -
cocks cost 5, hens 3 and chicks 1/3. How many of each did he buy?
This first appears in Zhang Qiujian's Zhang Qiujian Suan Jing. Libbrecht
gives a long discussion of this topic, but there may be other material.
THE CHINESE REMAINDER THEOREM. This appears in Sun Zi's Sun Zi Suan
Jing, but I have no other Chinese references until Qin Jiushao's Shushu
Jiuzhang of 1247.
CONJUNCTION OF PLANETS. This is an extension of the above. Sun Zi
gives a problem where sisters visit home every 5, 4, 3 days - when do they
all come together? I have very little on this topic as a recreational
ALL CRETANS ARE LIARS, ETC. In the History of the Warring States,
c-2C, is the story of the Elixir of Death, which is also said to have
occurred to Tung-Fang So, -2C. Are there any other Oriental versions of
paradoxical statements or situations?
OVERTAKING AND MEETING PROBLEMS. The earliest examples I know are in
the Chiu Chang Suan Ching (Jiu Zhang Suan Shu), the Bakhshali MS, Zhang
Qiujian's Zhang Qiujian Suan Jing, then in India and Europe. I would be
interested in other early Chinese examples.
BINARY DIVINATION. In the booklet by Shimodaira sent by Takagi, there
is some discussion of 'metsukeji' (magic cards), one version of which are
the binary divination cards which he says were known in Japan in the 14C or
earlier. If so, this would considerably precede my earliest known example
of Luca Pacioli, c1500. Actually it is not surprising to find these in the
Orient as the logical binary arrangement of the I Ching hexagrams was done
by Shao Yung, c1060. Can anyone send details and/or copies of early
versions? It would be nice to have some modern examples.
In Wakoku Chie-Kurabe, I cannot make out what is happening
on pp. 6 & 24.
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