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QUERIES ON MIDDLE-EASTERN SOURCES IN RECREATIONAL MATHEMATICS
by 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 Egyptian, Indian, Persian, Arabic or Turkish
connections which I am gathering here for convenience in correspondence.
Separate letters deal with Russian questions and with Oriental questions,
i.e. China and Japan.
Some questions relate to the whole of the Middle East, while others are
more specific to parts of it. I give the general questions first. Many of
these concern situations where problems are known from China (and/or India)
and Europe, but there are no Indian and Arabic (or just Arabic) sources
known to me. That is, the apparent transmission from the Orient has a gap
in it. There are also problems which appear quite fully developed in
Fibonacci or Alcuin which seem as though they must have some earlier
appearances, probably in Arabic, though Byzantine sources are also possible.
There are also a few cases where transmission seems to have gone from
Europe to the Orient. It is possible that transmission may have taken place
along the Silk Roads of Central Asia and consequently left no trace in the
Indian and Arabic cultures. I would be most grateful to anyone who can shed
light on such transmission.
The recreational questions are discussed more fully in my Sources or
the Queries thereto. I am currently working on the sevent preliminary
edition of this.
MORRIS GAMES. The game known as Nine Men's Morris, Mill, Moule, Mühle,
etc. is said to have been played at various places in the ancient world. A
game board from -14C was found at Kurna, Egypt and boards from 1C occur at
Mihintale, Ceylon. Are there other early examples? Are there any ancient
written references? Murray's History of Chess mentions the Arabic game of
qirq as (a kind of) morris and gives some 8C and 10C references. This
becomes alquerque in Spain.
RIVER CROSSING PROBLEMS. These are first known in Alcuin, 9C, but I
wonder if he learned these from some Middle Eastern source. Marcia Ascher
has described many African versions, but it is not clear if these are
EULER CIRCUITS. Ascher has also described Euler Circuits in many
cultures, but again we do not know if these are ancient. One pattern, of
two crescents facing opposite directions but overlapping, is known as the
'Seal of Mahomet'. Does it occur in Arabic sources?
THE EXPLORER'S OR JEEP PROBLEM. Alcuin is also the earliest known
example of trying to cross a desert - the Explorer's or Jeep Problem. His
version is not too clear and the problem does not seem to reappear until
Pacioli (c1500) and then the 20C! Alcuin's problem involves camels and
again I wonder if there was a Middle Eastern source.
KNIGHT'S TOURS. The earliest known example is in an Sanskrit MS poem
K_vy_lank_ra, by Rudrata, c900, described in Murray's A History of Chess.
However, an Arabic MS of 1141 gives tours which may derive from lost works
of al-`Adl_ (c850) or as-S_l_ (c920). Rudrata seems to be the only known
early Indian source. Since chess originated in India, there ought to be
more examples in early chess literature in India? Many problems based on
the chessboard appear in the 19C, but in view of their naturalness, I wonder
if some of these appear in earlier chess literature.
TANGRAMS. These became a fad in China and Europe c1805 and the
earliest appearance in the Far East appears to be early 18C in Japan, but
with a different set of pieces. The only other early tangram-like puzzle
seems to be the Loculus of Archimedes, known in the classical world from -3C
to at least 6C. It has 14 pieces and is rather more complex than the
Tangrams. Some of the Archimedes sources are two 17C Arabic MSS, so the
Loculus may have been known to the Arabs and possibly they transmitted it to
PYTHAGOREAN PROBLEMS. Fibonacci gives a Pythagorean problem of
locating a Well Between Two Towers which shall be equidistant from the tops
of the towers. Most Pythagorean problems already appear in Babylonian,
Chinese or Indian sources, but I haven't seen this one before Fibonacci.
Also the Broken Bamboo problem appears in Chinese and Indian sources, but I
feel it may appear in Babylonian or Greek sources.
THE JOSEPHUS OR SURVIVOR PROBLEM. This has only a vague connection
with Josephus, but may have its origins in the Roman custom of decimation.
I have recently seen an article which claims an Irish origin of the problem,
c800, and which gives early medieval forms called the Ludus Sanct Petri. It
was certainly very popular in medieval and Renaissance Europe. 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. I have a reference to a 1659 Arabic version. Are there any
other early Arabic sources?
EGYPTIAN FRACTIONS. Is there any attempt to show that every fraction
can be written as a sum of distinct unit fractions before Fibonacci?
THE APPLE ORCHARD WITH THREE GUARDS PROBLEM involves a man taking
apples from a garden and having to pay each of three guards something like
half of his apples and half an apple more, leaving him with a given amount.
An alternate version involves a travelling merchant who doubles his money
and spends 1000 at each of three fairs, ending with no money. Both versions
are common throughout medieval European mathematics and some occur in the
Chiu Chang Suan Ching (c-150), in Sridhara (c900) and Bhaskara II (1150).
An indeterminate form, usually known as the Monkey and Coconuts Problem,
where we only know the final result (mod m), occurs in Mahavira (850) and
then in Ozanam (1725) and reappears about 1900. Are there any Arabic or
Persian versions of this problem? A different version has the i-th child
receiving i plus 1/n of the remainder with all children getting the same
amount. This appears in Fibonacci but it might be Arabic?
HALF + THIRD + NINTH, ETC. A common version of this is called The 17
Camels. Originally this problem was solved by dividing the camels in the
proportion 1/2 : 1/3 : 1/9 and the fact that the fractions did not add up
to 1 was ignored. Such problems occur already in the Rhind Papyrus and in
the Bakhshali MS and in Chaturveda. Tartaglia is claimed to be the earliest
to borrow an 18th camel, but I can't find it there. The problem in this
form is often claimed to be of Arabic or Hindu origin.
CISTERN PROBLEMS. These go back to the Chiu Chang Suan Ching and
Hero(n) of Alexandria. When do fanciful versions - e.g. ship with two sails
or three animals eating a sheep - originate? Fibonacci gives the latter and
the Byzantine Rechenbuch gives the former (with 5 sails), which Vogel states
to be the earliest of this form. I have no earlier examples. However, the
equivalent 'assembly' problems of the Chiu Chang Suan Ching also occur in
EACH DOUBLES THE OTHER'S MONEY TO MAKE ALL EQUAL. A version appears in
Diophantos, then the problem occurs in Mahavira and Fibonacci. Are there
other early examples?
SHARING COSTS. There are two common forms. The earlier involves men
who work for part of the time and this occurs in Mahavira and Sridhara. The
other concerns a man who digs part of a well - how much should he be paid?
This occurs in al-Kazwini (1262) with several possible answers, but no
resolution. Dell-Abbaco gives both forms with a resolution of a well
problem. Are there other early examples?
CASTING OUT NINES. This is often attributed to the Hindus, but I have
some references to Greek special uses of it by St. Hippolytus (c200) and
Iamblichus (c325) though I haven't seen either source. I have read that
Avicenna's attribution of it to the Hindus is a dubious interpretation. It
appears in al-Khw_rizm_ (c820) and al-Uql_dis_ (952/953) as well as in
Aryabhata II's Mah_-siddh_nta
THE CHESSBOARD PROBLEM. (which leads to 1 + 2 + 4 + 8 + ...) is often
attributed to India, but my earliest sources are Arabic: al-Ya`q_b_ (c875)
(described in Murray's History of Chess), al-Mas`_d_'s Meadows of Gold
(943), which doesn't relate the series to the chessboard) and al-B_r_n_'s
Chronology of Ancient Nations (1000). Murray cites a 9 or 10 C treatise on
the problem by al-Missis_ - does this exist? Is there any Indian or Persian
material of relevance?
The use of 1, 2, 4, ... and 1, 3, 9, ... as weights occurs in
Fibonacci but I feel there must be earlier examples. I have a reference to
CHINESE RINGS. These are claimed to occur in Sung China (c11C) - but I
have no references to such material. My first reference is Cardan (1550),
but it must have been transmitted to Europe in some way.
MAGIC SQUARES. The Indian and Arabic history of Magic Squares is quite
confused. A. N. Singh (Proc. ICM, 1936, pp. 275-276) refers to a c1C order
4 square by N_g_rjuna, but I know nothing more about this. There are many
Indian and Arabic references that I have not been able to find - indeed some
apparently are not extant - but it would take too long to list them here.
THE 100 FOWLS PROBLEM. appears in China c475 and recurs in Chinese
works over the next centuries. In the 9C, it appears as a well developed
problem in Alcuin (c800?), Mahavira (850) and Abu Kamil (c900). This is a
remarkably rapid transmission. Abu Kamil implies the problem was well
known. His comments and the rapidity of transmission lead me to wonder if
there are earlier examples. The problem appears in the Bakhshali MS which
may have been early enough to show transmission across north India??
SELLING DIFFERENT AMOUNTS AT THE SAME PRICES TO YIELD THE SAME AMOUNT
appears in one form in Mahavira, Sridhara and Bhaskara. A simpler form
occurs in Fibonacci and later European books. Are there any Arabic versions
or other Indian ones?
CONJUNCTION OF PLANETS. This is a variant of the Chinese Remainder
Theorem which appears in China and India. Are there any Arabic forms?
THE BLIND ABBESS AND HER NUNS. This occurs in an Arabic MS on chess in
c1370. I then have Pacioli (c1500) and van Etten (1653 ed.). Are there
other Arabic or medieval forms?
DILUTION PROBLEMS. A one stage version is in the Rhind Papyrus and a
four stage version is in the Bakhshali MS. My next example is Tartaglia.
THE APPLESELLER'S PROBLEM, involving combining amounts and prices
incorrectly, appears in Alcuin (9C) and Ibn Ezra (c1150), then Fibonacci,
etc. Are there Arabic versions?
THE LAZY WORKER, who gains for each day he works and forfeits for days
he doesn't work, appears in al-Karkhi (c1010) and Fibonacci, etc. Are there
other Arabic or perhaps Greek sources?
LIAR PARADOX. Do the Liar or similar paradoxes occur in Arabic or
THREE MEN WITH SPOTS ON FOREHEADS. This problem appears in the US,
c1935, attributed to Alonzo Church. A book on Palestinian stories says it
was a well known folk story when it was heard before 1948, though the
solution was based on symmetry rather than logic. The version known as
Forty Unfaithful Wives has a Central Asian setting but may be a 20C
STRANGE FAMILIES - e.g. men marrying each other's sister or daughter -
occur in Alcuin, Abbott Albert and a c1430 Hebrew text. Are there earlier
SNAIL CLIMBING OUT OF A WELL. This seems to derive from the habit of
expressing velocity by unit fractions - e.g. the snail goes at a rate of
+1/2 -1/3. Such problems occur in Chaturveda, Mahavira, Sridhara and
Fibonacci. But in Europe of c1370, the idea that the 1/2 was in the day
and the 1/3 was in the night was being treated carefully. Are there
earlier examples where the rates are treated alternately?
SOLOMON'S SEAL. I have recently seen the string and bead puzzle known
as Solomon's Seal claimed as an African puzzle, but with no reference. Can
anyone supply details?
WIRE PUZZLES first seem to appear in the West in the late 19C. They
were and are popular in India and China, but I don't know any early sources
except for the Chinese Rings.
PUZZLE RINGS are believed to be Middle Eastern in origin. Are there
any sources? There is a 17C example in the British Museum.
I have had difficulty in communicating with the National Institute of
Sciences, New Delhi. They haven't responded to my letters - do they exist?
I wanted to get the bibliography by Sen from them. This may have been
obsoleted by Rahman's Bibliography, published by the Indian National Science
Academy in New Delhi. I have recently received a copy of Rahman.
I'd like to get second-hand copies or reprints of the following:
Colebrooke, H, T. Algebra, with Arithmetic and Mensuration from the
Sanscrit of Brahmagupta and Bhascara. (Originally Murray, London, 1817.)
Annotated by H. C. Banerji, Calcutta, 1893 & 1927; Kitab Mahal, Allahabad,
1967. (I have a reprint of the original ed. I would like a copy of the
Kaye, G. R. The Bakhshali Manuscript - A Study in Medieval Mathematics.
Archaeological Survey of India - New Imperial Series 43:1, 1927. (Part 1 is
the item of main interest, but parts 2 and 3 are of some interest.)
Kaye, G. R. Indian Mathematics. Thacker & Spink, Calcutta, 1925.
Mahavira. The Ganita-Sara-Sangraha of Mahaviracarya. Translated by M.
Rangacarya. Government Printing Press, Madras, 1912.
Aryabhata II, Maha-Siddhanta. Is there an English version? Does it
have anything of interest to me?
Bhaskara I, Aryabhatiyabhasya. Same questions as for the last item.
Narayana, Ganita-Kaumudi. Same questions.
Satya Prakash. He translated two of the Sulbasutras, published at New
Delhi in 1968. Are these available?
Bag cites a Manava Srautasutra belonging to the Maitrayani Samhita,
edited by Van Gelder. I haven't been able to find this - is it of any
interest to me?
I have recently bought some modern puzzle books by Shakuntala Devi and
a book of every day science puzzles. I found these interesting because they
had quite different collections of problems than English and American books,
so I would be interested in copies of similar Indian books. There may also
exist reprints of older English works which would interest me.
I am also interested in books on Ramanujan - e.g. S. R. Ranganathan;
Ramanujan: The Man and the Mathematician - and in works by J. N. Kapur -
e.g. The Fascinating World of Mathematics; S. Chand, 1970 and The Nature
of Mathematics and Thoughts on Mathematical Education; Atma Ram & Sons,
CROSSED LADDERS. Bhaskara gives an easy problem with crossed ladders
in a street. Are there any similar problems in Hindu or Arabic literature?
Bharava Tantra & Siva Tandava Tantra. Dvivedi, Intro. to Narayana's
Ganita Kaumudi, cites these as early sources for Magic Squares, but I cannot
find out anything about them.
Varahamihira, Pancasiddhantika - Bag implies this says something about
Magic Squares, but I looked through the English edition by Neugebauer and
Pingree and couldn't find anything??
MEMORY WHEELS = CHAIN CODES. The 'Memory Wheel'
ya má tá rá ja bhá na sa la gám gives all the sequences of three short or
long syllables. It is supposed to have been used as a mnemonic by Sanskrit
poets and musicians, c1000. Can anyone give some references?
LATIN SQUARES. Ahrens discusses and cites early examples from medieval
Islamic times, c1200, when they were used on amulets. Are there other
sources, discussions, examples, etc.?
TESSELLATIONS. In view of the Arabic emphasis on geometric design, was
there any work on the Archimedean or other Tessellations of the Plane?
THE PROBLEM OF THE PANDECTS is the problem of two persons sharing
unequal resources with a third person. This is sometimes phrased as though
it was a Roman problem, but my earliest sources are Fibonacci (1202) and al-
Kazwini (1262). Are there any early Arabic sources?
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.
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