ÿWPCL ûÿ2BJ|xÕ)Ð °x ÐÐа¤˜ˆXÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐÁ`ÁMideast Queries ÀMÀ p.)ÕÕÐ °x ÐÐа¤˜ˆXÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐÕÐ °x ÐÐа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐ ÁàÁà ÃQUERIES ON MIDDLE©EASTERN SOURCES ÁàÁ IN RECREATIONAL MATHEMATICSÄ Ä ÁàÁà ÃDAVID SINGMASTERÄ Ä ÁàÁ87 Rodenhurst Road, London, SW4 8AF, UK ÁàÁTel/fax: 020©8674 3676; email: ZINGMAST @ LSBU.AC.UK ÁàÁLast revised on 10 Oct 2002. ÁàÁLast updated on ØD1 3 4. DØ ÁÁI 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. ÁÁSome 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. ÁÁThe recreational questions are discussed more fully in my à ÃSourcesÄ Ä or the à ÃQueriesÄ Ä thereto. I am currently working on the seventh preliminary edition of this. ÁÁÁÁÁÁà ÃGENERALÄ Ä ÁÁÃÃMORRIS GAMES.ÄÄ The game known as Nine Men's Morris, Mill, Moule, MÀGÀhle, 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 à ÃHistory of ChessÄ Ä mentions the Arabic game of qirq as (a kind of) morris and gives some 8C and 10C references. This becomes alquerque in Spain. ÁÁÃÃRIVER CROSSING PROBLEMS.ÄÄ 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 ancient. ÁÁÃÃEULER CIRCUITS.ÄÄ 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? ÁÁÃÃTHE EXPLORER'S OR JEEP PROBLEM.ÄÄ Alcuin is also the earliest known example of trying to cross a desert ÀMÀ 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. ÁÁÃÃKNIGHT'S TOURS.ÄÄ The earliest known example is in an Sanskrit MS poem à ÃKÀ]ÀvyÀ]ÀlankÀ]ÀraÄ Ä, by Rudrata, c900, described in Murray's à ÃA History of ChessÄ Ä. However, an Arabic MS of 1141 gives tours which may derive from lost works of al©ÀÀAdlÀ…À (c850) or as©SÀÁÀlÀ…À (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. ÁÁÃÃTANGRAMS.ÄÄ These became a fad in China c1810 and in Europe c1817 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 China?? Jerry Slocum has managed to track this back considerably in China and a book that I have just acquired indicates it was introduced to Europe in 1817. ÁÁIn 1996 I noticed the ceiling of the room to the south of the Salon of the Ambassadors in the Alcazar of Seville. This 15C? ceiling was built by workmen influenced by the Moorish tradition and has 112 square wooden panels in a wide variety of rectilineal patterns. One panel has some diagonal lines and looks like it could be used as a 10 piece tangram©like puzzle. Since geometric patterns and panelling are common features of Arabic art, I wonder if there are any instances of such patterns being used for a tangram©like puzzle? ÁÁÃÃPYTHAGOREAN PROBLEMS.ÄÄ 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. ÁÁÃÃTHE JOSEPHUS OR SURVIVOR PROBLEM.ÄÄ This has only a vague connection with Josephus, but may have its origins in the Roman custom of decimation. An article claims an Irish origin of the problem, c800, and gives early medieval forms called the Ludus Sanct Petri. It was certainly very popular in medieval and Renaissance Europe. Murray's à ÃHistory of ChessÄ Ä 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? ÁÁÃÃEGYPTIAN FRACTIONS.ÄÄ Is there any attempt to show that every fraction can be written as a sum of distinct unit fractions before Fibonacci? ÁÁÃÃTHE APPLE ORCHARD WITH THREE GUARDS PROBLEMÄÄ 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 à ÃChiu Chang Suan ChingÄ Ä (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? ÁÁÃÃHALF + THIRD + NINTH, ETC.ÄÄ 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 à ÃRhind PapyrusÄ Ä and in the à ÃBakhshali MSÄ Ä 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. ÁÁÃÃCISTERN PROBLEMS.ÄÄ These go back to the à ÃChiu Chang Suan ChingÄ Ä and Hero(n) of Alexandria. When do fanciful versions ÀMÀ e.g. ship with two sails or three animals eating a sheep ÀMÀ 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 à ÃChiu Chang Suan ChingÄ Ä also occur in Old Babylonian. ÁÁÃÃEACH DOUBLES THE OTHER'S MONEY TO MAKE ALL EQUAL.ÄÄ A version appears in Diophantos, then the problem occurs in Mahavira and Fibonacci. Are there other early examples? ÁÁÃÃSHARING COSTS.ÄÄ 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 ÀMÀ how much should he be paid? This occurs in al©Qazwini (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? ÁÁÃÃCASTING OUT NINES.ÄÄ 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©KhwÀ]ÀrizmÀ…À (c820) and al©UqlÀ…ÀdisÀ…À (952/953) as well as in Aryabhata II's à ÃMahÀ]À©siddhÀ]ÀntaÄ Ä ÁÁÃÃTHE CHESSBOARD PROBLEM.ÄÄ (which leads to 1 + 2 + 4 + 8 + ...) is often attributed to India, but my earliest sources are Arabic: al©YaÀÀqÀÁÀbÀ…À (c875) (described in Murray's à ÃHistory of ChessÄ Ä), al©MasÀÀÀÁÀdÀ…À's à ÃMeadows of GoldÄ Ä (943), which doesn't relate the series to the chessboard) and al©BÀ…ÀrÀÁÀnÀ…À's à ÃChronology of Ancient NationsÄ Ä (1000). Murray cites a 9 or 10 C treatise on the problem by al©MissisÀ…À ÀMÀ does this exist? Is there any Indian or Persian material of relevance? ÁÁThe 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 Tabari (c1075). ÁÁÃÃCHINESE RINGS.ÄÄ These are claimed to occur in Sung China (c11C) ÀMÀ 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. ÁÁÃÃMAGIC SQUARES.ÄÄ The Indian and Arabic history of Magic Squares is quite confused. A. N. Singh (à ÃProc. ICMÄ Ä, 1936, pp. 275©276) refers to a c1C order 4 square by NÀ]ÀgÀ]Àrjuna, but I know nothing more about this. There are many Indian and Arabic references that I have not been able to find ÀMÀ indeed some apparently are not extant ÀMÀ but it would take too long to list them here. ÁÁÃÃTHE 100 FOWLS PROBLEM.ÄÄ 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 à ÃBakhshali MSÄ Ä which may have been early enough to show transmission across north India?? ÁÁÃÃSELLING DIFFERENT AMOUNTS AT THE SAME PRICES TO YIELD THE SAME AMOUNTÄÄ 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? ÁÁÃÃCONJUNCTION OF PLANETSÄÄ. This is a variant of the Chinese Remainder Theorem which appears in China and India. Are there any Arabic forms? ÁÁÃÃTHE BLIND ABBESS AND HER NUNS.ÄÄ 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? ÁÁÃÃDILUTION PROBLEMS.ÄÄ A one stage version is in the Rhind Papyrus and a four stage version is in the à ÃBakhshali MSÄ Ä. My next example is Tartaglia. ÁÁÃÃTHE APPLESELLER'S PROBLEMÄÄ, involving combining amounts and prices incorrectly, appears in Alcuin (9C) and Ibn Ezra (c1150), then Fibonacci, etc. Are there Arabic or Indian versions? ÁÁÃÃTHE LAZY WORKERÄÄ, who gains for each day he works and forfeits for days he doesn't work, appears in al©Karkhi (c1010), Tabari (c1075) and Fibonacci, etc. Are there other Arabic or perhaps Greek or Indian sources? ÁÁÃÃLIAR PARADOX.ÄÄ Do the Liar or similar paradoxes occur in Arabic or Indian works? ÁÁÃÃTHREE MEN WITH SPOTS ON FOREHEADS.ÄÄ 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 invention. ÁÁÃÃSTRANGE FAMILIESÄÄ ÀMÀ e.g. men marrying each other's sister or daughter ÀMÀ occur in Alcuin, Abbott Albert and a c1430 Hebrew text. Are there earlier versions? There is a riddle about a strange family attributed to the Queen of Sheba, but I can't find any version of her riddles. ÁÁÃÃSNAIL CLIMBING OUT OF A WELL.ÄÄ This seems to derive from the habit of expressing velocity by unit fractions ÀMÀ 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? ÁÁÃÃSOLOMON'S SEAL.ÄÄ 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? ÁÁÃÃWIRE PUZZLESÄÄ 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. ÁÁÃÃPUZZLE RINGSÄÄ are believed to be Middle Eastern in origin. Are there any sources? There is a 17C example in the British Museum. ÁÁÁÁÁÁà ÃINDIAÄ Ä ÁÁI have had difficulty in communicating with the National Institute of Sciences, New Delhi. They haven't responded to my letters ÀMÀ do they exist? I wanted to get the bibliography by Sen from them. This may have been obsoleted by Rahman's à ÃBibliographyÄ Ä, published by the Indian National Science Academy in New Delhi. I have recently received a copy of Rahman. ÁÁI'd like to get copies of the following: ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐColebrooke, H, T. à ÃAlgebra, with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhascara.Ä Ä (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 annotated ed.) Kaye, G. R. à ÃIndian Mathematics.Ä Ä Thacker & Spink, Calcutta, 1925. Mahavira. à ÃThe Ganita©Sara©Sangraha of Mahaviracarya.Ä Ä Translated by M. Rangacarya. Government Printing Press, Madras, 1912. Aryabhata II, à ÃMaha©SiddhantaÄ Ä. Is there an English version? Does it have anything of interest to me? Narayana, à ÃGanita©KaumudiÄ Ä. Same questions. Satya Prakash. He translated two of the à ÃSulbasutrasÄ Ä, published at New Delhi in 1968. Are these available? Bag cites a à ÃManava SrautasutraÄ Ä belonging to the à ÃMaitrayani SamhitaÄ Ä, edited by Van Gelder. I haven't been able to find this ÀMÀ is it of any interest to me? Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁI 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. ÁÁI am also interested in some works by J. N. Kapur ÀMÀ e.g. à ÃThe Fascinating World of MathematicsÄ Ä; S. Chand, 1970 and à ÃThoughts on Mathematical EducationÄ Ä; Atma Ram & Sons, Delhi, 1973. ÁÁÃÃCROSSED LADDERS.ÄÄ Bhaskara gives an easy problem with crossed ladders in a street. I have recently discovered some examples in Mahavira. Are there any other similar problems in Hindu or Arabic literature? ÁÁÃÃMAGIC SQUARES.ÄÄ ÁÁà ÃBharava TantraÄ Ä & à ÃSiva Tandava TantraÄ Ä. Dvivedi, Intro. to Narayana's à ÃGanita KaumudiÄ Ä, cites these as early sources for Magic Squares, but I cannot find out anything about them. ÁÁVarahamihira, à ÃPancasiddhantikaÄ Ä ÀMÀ Bag implies this says something about Magic Squares, but I looked through the English edition by Neugebauer and Pingree and couldn't find anything?? ÁÁÃÃMEMORY WHEELS = CHAIN CODES.ÄÄ The 'Memory Wheel' ya mÀÀ tÀÀ rÀÀ ja bhÀÀ na sa la gÀÀm 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? ÁÁÁÁÁÁà ÃARABICÄ Ä ÁÁÃÃLATIN SQUARES.ÄÄ Ahrens discusses and cites early examples from medieval Islamic times, c1200, when they were used on amulets. Are there other sources, discussions, examples, etc.? ÁÁÃÃSOLITAIRE OR FROGS AND TOADS.ÄÄ Thomas Hyde; à ÃHistoria Nerdiludii, hoc est dicere, Trunculorum; ....Ä Ä (= Vol. 2 of à ÃDe Ludis OrientalibusÄ Ä); from the Sheldonian Theatre (i.e. OUP), Oxford, 1694, p. 233; has: De Ludo dicto Ufuba wa Hulana. This has a 5 x 5 board with each side having 12 men, but the description is extremely brief. It seems to have two players, but this may simply refer to the two types of piece. I'm not clear whether it's played like solitaire (with the jumped pieces being removed) or like frogs & toads. I would be grateful if someone could read the Latin carefully. The name of the puzzle is clearly Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the pages I have) ÀMÀ I would be grateful to anyone who can track down and translate Arabic sources. ÁÁÃÃTESSELLATIONS.ÄÄ In view of the Arabic emphasis on geometric design, was there any work on the Archimedean or other Tessellations of the Plane? ÁÁÃÃTHE PROBLEM OF THE PANDECTSÄÄ 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©Qazwini (1262). Are there any early Arabic sources? ÙÙ ÁÁÁÁÁÁà ÃPERSIANÄ Ä ÁÁTropfke cites: Mohammed ibn AyyÀÁÀb TabarÀ…À; à ÃMiftÀ]Àh al©muÀÀÀ]ÀmalÀ]ÀtÄ Ä; c1075; ed. by Mohammed Amin RiyÀ]Àhi, Teheran, 1970; many times and some of these are among the earliest known examples. A Persian girl is working on translating part of this into English, but is there any western version of it? ÁÁÃÃDEAD DOGS AND TRICK PONIES.ÄÄ 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 Renaissance Europe (c1600), medieval Persia (Rza Abbasi, (1587©1628)), Edo period Japan (17©19C) and China (17©20C). There are examples from 18C India. I would like to know of early examples. I have recently discovered 14 & 15C English versions with three rabbits having three ears in total and 14 & 16C German versions and an example in a Buddhist cave at Dunhaung from c600 !! However, I am told that the pattern only occurs in Chinese art at this time and place © might there be Buddhist examples in India? Another source has found more examples, including more in China and Tibet. The 'two heads, four horses' pattern occurs in the first Peterborough Psalter of the end of the 13C (Fig. 99 of BaltruÀ±Àaitis, who also gives another version of the Abbasi drawing), but an expert on Iranian art says the Iranian tradition goes back far before Abbasi and could have been the inspiration for the Peterborough example. Further, a sculpted relief version of 1290©1300 occurs at Rouen (Fig. 100 of BaltruÀ±Àaitis). ÁÁJurgis BaltruÀ±Àaitis. Le Moyen Age Fantastique, AntiquitÀ)Às et Exotismes Dans l'Art Gothique. A. Colin, Paris, 1955, 299 pp. Reprinted by Flammarion, Paris, 1981, 281 pp. Supposedly an English edition was published in 1998, but we have not yet found it. Pp. 132-139 of the 1981 edition have many examples of three and four rabbits, four boys, etc.