ÿWPCL ûÿ2BJ|xÕ)Ð x ÐÐÐüð ä بÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÐЊ‚ÐÈÐÁ`ÁRussian Queries ÀMÀ p.)ÕÐ °x ÐÐа¤˜Œ € tDÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐ ÁàÁà ÃQUERIES ON RUSSIAN SOURCES 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 14 Jan 2002. ÁàÁLast updated on ØD1 3 4. DØ ÁÁI am working on a history of recreational mathematics. I have found a few topics which have Russian connections which I am gathering here for convenience in correspondence. Separate letters deal with Oriental questions and with Middle Eastern questions, i.e. Egyptian, Babylonian, Indian, Arabic, Persian and Turkish. ÁÁThere is a general question about some problems which are known from China and Europe, but for which 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 seems inadequate to explain the early history. ÁÁFor example, the cistern problem appears almost simultaneously in China and Alexandria. Heron's work gives two problems, both incorrectly solved, while the à ÃChiu Chang Suan Ching (Jiu Zhang Suan Shu)Ä Ä gives a clear example with 5 pipes and several related problems. However, an equivalent type of problem appears in Old Babylonian and is probably the ancestor of both the Chinese and western forms. ÁÁAs 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. ÁÁTait's Counter Puzzle and the Chinese Rings are further examples where there is no sign of the usual transmission through India and the Arabs. Anatoli Kalinin says that the Chinese Rings are a old folk puzzle called À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ À [Meleda], especially popular among the Kalmyks near the Caspian Sea, where it is called À ÀÀ ÀÀ# ÀÀ ÀÀ À©À3 ÀÀ ÀÀ ÀÀ À [Naran©shina] (stirrup ring toy). The name À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ À is derived from a verb which is no longer in Russian. ÁÁTangrams and the Josephus Problem are examples where there is no sign of the usual transmission, and the transmission may well have gone the other way. Kalinin informs me that Tangrams were unknown in Russia before the Richter puzzles of the late 19C. Jerry Slocum has recently found Chinese versions of tangrams going back to the 11C and a recently acquired book indicates they were introduced to Europe in 1817. However, the Chinese version may derive from a classical Greek version. ÁÁI wonder if there was some transmission over the Silk Road or other central Asian trade routes which could have carried some information directly between the Orient and Europe, bypassing the Indians and Arabs ÀMÀ see Dead Dogs and Trick Ponies on p. 4 for an example. If so, there may be some evidence for this in the folk cultures of the central Asians and Russians. I can find nothing about this and would be delighted to hear from anyone who does know about this. Are there any collections of early folk mathematics in the USSR? 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. ÁÁÁÁÃÃPERELMAN.ÄÄ ÁÁTatiana Matveeva has kindly searched the Russian State Library which has provided original publishers and dates. David Calinsky has added useful details. ÁÁI am particularly interested in the works of Yakov (Jacob) Isidorovich Perelman (À@ À. À À. À ÀepeÀ ÀÀ ÀaÀ À). All of the copies of his works that I have seen are quite recent editions (1950s onward), but he began publishing in 1913 and died in 1942. His first mathematical puzzle book was Vesjolye Zadachi [Funny Questions] of 1916. Prior to that he published in a weekly in St. Petersbourg, ÃÃPriroda i LjudiÄÄ [ÃÃNature and PeopleÄÄ]. A number of the problems in his books are quite interesting as they seem to originate in this century. If Perelman published them in his early books, he might be the originator or first publisher of them. As will be seen below, several of these problems lead to questions of priority between Perelman and H. E. Dudeney in the period 1915©1930. Calinsky notes that Perelman knew English but it is unlikely that Dudeney knew Russian! Consequently, I am very keen to obtain information about Perelman's books (especially the original dates) or even to obtain copies of them. I can struggle through Russian, but early translations into English, German, French or Italian (or even Spanish) might be more useful. ÁÁAn especially interesting point arises in à ÃFun with Maths and PhysicsÄ Ä [FMP] (À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ' ÀÀ ÀÀ ÀÀ; ÀÀ ÀÀ9 ÀÀ À À ÀÀ ÀÀ ÀÀ ÀÀ1 ÀÀ À À À À ÀÀ# ÀÀ9 ÀÀ' ÀÀ9 À), MIR, Moscow, 1984. (There was a À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ' ÀÀ ÀÀ ÀÀ; ÀÀ ÀÀ9 ÀÀ À À ÀÀ ÀÀ ÀÀ ÀÀ1 ÀÀ À, with 4th ed. in 1935 ÀMÀ no earlier version in Russian State Library. This title originally published by À ÀÀ ÀÀ' ÀÀ ÀÀ ÀÀ= À, 1959.) This is a collection compiled from Perelman's works by I. I. Prusakov. If this is mainly based on the earlier book of the similar title, this would date the material to c1935? On p. 194, we find the following. ÂÂÁÁMany experts in Russian literature don't suspect that the poet V. G. Benediktov (1807©1873) was also the author of the first collection of mathematical brain©twisters in the language. The collection wasn't printed and remained in a manuscript form to be found only in 1924. I had the opportunity to get acquainted with the manuscript and even established, based on one of the problems, the year it was compiled, namely 1869 (the manuscript wasn't dated). Perelman then gives one of the problems, a version of 'selling different amounts at the same prices but making the same'. This problem derives from India, c850, and the exact same numbers already occur in 14C Europe. It is not clear how useful this collection would be ÀMÀ it might contain some problems which Benediktov attributes to earlier authors. I would like to know if this collection has ever been printed. If not, where might the manuscript be? Prof. Boltyanski suggested the Library in St. Petersburg. Would it be possible to get a copy of the manuscript? Would any Russian speaker be interested in producing a rough translation or summary of it in English? Kalinin has made inquiries in Moscow and St. Petersburg but has not located this manuscript. ÁÁI have three versions of à ÃFigures for FunÄ Ä [FFF] (À ÀÀ ÀÀ ÀaÀA À À ÀaÀ' ÀeÀ ÀaÀ' ÀÀ ÀÀ Àa), (Originally published as Living Mathematics by À ÀÀ ÀÀ% À. À' ÀÀ ÀÀ- ÀÀ À.©À' ÀÀ ÀÀ ÀÀ# À. À ÀÀ ÀÀ ÀÀ ÀÀ' À., Leningrad©Moscow, 1934, 2nd ed, 1936) all translated by G. Ivanov©Mumjiev. [Calinsky says the first English translation was an abridged version published in the USA in c1950.] ÁÁForeign Languages Publishing House, Moscow, 1957; ÁÁ3rd ed., MIR, 1979; ÁÁas the first part of à ÃMathematics can be FunÄ Ä [MCBF], MIR, 1985, apparently based on the 2nd Russian ed. of 1970, translated 1973. (The second part is à ÃAlgebra Can be FunÄ Ä, translated from À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ' ÀÀ ÀÀ ÀÀ; ÀÀ ÀÀ ÀÀA À À ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ# ÀÀ À (3rd ed. was published by À ÀÀ ÀÀ& ÀÀ À, LeningradªMoscow, 1937 ÀMÀ The Russian State Library has no earlier edition), edited and supplemented by V. Boltyansky, À ÀÀ ÀÀ) ÀÀ ÀÀ À, Moscow, 1976, translated by G. Yankovsky, 1976. References to MBCF will be to material not in FFF, so will be dated 1937.) ÁÁ(Schaaf's à ÃBibliography of Recreational MathematicsÄ Ä, vol. 1, p. 9, refers to à ÃRecreational ArithmeticÄ Ä, 6th ed., Leningrad, 1935. Calinsky says it is different than à ÃFigures for FunÄ Ä and was first published in 1926, with 8th ed, 1938. There is no English version of this, but there is a Spanish version. ÁÁThe 2nd and 3rd editions seem to be almost identical, with minor changes to the English wording. The version in à ÃMathematics can be FunÄ Ä is on larger pages, but with more diagrams, so the pagination is almost the same as in the 3rd ed. ÀMÀ the 3rd ed. ends on p. 183, while the other ends on p. 186. The 1957 edition has 120 problems, while the 2nd and 3rd editions omit 3 problems and add 6 others giving 123 problems. In re©examining this, I was surprised to find that I had forgotten that the references to Benediktov and the above problem appear as Problem 120, pp. 141©142 & 150©153. ÂÂÁÁ120. The Benediktov Problem. ©© A great many lovers of Russian literature probably do not even suspect that the poet Benediktov (1807©1873) collected and compiled a whole volume of mathematical conundrums. Had it been published, it would have been the first Russian book of this type. But it never was and the manuscript was only found in 1924. I had the good fortune to study the manuscript and even established ÀMÀ by solving one of the brainteasers contained therein ÀMÀ that the collection was completed in 1869 (the manuscript itself was not dated). Perelman then gives the same problem as mentioned before, but in much more detail. My memory was that Benediktov was a collector rather than an author and probably I was recalling the phrasing here. ÁÁÁÁÃÃQUERIES ON PERELMAN PROBLEMS.ÄÄ ÁÁI will cite the above mentioned works as FMP with page numbers, FF 1957 and FF 1979 with problem numbers and MBCF, 1937, with problem numbers. I would be particularly grateful for the original dates of the FMP and MCBF items. ÁÁÃÃPIGEONHOLE RECREATIONS.ÄÄ FMP, p. 277: Socks and gloves. This is my earliest example of a pigeonhole problem with handed objects. I have another example from 1943 with shoes and socks. ÁÁÃÃSPIDER & FLY PROBLEMS.ÄÄ Dudeney and Loyd give several versions of the problem in a rectangular room. In 1926 Dudeney gives a version on a cylindrical glass with the source and the target on opposite sides. FF 1957, prob. 68 = FF 1979, prob. 73 is a cylindrical version with different numbers than Dudeney. I don't have any other early versions of the cylindrical problem. ÁÁÃÃSILHOUETTE AND VIEWING PUZZLES.ÄÄ The best known, though not the earliest, example is to find an object which will plug holes in the shape of a circle, a square and a triangle. FMP, p. 340 = FF 1957, probs. 70 & 71 = FF 1979, probs. 74 & 75 are problems with circle, square, cross and with triangle, square, tee. I have only seen one slightly earlier version of the first of these forms and no other versions of either form. ÁÁÃÃNETS OF POLYHEDRA.ÄÄ In 1926, Dudeney gives the problem of finding all ways of unfolding a cube into a flat network and he correctly finds 11 ways. In FMP, p. 179: To develop a cube, Perelman asks the same question, but his answer says there are 10 solutions, but two can be turned upside down, increasing the total to 12. The fact that Perelman has the wrong answer has two interpretations. First, he had created the problem and failed to get the right answer. Second, he had vague information about Dudeney's problem and answer and was misled by it. Both cases are possible. ÁÁÃÃWHAT COLOUR WAS THE BEAR?ÄÄ These are problems involving travel near a pole ÀMÀ e.g. man goes 10 miles south, then 10 miles east, then 10 miles north to return to his starting point. Simpler versions ÀMÀ e.g. man starts at the North Pole, goes 40 miles south and 30 miles east, how far is he from his starting point? ÀMÀ occur in 1907, 1925, 1930s, but those with triangular circuits occur in the 1940s. FF 1957, prob. 6: A dirigible's flight = FF 1979, prob. 7: A helicopter's flight concerns a square circuit and notes that going 500 km N, E, S, W doesn't get you back to where you started. This may be the earliest version of the problem with a circuit, though he doesn't ask the more interesting question of where you could be if the square circuit does return to the origin. My next example with a square circuit does ask this in 1958/59 but fails to get the complete solution, which I have recently found. ÁÁÃÃCUTTING UP IN FEWEST CUTS.ÄÄ FF 1979, prob. 122: Sectioning a cube and prob. 123: More sectioning, ask for the minimum number of cuts to divide a 3 x 3 x 3 cube into unit cubes and a 8 x 8 chessboard into unit squares. Surprisingly, I don't recall seeing any other versions of these problems before relatively recent times. These are not in the 1957 ed. ÀMÀ they are two of the problems added in the 2nd ed. ÁÁÃÃCHESSBOARD PROBLEM.ÄÄ FF 1957, prob. 52 = FF 1979, prob. 55. This describes a Roman version where the general Terentius can take 1 coin the first day, 2 the second day, 4 the third day, ..., until he can't carry any more, which occurs on the 18th day. A footnote says this is a translation "from a Latin manuscript in the keeping of a private library in England." Does any Russian version give more precise details ?? ÁÁÃÃHUNDRED FOWLS.ÄÄ FF 1957, prob. 37 = FF 1979, prob. 40: Hundred rubles for five. Using 50, 20 & 5 kopeck coins, it is impossible to make 5, 3 or 2 rubles in 20 coins. Perelman describes this as a magician's come©on. I have not seen any other version of the problem which uses so many impossible forms, nor any which use it in this way. ÁÁÃÃWATER IN WINE VERSUS WINE IN WATER.ÄÄ This has been a popular problem since the late 19C. FMP has it on p. 215. ÁÁÃÃSKELETON ARITHMETIC.ÄÄ Again, these are popular problems, first appearing c1900. FMP, p. 256, has the skeleton division of 11268996 by 124 yielding 90879 with only the 7 of the quotient given. This has 11 solutions. ÁÁÃÃARITHMETIC PROGRESSIONS.ÄÄ MCBF, 1937 part, prob. 195: A team of diggers has a team which can dig a ditch in 24 hours, but just one digger begins and then the others join in at equal intervals, with the work finished in one interval after the last man joined. The first man works 11 times as long as the last man. How long did the last man work? Perelman finds this noteworthy (and I agree) because the number of men in the team cannot be determined! ÁÁÃÃFLOATING BODY PROBLEMS.ÄÄ Surprisingly, I haven't noticed any of these prior to a possible Dudeney in the ÃÃDaily MailÄÄ in 1905. I have some forms from 1925 and 1931. Hence FMP, pp. 114 & 199 are among the earliest popular versions I know of. P. 114 asks whether a bucket full of water is heavier or lighter than a similar bucket with a floating block of wood in it. This is conceptually the same as asking whether a glass full of water with floating ice will overflow when the ice melts, which is the 1905 problem. FMP, p. 199 has a balanced balance with iron versus stone ÀMÀ what happens when it is submerged? ÁÁÁÁÃÃOTHER QUERIES.ÄÄ ÁÁÃÃNIM GAMES.ÄÄ 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 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. ÁÁWythoff'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. 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. à ÃWinning WaysÄ Ä says it is called Chinese Nim or Tsyan©shizi, but Richard Guy says he recalls this was based on Domoryad. Is there any evidence for any games of this kind in China or eastern Russia, etc.? ÁÁÃÃFOX AND GEESE, ETC.ÄÄ These are board games with asymmetric forces. Fox and Geese is supposed to be medieval, even 1st millennium, but Murray's à ÃHistory of ChessÄ Ä cites a North Asian version of Bouge©Skodra (Boar's Chess). Are there other early forms in this area? ÁÁÃÃTANGRAMS.ÄÄ 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. A slightly different form of the game appears in Japan by 1742 and there is an Utamaro woodcut of 1780 showing some form of the game (Jerry Slocum has finally found this © it is from 1807(?)). Needham says there are some early Chinese books, and van der Waals' historical chapter in Elffers' book à ÃTangramÄ Ä cites a number, but many of his citations are dubious. Slocum has now tracked the puzzle back to an early Chinese form in the 11C! A recently acquired book and Slocum's work indicate it was introduced to Europe in 1817. ÁÁI would 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 Classical world. Could it have travelled to China over the Silk Road? I found a plastic version of the Loculus on sale in Xian, made in Liaoning province. 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 have recently discovered a 15C version of riders with a rotating piece. I would like to know of early examples. Perhaps some exist in central Asia. ÙÙÁÁA related puzzle is the Three Rabbits, where each has two ears, but there are three ears all together. I have recently discovered several medieval European versions of this and a c600 Chinese version in a Buddhist cave at Dunhuang (c600). There is a 12C Oriental chalice with the pattern on its base in The Hermitage, Leningrad. Again are there examples in Central Asia? ÁÁÃÃTHE JOSEPHUS OR SURVIVOR PROBLEM.ÄÄ This is the problem of counting out every k©th person from a circle. It was a common medieval European problem from the 10C 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 to lighten the ship and one of them says that everyone should get in a circle and be counted off by 9s. I have 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. Cardan suggested that this might be the way in which Josephus survived, though there, the counting goes to the last man. ÁÁIt 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 on the recreational problems in à ÃJingÀ¥ÀkiÄ Ä, 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 à ÃTsurezuregusaÄ Ä by KenkÀ¥À Yoshida, written c1331, where it is used as a metaphor for death in a way which shows the author knew the problem of counting to the last man and expected his readers to know it. (There is a nice modern translation of the à ÃTsurezuregusaÄ Ä as à ÃEssays in IdlenessÄ Ä by Donald Keene.) 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 à ÃJingÀ¥ÀkiÄ Ä has the stepmother doing this by accident and the bit about the 15th stepchild first occurs in Miyake KenryÀÁÀ, 1795. Michael Dean has found illustrations of the problem on inro boxes going back to the 17C and has a nice article in ÃÃInternational Netsuke Society JournalÄÄ 17:2 (Summer 1997) 41©53. à ÃWakoku ChieªkurabeÄ Ä 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. ÁÁMurray'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. ÁÁIs there any material from Russia or central Asia on this problem?