Computing, Information Systems and Mathematics 87 Rodenhurst Road South Bank University London, SW4 8AF, England London, SE1 0AA, England Tel/fax: 0181-674 3676 Tel: 0171-815 7411 Fax: 0171-815 7499 E-mail: ZINGMAST@VAX.SBU.AC.UK## QUERIES ON RUSSIAN 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 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. There 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. For example, the cistern problem appears almost simultaneously in China and Alexandria. Heron'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. Anatoli Kalinin says that the Chinese Rings are a old folk puzzle called [Meleda], especially popular among the Kalmyks near the Caspian Sea, where it is called - [Naran-shina] (stirrup ring toy). The name is derived from a verb which is no longer in Russian. Tangrams 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. I 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. 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? The recreational questions are discussed more fully in my Sources or the Queries thereto. I am currently working on the seventh preliminary edition of this. PERELMAN. I am particularly interested in the works of Yakov (Jacob) Isidorovich Perelman (. . epea). 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. 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. 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. An especially interesting point arises in Fun with Maths and Physics [FMP], MIR, Moscow, 1984. This is a collection compiled from Perelman's works by I. I. Prusakov. On p. 194, we find the following. Many 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 - 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 Lenin(?) Library in 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. I have three versions of Figures for Fun [FF] (a aeaa), all translated by G. Ivanov-Mumjiev: Foreign Languages Publishing House, Moscow, 1957; 3rd ed., MIR, 1979; as the first part of Mathematics can be Fun, MIR, 1985, apparently based on the 2nd Russian ed. of 1970, translated 1973. (Schaaf's Bibliography of Recreational Mathematics, vol. 1, p. 9, refers to Recreational Arithmetic, 6th ed., Leningrad, 1935. Is this the same book as translated as Figures for Fun??) The 2nd and 3rd editions seem to be almost identical, with minor changes to the English wording. The version in Mathematics can be Fun is on larger pages, but with more diagrams, so the pagination is almost the same as in the 3rd ed. - 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 reexamining this, I was surprised to find that I had forgotten that the reference to Benediktov and the above problem appear as Problem 120, pp. 141-142 & 150- 153. 120. 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 - by solving one of the brainteasers contained therein - 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. QUERIES ON PERELMAN PROBLEMS. I will cite the above mentioned works as FMP with page numbers and FF 1957 and FF 1979 with problem numbers. I would be particularly grateful for the original dates of these items. PIGEONHOLE RECREATIONS. 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. SPIDER & FLY PROBLEMS. 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. SILHOUETTE AND VIEWING PUZZLES. 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 give problems with circle, square, cross and with triangle, square, tee. I haven't seen earlier versions of these forms. NETS OF POLYHEDRA. 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. WHAT COLOUR WAS THE BEAR? These are problems involving travel near a pole - e.g. man goes 10 miles south, then 10 miles east, then 10 miles north to return to his starting point. Simpler versions - e.g. man starts at the North Pole, goes 40 miles south and 30 miles east, how far is he from his starting point? - 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. CUTTING UP IN FEWEST CUTS. 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. - they are two of the problems added in the 2nd ed. HUNDRED FOWLS. 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 don't recall any other versions of the problem which depend on there being no solutions. WATER IN WINE VERSUS WINE IN WATER. This has been a popular problem since the late 19C. FMP has it on p. 215. SKELETON ARITHMETIC. 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. PAN_DIGITAL PRODUCTS. FF 1957, prob. 45 = FF 1979, prob. 48: Tricky multiplication finds all the pan-digital (without 0) products. There are 2 of the form A x BCDE = FGHI and 7 of the form AB x CDE = FGHI. Loyd gives a 10-digital problem involving the smallest result, but my next version of the problem is 1934 when all 9 solutions are given. Hence Perelman may be the earliest source for this, though I suspect some 19C versions may turn up or it may appear in Dudeney. FLOATING BODY PROBLEMS. Surprisingly, I haven't noticed any of these prior to a possible Dudeney in the Daily Mail in 1905. 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 - what happens when it is submerged? OTHER QUERIES. 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 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. 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 or eastern Russia, etc.? FOX AND GEESE, ETC. These are board games with asymmetric forces. Fox and Geese is supposed to be medieval, even 1st millenium, but Murray's History of Chess cites a North Asian version of Bouge-Skodra (Boar's Chess). Are there other early forms in this area? 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. 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 (not yet seen by me). Needham says there are some early Chinese books, and van der Waals' historical chapter in Elffers' book Tangram cites a number. I 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 Greek 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. 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. Perhaps some exist in southern USSR. 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. Is there any Chinese or central Asian material on this? 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. 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. David Singmaster

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