Queries on Sources in Recreational Mathematics

QUERIES - p.1

QUERIES ON "SOURCES IN RECREATIONAL MATHEMATICS"

DAVID SINGMASTER

The version of 4 Aug 1996 is on the following. http://anduin.eldar.org/~problemi/singmast/queries.html http://www.g4g4.com/david.htm Word version made on 21 Mar 2004, but I haven't checked on details, e.g. diacritical marks.

I would be grateful for any information you can provide on the problems sketched below. My assertions should be treated as implied questions - a statement like "The earliest reference is ...." should be considered as "The earliest reference I know is .... Do you know any earlier ones or can you confirm this is the earliest?" Many of these topics have originated in publications which are difficult to search for - e.g. puzzle columns in newspapers and magazines, medieval and oriental books and manuscripts, unpublished letters, etc. I would be especially grateful for detailed references to such items, and photocopies. In particular, I have found a number of references which are not sufficiently detailed to locate the item of interest. Any help with such items would be appreciated. Any information on translations of sources would be useful. I would also be grateful for references to biographical and bibliographical articles, especially on particular authors or problems. Interviews, obituaries and photographs are of especial interest. The sections below cover a number of topics. A very few topics are well enough known that I have no questions (yet). Suggestions for other topics would be useful. The first version of this letter generated so much response that I had to make this into a computer file. Many thanks to all of you who have helped resolve questions. This version is ordered in the ad hoc arrangement I have used in my Sources in Recreational Mathematics, presently about 600pp long. The 'Abbreviations and Common References' section of Sources may be attached if the entire document is not.

General 1. Biographical Items 2. General Puzzle Collections and Surveys 3. General Historical Articles 4. Mathematical Games 5. Combinatorial Recreations 6. Geometric Recreations 7. Arithmetical & Number-theoretic Recreations 8. Probability Recreations 9. Logical Recreations 10. Physical Recreations 11. Topological Recreations QUERIES - p.2

GENERAL

I have found a number of topics which have particular geographic connections and I have prepared special letters covering: Oriental queries; Middle Eastern queries (i.e. Egyptian, Babylonian, Indian, Arabic, Persian and Turkish) and Russian queries. I have some recent books and compilations from the works of Y. I. Perelman. He began writing in 1913 and died in 1943. However, the books are from the late 70s and the 80s. Some of the problems would have been quite new if they appeared about 1920. Hence I am very interested in obtaining information about earlier versions of his works - in any language. I have a vague reference to an Arabic book: Ketabol Algaz. Can anyone identify this? I have a recent Spanish edition of a Russian work by Ignatiev, 1908. I'd be interested in seeing an early edition in any language. I would like to see copies of Our Puzzle Magazine by Sam Loyd. In the mid-19C, there was a profusion of books, beginning with The Magician's Own Book (1857), many of which are pseudonymous and I have been unable to determine the authors. One version of The Secret Out (1859) says it is based on Le Magicien des Salons (this may be Le Magicien de Société, Delarue, Paris, c1860? - I have seen it advertised in another of their books from 1861), and I have a reference to Le Manuel des Sorciers (various Paris editions from 178?-1825), so perhaps this era was inspired by some earlier French books?? One bibliographer doubts whether Le Magicien des Salons exists. I have recently discovered some earlier appearances of the same material in The Family Friend, a periodical which ran in six series from 1849 to 1921 and which I have not yet tracked down further. However, vol. 3 of 1850 and the volume for Jul-Dec 1859 both contain a number of the problems which appear repeatedly and identically in the above cited books. Toole Stott cites an edition of The Illustrated Boy's Own Treasury of c1847 but the BM copy was destroyed in the war and the other two copies cited are in the US. Any help in clarifying the bibliographic confusion here would be much appreciated.

1. BIOGRAPHICAL ITEMS

I am looking for relevant pictures. I have located pictures of Ahrens, Bachet, Ball, Carroll, Dudeney, Hoffmann, Loyd, Lucas, Phillips, Schubert. Does anyone know of any published picture of Alcuin, Leurechon, Montucla or Ozanam? I have discovered a pencil sketch in a 1696 copy of Ozanam's book which claims to be him.

2. GENERAL PUZZLE COLLECTIONS AND SURVEYS

Y. I. Perelman [Fun with Maths and Physics] describes a Russian MS by the poet V. G. Benediktov (1807-1873), dated c1869, as the first puzzle collection in Russian. Has this ever been published? Does the MS still exist? Where?

3. GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL

3.A. BIBLIOGRAPHICAL MATERIAL

Did Part II: Angewandte Mathematik of: Ernst Wölffing; Mathematischer QUERIES - p.3

Bücherschatz. Systematisches Verzeichnis der wichtigsten deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts auf dem Gebiete der mathematischen Wissenschaften; AGM 16, part II, 1903(?) ever appear??

4. MATHEMATICAL GAMES

4.A.1. One Pile Game. This is the game where one can take 1 to m at a time from a pile. A version and a general discussion are in De Viribus and in Bachet (Prob. 22), then in Ozanam and Alberti. Loyd's "Blind Luck" may be a version of it?? E. Ducret, Recreations Mathematiques [1900?], calls it 'le jeu du piquet a cheval', and gives some explanation of the name. Les Amusemens (1749) calls it Piquet des Cavaliers and says it was played while riding. When does this name originate? There are versions using a die (Loyd, 1914) and with a restriction on the number of times you can use a number (Ball, MRE 5th ed., 1911). There is a 20C(?) version where the winner is the one who takes an odd number of matches in total. I think I recall this in Dudeney?? 4.A.1.a. The 31 Game. It is claimed that this was invented by Charles James Fox (1749-1806), but my earliest reference is 1857. 4.A.2. Symmetry Arguments. This refers to the idea that a player can win by playing symmetrically in some way. It appears both in Loyd's Cyclopedia and Dudeney's AM. I am inclined to believe Dudeney may have discovered it. Dudeney gave it as his 500th problem in The Weekly Dispatch (7 Jun 1903) and it occurs in a version of Kayles, ibid., (26 Jan 1902). Kraitchik (Math. des Jeux & Math. Rec.) gives the problem of the child who agrees to play chess with two masters and bets she will score a point. She plays white against one and black against the other! 4.A.3. Kayles. This occurs in both Dudeney and Loyd. Gardner (MPSL 2) asserts that Dudeney invented it, though it is Loyd's column in Tit-Bits. 4.A.4. Nim. Is there any reference prior to Bouton (1901/2)? David Parlett says 16C references to 'Les luettes' seem to suggest a game of the Nim family. Many popular accounts claim it is Chinese but this is unsubstantiated. Ahrens (Naturwissenschaftliche Wochenschrift (1902)) says Bouton admits that Nim is not the same as Fan-Tan, as he had claimed in his paper. I have seen descriptions of Fan-Tan and it is much different. I have recently seen a letter in MG asserting that Bouton invented the word 'Nim'. Domoryad, Brandreth and Winning Ways assert that Wythoff's Nim is an ancient oriental game. 4.A.5. General Theory. The origin of the result that a game without ties has a strategy for one player or the other seems to be due to Zermelo (1912). Ahrens (A&N, 1918) mentions the idea fairly clearly. Steinhaus attributes the proof using the fact that a first player strategy is the negation of a second player strategy to Mycielski, but without a reference (AMM 72 (1965) 400; see also Kac, AMM 81 (1974) 577). Babbage's MSS of 1820-1860 show some general analysis, including the tree of a game. 4.B.1. Tic-Tac-Toe = Noughts and Crosses. Are there any ancient references? The earliest mathematical approach, indeed my earliest reference at all, is in Babbage's unpublished MSS The Philosophy of Analysis, c1820, though Wordsworth may refer to it in his The Prelude (1805). The first complete analysis seems to be 1892 in Boys' Own Magazine. A. C. White gives an analysis in British Chess Magazine (Jul 1919). 4.B.1.a. In Higher Dimensions. The first I have is Funkenbusch & Eagle, National Mathematics Mag. (1944) NYR, but Eureka 11 (1949) 5-9 says the game was at Cambridge in 1940 and some people recall playing 4-D versions about that time, while others recall the 3 x 3 x 3 version where you have to get the most rows from the 1930s. 4.B.2. Hex. I have a reference to the Danish newspaper Politiken (26 Dec 1942) - QUERIES - p.4

Mogens Larsen has sent me copies, NYR. The Parker Brothers instructions are from about 1952, but I have not yet seen them. Are there any other early Danish references? John Nash also invented the game - did he ever write anything on it? Who originated the strategy stealing argument? 4.B.3. Dots and Boxes. Lucas, RM2, 90-91, calls it Le jeu de l'Ecole Polytechnique and implies it is fairly recent. In his L'Arithmetique Amusante (1895), the editors have included his booklet: La Pipopipette - Nouveau Jeu de Combinaisons (1889). I think he may be asserting that he invented the game ?? Lucas also has an article in La Nature (1889): 'un nouveaux jeu ... dedie aux eleves de l'ecole Polytechnique'. Ahrens, A&N, describes it as recent. Ahrens's version has the boundary lines already in place. This is not clear to me from Lucas' description. Conway told me that it came from the early 19C, but perhaps this is a misremembrance?? Sam Loyd has it as 'The Boxer's Puzzle' in the Cyclopedia and says it is 'from the East'. Has it been played on other lattices? 4.B.4. Sprouts. So far as I know, this appeared first in Gardner's SA column (Jul 1967). 4.B.5. Ovid's Game and Nine Men's Morris. I am gathering information on this type of game. 4.B.6. Phutball. Did Conway describe this anywhere before Winning Ways? 4.B.9. Snakes and Ladders. The 7C Chinese Game of Promotion appears to be a version of this - can anyone provide more details? I am interested in the mathematics - the game can be viewed as a Markov process and the expected length can be worked out. Reference to such work would be appreciated. 4.B.11. Mastermind, etc. Mastermind is said to be a modern development of various older games, but I have no references to such games, except Reddi in JRM (1975). 4.B.13. Mancala Games. I am looking for early material on this - perhaps in anthropological journals??

5. COMBINATORIAL RECREATIONS

5.A. The 15 Puzzle. Dudeney ("The Psychology of Puzzle Crazes" (1926)) dates this to 1873. Loyd's Cyclopedia says 'early seventies'. Lucas, Ahrens and Schaaf give 1878. The earliest mathematical articles are in the Amer. J. Math. (1879), though this may not have appeared until early 1880?? The next I have is a New York Times piece on 23 Feb 1880 where it is described as a mental epidemic. Ahrens cites some 1879 items and many 1880 items. I have several popular articles and a solution booklet from 1880. SLAHP says early 80s. The New York Times (22 Mar 1880) says it appeared in Boston a few months ago. Do you know any other contemporary references? Many of the early references in Ahrens are obscure and I would appreciate copies of any that you come across. Edward Hordern and Jerry Slocum now believe that Loyd did not invent the puzzle, but merely the 15-14 problem, but we don't know where or when. The earliest example is one by the Embossing Co. which claims to be patented on 24 Oct 1865, but the patent has not been found. I'd like to get an example or photocopy of it. Consider a 9-puzzle in the usual arrangement: 1 2 3, 4 5 6, 7 8 x. Move the 1 to the blank position in the minimal number of moves, ignoring what happens to the other pieces. I call these 'one-piece problems' and I don't know any earlier example. 5.A.2. Three Dimensional Version. P. G. Tait describes this as "conceivable, but scarcely realisable" in PRSE 10 (1880) 664-665. Hordern cites 1889 and 1905 patents for the idea. Rouse Ball, MRE (1st ed., 1892) mentions the idea. Gardner is the first to describe Hein's Bloxbox in SA (Feb 1973). Did Hein write anything? Is there a patent? Are there any earlier references in Europe? 5.A.4. Panex Puzzle. Can anyone provide more details about the origin of this - QUERIES - p.5 particularly the date. Also, what is the date of the paper by Manasse, et al. - and was it ever published? 5.B. Crossing Problems. Wolf, goat, cabbages; 3 couples; man, wife and 2 small children are all in Alcuin. Pacioli, De Viribus, says 4 or 5 couples requires a 3 person boat. Tartaglia gives 4 couples, erroneously. Bachet points this out and shows that 4 couples can be done with a 3 person boat. What is the origin of the missionaries and cannibals version? H&S (1927) says it is 'a modern variant'. Jackson (1821) and Mittenzwey (1879?) give equivalent versions with masters and servants. The island in the river version is due to De Fonteney or De Fontenay - can anyone provide the correct spelling of the name?? Pressman and I found more efficient solutions for n couples and an island, but later contemplation reveals that the jealousy conditions are not clear and our solutions are perhaps not acceptable. 5.C. False Coins with a Balance. Schell's Problem E651, AMM 52:1 (Jan 1945) 42 appears to be the source of this problem though it deals with finding at most one light coin among 8 coins in two weighings. There is a letter in MM (1978) stating that Schell said he had invented the problem. However Paul Campbell has corresponded with Schell, who says he did not invent the problem and that he submitted the problem of finding at most one light coin among 26. In this form, it is beginning to look like a development from ternary or weighing with 1, 3, 9, .... If there is known to be one one light coin, it can be located among 3n in n weighings. This version is discussed by Karapetoff in SM 11 (1945) 186-187, where he cites AMM 52, p. 314, but this seems to be an erroneous citation, as I can't find anything in all of vol. 52, except Schell's problem. If there is at most one light coin, it can be located among 3n - 1 in n weighings - this is the form submitted by Schell, for n = 3 and simplified to n = 2 by the AMM problem editor (who was he??). Schell says he heard the problem from Walter W. Jacobs, who replied to Campbell that he had heard it by late 1943 and he would try to contact the two people who might have told it to him, but he has not written further. There are rumours that Eilenberg says it dates back to 1939. From the above two forms of the problem, it would be easy to change to one false coin, either heavy or light. Schell solved Problem E712 in AMM 54 (1947) 46-48, which is the 12 coin case with one false coin, but it had appeared elsewhere by then, e.g. in the Graham Dial (Oct 1945), SM (Dec 1945), MG (1946). The general problem is first tackled by R. L. Goodstein in MG (Dec 1945) but required a correction in 1946. I have been sent a copy of Jack Sieburg; Problem Solving by Computer Logic; Data Processing Magazine – but the date is cut off – can anyone provide details?? 5.C.1. Ranking Coins with a Balance. Steinhaus Math. Snapshots, 1950) describes this and cites Schreier (1932, NYS). 5.D.1. Measuring With Jugs. I have reorganised the notation of these problems. Tweedie, MG 23 (1939) 278-282, is the source of the triangular graphical method of solution. Halving 8 using 5 and 3 is in Abbot Albert which seems to be the earliest version. Pacioli, De Viribus, seems to be the first to use any other values, e.g. halving 12 using 7 and 5. Tartaglia seems to be the first to divide in thirds, e.g. divide 24 in thirds using 5, 11, 13. The general problem of what can be obtained from a, b, c with c full to start seems to be first treated by A. Labosne is his 5th ed. of Bachet's Problemes and appears to still be unsolved when c < a+b. 5.D.2. Ruler With Minimal Number of Marks. Dudeney gives this in MP, 1926. Gardner, in his note in 536, says Dudeney invented the problem. I have found Dudeney's Strand articles giving the linear and circular problems in 1920 & 1921 (c= MP, no. 180 & 181). MacMahon has a paper on the infinite case in 1922-23. A. Brauer (1945) seems to be the first paper on the general finite case and is an independent invention based on a resistor problem. 5.D.3. False Coins with a Weighing Scale. I have an AMM problem of 1960 and problems by Fujimura and Hunter in RMM (1961 & 1962). QUERIES - p.6

5.D.4. Timing with Hourglasses. New section - my examples are from 1962 and 1983. 5.D.5. Measure half a barrel. New Section. My earliest example is 1904. 5.E. Euler Circuits and Mazes. My earliest version of the 'five-brick puzzle' is 1844. A planar Eulerian graph has a non-crossing Euler circuit. When was this discovered? Lewis Carroll used to give such a problem, but I doubt if he had a proof of the general result. There are other examples back to 1826? 5.E.1. Maze Algorithms. BLW discuss this and cite Wiener, Math. Annalen 6 (1873) 29-30, as the first solution. Tremaux gave a simpler solution, but Robin Wilson says they could never find anything by or about Tremaux. I find that he was a telegraph engineer in Paris. Tremaux's algorithm is described in Lucas, Rouse Ball (1st to 11th eds.) and Dudeney, AM. 5.E.2. Memory Wheels = Chain Codes. These were rediscovered by Baudot in 1882, but I have no references before 1956. 5.F.1. Knight's Tours. I have references from R. Wieber, Das Schachspiel in Arabische Literatur..., and van der Linde, Geschichte und Literatur des Schachspiels, back to 1141, but Murray's History of Chess gets back to 9C India. These are very difficult to track down. Any help with these would be appreciated. Kraitchik's Math. des Jeux says the Diderot-D'Alembert Encyclopedie asserts that the problem was known very anciently to the Hindus. 5.F.2. Other Hamiltonian Circuits. Was there ever a solid dodecahedral version of Hamilton's Icosian Game? Lucas, RM 2, pp. 208-209, describes a solid wooden version. Ahrens, Mathematische Spiele, 2nd ed., 1910, p. 44, says a Dodekaederspiel is available from Firma Paul Joschkowitz - Magdeburg (.65 mark). This is not in the first ed. and was dropped in later eds. Are there any actual examples anywhere? 5.G.1. Gas, Water and Electricity. Dudeney's AM describes this as being 'as old as the hills', but was the earliest known reference until I found Dudeney's column in Strand Magazine 46 (Jul 1913) 110, but this still says: "It is much older than electric lighting, or even gas ...." In SLAHP (1928), Sam Loyd Jr says he (??) brought out the puzzle about 1900. Dudeney's Strand column says he has recently had four letters from the US about it. 5.H.1. Instant Insanity = The Tantalizer. Moffat's UK Patent (1900) is for 6 cubes considering 4 sides. S&B cites some other early examples, including Schossow (1900) for four cubes and this was applied for about a year before Moffat's application.. Meek's UK Patent (1909) is for 4 cubes. Wyatt's Puzzles in Wood (1928) gives a 6 cube version considering all 6 directions. The graphical method of solution is due to Carteblanche (1947). I am collecting examples of this puzzle and would be grateful for examples or descriptions of any you have. 5.H.2. MacMahon Pieces. Can anyone supply a copy of F. Winter, Das Spiel der 30 Bunte Würfel? (Not in British Library.) 5.I. Latin and Euler Squares. Euler's "Recherches sur une nouvelle espece de Quarres Magiques" (1782) appears to be the modern source. But there are pairs of orthogonal 4 by 4 squares in Ozanam (1725) and Alberti (1747). (The pair in Bachet is due to the 1874 editor.) Ahrens says Latin square amulets go back to medieval Islam (c1200) and a magic square of al-Buni, c1200, indicates knowledge of two orthogonal 4 x 4 Latin squares. I have recently found a 7 x 7 Latin square in a Venetian book of 1541 and references to a 4 x 4 Latin square epitaph from Cornwall, 1708, but the references are different - to Meneage parish church, St. Mawgan and to Cunwallow, near Helstone. Can anyone provide clear information on this? Are there any other early western usages of Latin squares?? 5.I.1. Eight Queens Problem. Can anyone supply copies of (Berliner) Schachzeitung 3 (Sep 1848) 363 & 4 (Jan 1849). (Not in British Library.) 5.I.2. Colouring Chessboards with no Repeats in a Line. I recall a result that an n x n board can be thus n-coloured if and only if n  1 or 5 (mod 6), but I can't find it. QUERIES - p.7

5.J. Squaring the Square. A. Mąkowski has kindly provided a translation of Z. Moron's 1925 paper. (M. Goldberg has also done a translation.) There is a Russian book by I. M. Yaglom on the subject - is it worth translating? Federico's survey article in Graph Theory and Related Topics, has its earliest reference being Dehn (1903). I find that Dudeney's 'Lady Isabel's Casket' appeared in Strand Mag. 7 (Jan 1902) 584. It later appeared as prob. 40 in his CP (1907). Both Sprague and Tutte cite this problem as a source. 5.J.1. Mrs Perkins's Quilt. This is the problem of cutting a square into squares, possibly equal. Loyd and Dudeney both give it. Gardner thinks Loyd was first. 5.J.2. Cubing the Cube. This was apparently first posed by S. Chowla (1939). 5.K. Derangements. This is derived in de Montmort (1713) who had mentioned the results without proof in the 1708 ed. 5.K.1. Deranged Boxes of A, B and A+B. New section - my example is 1962. 5.K.2. Other Logic Puzzles Based on Derangements. These typically have a butcher, a baker and a brewer whose surnames are Butcher, Baker and Brewer, but no one has the profession of his surname. 5.M. Six People at a Party = Ramsey Theory. The earliest source seems to be the Putnam exam of 1953. Who posed the problem? It is a special case of a result in the Erdös & Szekeres paper of 1935. 5.N. Jeep or Explorer's Problem - Crossing a Desert. Ball (5th ed., 1911) describes two versions, one of which is in Pearson (1907). The other, more common, form appears in Dudeney, MP. The earliest mathematical paper appears to be Helmer's Rand report (Dec 1946). There is a version in Alcuin, but the answer is obscure. Folkerts corrects an error, but it still assumes camels only eat while loaded - or perhaps they are abandoned and not reused. Pacioli gives 4 versions in De Viribus, but Agostini's description does not give the solutions. Both these want to get the most goods across a finite desert, where the transport eats some of the goods, so this is not quite the same as the 20C versions. Mahavira and Sridhara (c900) give problems where a porter gets part of the load, but they are concerned with interpolating from a given distance and payment to find the payment for a different distance. 5.O. Tait's Counter Puzzle: HHHHTTTT.. to HTHTHTHT.. , by Moving Coins in Pairs. Tait, in his review of Listing's Topologie, (1884), says he saw it recently in a railway train. Hayashi (Bibl. Math., (3) 6 (1905) 323 says it is in a Japanese book by Genjun Nakane (1743) and in books by Tasoku Takeda (1844) and Riken Fukuta (1879). Can anyone supply copies or translations? 5.P.1. Shunting Puzzles. Passing with a siding or a side-line are in RM, 1883. Reversing with a turntable was patented in 1885. The Great Northern Puzzle (with two cars, one engine, with a Y off a main line, i.e. a 'delta' configuration) was patented in 1890. The Chifu-Chemulpo puzzle appeared in 1903, according to Hordern. See E. Hordern, Sliding Piece Puzzles for other early versions. 5.P.2. Taquin. Can anyone supply early references? Lucas, RM, uses the term generically. Did it refer to any puzzles before the 15-puzzle? 5.Q. Number of Regions Determined by N Lines or Planes. Steiner, 1826, says the plane problem has been raised before, even in a Pestalozzi school book, but he believes he is the first to consider 3-space. He allows lines or circles and parallel families, but no three coincident. Schlafli (1901) does the general problem. 5.R.1. Peg Solitaire. Ahrens, MUS, cites the legend that a Frenchman saw American Indians doing a version of this with arrows. The earliest published references are Leibniz, 1710 & 1716, but material has been found in his MSS from c1678. Beasley has discovered an engraving from 1697 showing the game. S&B cite a 1698 engraving. Can anyone send copies of these? 5.R.2. Frogs and Toads: BBB_WWW to WWW_BBB by Moving or Jumping. Shortz reports it is in American Agriculturist (1867), NYR. QUERIES - p.8

5.R.3. Fore and Aft - 3 by 3 Squares Meeting at a Corner. This is in MRE (1st ed., 1892) and Loyd's Cyclopedia. Ball, MRE (3rd ed.) says he was the first to publish it. MRE (5th ed., 1911) gives a 46 move solution (which is optimal) due to Dudeney, but I can't find it in any of Dudeney's books. I have seen an example called "The English Sixteen Puzzle" from c1895. 5.R.4. Reversing Frogs and Toads: .12...n to .n...21. There are versions in Dudeney, AM and he gives general solutions. 5.R.6. Octagram Puzzle. My earliest version is Les Amusemens, 1749. 5.R.7. Passing Over Counters. Babbage discusses this in his MSS of c1820 and seems to say the general version was posed by Roget. I have also 1826?, 1857ff. 5.S. Chain Cutting and Joining. This is in Loyd's Cyclopedia and Dudeney's MP. Dudeney, "World's best puzzles" (1908), attributes it to Loyd, though I find it in Pearson (1907). 5.S.1. Using Chain Links to Pay for a Room. My earliest source is 1939, saying he heard it in 1935. 5.T. Dividing a Cake Fairly. Steinhaus, Sur la division pragmatique, Econometrica (supplement) 17 (1949) 315-319, is cited as the source of the problems and as giving Banach & Knaster's solution, though the material occurs in English in Econometrica 16 (1948) 101-104. His Math. Snapshots discusses solutions but gives no references. I find two earlier papers by Knaster & Steinhaus in Ann. Soc. Pol. Math. 19 (1946) 228-230 & 230-231. Knaster's title is "Sur le probleme du partage pragmatique de H. Steinhaus", and says Steinhaus posed it in a letter to Knaster in 1944. 5.U. Pigeonhole Recreations. Two men have the same number of hairs, etc. is in van Etten, 1624. Fourrey refers to its occurring in c1660. Erdös is the source of the applications to n+1 numbers from first 2n have one dividing another (AMM 42 (1935) 396) and have two relatively prime (??). Erdös is also the source for n2+1 distinct numbers contain a monotonic subsequence of length n+1, which appears in the Erdös-Szekeres paper of 1935. L. Moser (AMM 55 (1948) 369) is the problem that n elements of a group of order n have a subinterval whose product is the identity. Willy Moser thinks Leo invented this. O. A. Sullivan (SM 9 (1943) 116) has the problem of choosing socks and shoes in the dark. 5.V. Think-A-Dot. This was popular in 1967-69, but I haven't found out the inventor. 5.W. Making Three Pieces of Toast. My earliest example is 1943. 5.X.1. Counting Triangles. My earliest example is Pearson (1907). 5.X.2. Counting Rectangles or Squares. My earliest examples are 1921, 1927, 1928. 5.X.3. Counting Hexagons. My earliest example is 1939. 5.Y. Number of Routes in a Lattice. An early form was the number of ways to spell a word in an array - I have examples from 1822, 1893, 1897. Loyd Jr., 1928, gives the problem in abstract form. 5.Z. Chessboard Placing Problems. There are many forms of these, but the seem to arise c1900, except for earlier versions with queens. 5.AA. Card Shuffling. Recently added. A number of articles were in magic journals and I haven't been able to get them - particularly articles by Jordan in The Bat (Nov 1948 - Mar 1949) and by Elmsley in Ibidem (1957). The faro or riffle or perfect shuffle seems to have arisen in the late 19C, but it was not used until the late 1950s. 5.AE. Reversing Cups. New section - I only have a 1978 example.

6. GEOMETRIC RECREATIONS QUERIES - p.9

6.B. Straight Line Linkages. I have located Watt's parallel motion in his 1784 patent. Sarrus was misprinted as Sarrut. I haven't yet got any of the material by Lipkin(e). 6.C. Curves of Constant Width. Euler, "De curvis triangularibus" (1778) seems to be the source for this, though he only considers the triangular case. I find a vaguely similar picture in his Intro. in Analysisin Infinitorum (1748), but he doesn't seem to study the width of the curve there. 6.D. Flexagons. Gardner (Second Book) says a hexatetraflexagon was copyrighted by Roger Montandon of the Montandon Magic. Co., Tulsa, under the name "Cherchez la Femme" in 1946, but a 1993 book attributes the invention to Gardner. Can anyone supply a copy? Margaret Joseph (1951) seems to be the first publication, although Willane's Wizardry (1947) shows the hexatetra. 6.E. Flexatube. A. H. Stone says he invented this as an outgrowth of the flexagons. Leech's note in MG (1955) seems to be the first publication. Steinhaus attributes it to the Dowkers and gives a solution different than the earlier solution. 6.F. Polyominoes, Etc. Prior to Golomb, AMM (1954), I had two references: Dudeney, CP, and Dawson & Lester, Fairy Chess Review (1937), but I have since found dozens of references in Fairy Chess Review and elsewhere, e.g. W. Stead in Fairy Chess Review 9 (1954) 2-4.. There is also a patent by Scrutchin (1908) for a polyiamond puzzle and a patent by Lester (1919) showing both polyomino and polyabolo puzzles. Grünbaum & Shephard give some other early references, NYS. Are there other pre-Golomb references? When were the number of solutions for pentomino rectangles found? Scott (1958) gives the 3 x 20, but both solutions were known in 1935; Miller (1960) gives the 6 x 10. By 1969, all the numbers were known, but they aren't in Golomb's book (1965). I had thought Gardner (SA, Sep 1958) was the first mention of solid pentominoes, pentacubes, tetracubes. However, the solid pentominoes occur in 1948 and polycubes in 1939. J. C. P. Miller [Eureka 23 (1960) 13-16] says van der Poel suggested assembling the 12 hexiamonds into a rhombus, but he doesn't show them or name them. Reeve & Tyrrell, MG 45 (Oct 1961) 97-99, are the first to show the 12 two-sided hexiamonds, but don't name them. O'Beirne shows them in New Scientist (26 Oct 1981) and discusses them the next week. He says he devised them some time ago. He considers the 19 one-sided pieces. R K Guy had already published solutions in Nabla. O'Beirne seems to be the originator of the names hexiamonds and polyiamonds. O'Beirne (New Scientist, 21 Dec 1961) is the first use of the word 'polyabolo', but some appear in Hooper (1774) and Lester's 1919 patent, etc. Gardner (SA, Jun 1967) is earliest mention of polyhexes?? 6.F.1. Other Chessboard Dissections. Jerry Slocum has an example of Luers' patent of 7 Sep 1880. 6.F.2. Covering Deleted Chessboard with Dominoes. Golomb's 1954 paper starts off with this. He can't remember the origin, but thinks it may go back to Dudeney?? It appears in L. A. Graham, Ingenious Math. Problems and Methods, but the original date is not given. In SM 14 (1948) 160, it is described as coming from Max Black's Critical Thinking (1946, NYS). I have seen it in the 2nd ed., (1952), but he gives no sign of having invented it. A 1969 reference claims it is due to von Neumann! The rook's tour method for the board with any two squares of opposite colour deleted is attributed to Gomory by Gardner. See also 6.U.2. 6.F.3. Dissecting a Cross into Zs and Ls. Les Amusemens (1749) has this. 6.F.4. Quadrisect an L-Tromino, etc. Les Amusemens (1749) has this, and I have three other references before 1825, though Dudeney asserts that Lord Chelmsford invented it in mid 19C. 6.G. Soma Cube. The earliest material I know is Gardner (SA, Sep 1958). There must be earlier material, perhaps in Danish by Hein, e.g. a patent?? Slocum indicates an QUERIES - p.10 invention date of 1936, but there is no US patent for it. 6.G.1. Other Cube Dissections. Hoffmann (1893) is the earliest example I know. Steinhaus, Math. Snapshots (1950) gives a cube due to Mikusinski, but with no references. Cundy & Rollett refer to Steinhaus' cube, but it is Mikusinki's. I am told that Mikusinski has a French patent on it. A set of 6 dissections called Impuzzables has been available since 1968 (at least) - Leisure Dynamics, the US distributor, says they are due to a Robert Beck of Minneapolis. Can anyone supply more details of their origin? Klarner's JRM article (1973) refers to Cubics, a book by J. Slothouber & W. Graatsma, Octopus Press, Holland, 1970, for the dissection of a 3 x 3 x 3 into 6 1 x 2 x 2 and 3 1 x 1 x 1. I have not yet located this book - is it in English or Dutch? Klarner also describes two dissections of the 5 x 5 x 5 cube, one due to Conway. Are these the only printed sources? 6.G.2. Dissection of 63 into 33, 43 and 53. This appears in Eureka 13 (1950). Gardner identifies the setter as John Leech. Cundy & Rollett cite this as the origin, but I have found it in MG 27 (1943) 142. 6.G.3. Dissection of a Die into Nine 1 x 1 x 3. This is in Hoffmann (1893), who says it is made by Wolff & Son. 6.H. Pick's Theorem. Peter Hajek has brought a copy of Pick's paper "Geometrisches zur Zahlentheorie", Sitzungsberichte des ... Vereines 'Lotos' in Prag 19 (1899) 311-319. Steinhaus' citation is to Ztschr. des .... Steinhaus also cites his own paper "O mierzeniu pol plaskich", Przeglad Mat-Fiz. 2 (1924) 24-29. Mąkowski has kindly send me an outline translation of this and it gives a version of the theorem , but doesn't cite Pick. I can find no other references to the theorem before Steinhaus' Math. Snapshots, though C. H. Hinton has a quadrilateral version in 1904. 6.I. Sylvester's Problem of Collinear Points. I now have this sorted out. Sylvester's problem appeared in the Educational Times 46 (NS, No. 383) (1 Mar 1893) 156. Most references cite the Math. Quest. with their Sol. from the Educ. Times, which is a separate publication. Two solutions (??) appear ibid. (No. 385) (1 May 1893) 231 and are printed with the problem in Math. Quest..., but well deserve their obscurity. E. Melchior found the solution in a different context, in 1940. Erdös reopened the problem as AMM Problem 4065 in 1943. A solution by R. Steinberg appears in AMM 51 (1944) 169-171. Editorial comment outlines the solution of T. Grunwald, who later changed his name to T. Gallai. L. M. Kelly's elegant solution appears to first be published in Coxeter's article, AMM 55 (1948) 26-28. 6.J. Four Bugs and Similar Pursuit Problems. The earliest source is a problem by R. Miller on the Cambridge Tripos exam (1871). His bugs are in general position, but the velocities are adjusted to make them meet. Lucas posed a problem of three dogs which was solved by Brocard in Nouv. Corr. Math 3 (1877) 175-176 & 280, NYS. (English versions in Bernhart, SM 24 (1959) 23-50.) L. A. Graham, Ingenious Math. Problems and Methods, attributes the square version to H. D. Grossman. 6.K. Dudeney's Square to Triangle Dissection. This first occurs in Dudeney's column in the Weekly Dispatch (6 Apr 1902). In CP, he says he presented it to the Royal Society at Burlington House on 17 May 1905, but I have no further details. The process extends to rectangles, but I recall an analysis finding the most unsquare rectangle for which it works, but I can't locate this. 6.L. Crossed Ladders Problem. A very simple version occurs in the Lilavati of Bhaskara II, c1150 and there are versions in Fibonacci, Pacioli's Summa and Loyd, but the first references to the standard problem are SSM problem 131 (1908) (= problem 1194 (1931)), AMM problem 2836 (1922) (special case), AMM problem 3173 (1926) (general case), SSM problem 1498 (1937) and AMM problem E433 (1940) (general case with integer solutions). QUERIES - p.11

6.L.1. Ladder over Box. My earliest references are Simpson (1745), then Pearson (1907). 6.M. Spider and Fly Problems. Dudeney first gives this in the Weekly Dispatch of 14 Jun 1903, in the simpler 4 wall form. The 5 wall form is given in an interview in the Daily Mail (1 Feb 1905). In MPSL 2, Gardner says Loyd has simplified Dudeney, but probably Gardner was unaware of the earlier Dudeney version which is identical to Loyd's version. However, Shortz says Loyd gives it in 1900. I have just started to include the problem on a cylinder - my first example is Dudeney, MP, 1926. 6.N. Dissection of a 1 x 1 x 2 into a Cube. My earliest references are to AMM problem E4, solutions in 1933 and 1935. 6.O. Passing a Cube Through an Equal or Smaller One - Prince Rupert's Problem. Schrek, SM 16 (1950) 73-80 & 261-267, gives the history of this and it is apparently really due to Prince Rupert. Wallis was the first to write on it and Pieter Nieuwland found the maximal cube which will pass through a cube. This appears in J. H. van Swinden, Grondbeginsels der Meetkunde, 1816 and in the German edition of C. F. A. Jacobi, Elemente der Geometrie. Can anyone supply copies? Hennessy, Phil. Mag. (1895), says he has a model which may be the example made for Philip Ronayne (18C), another inventor(?) of the problem. Where is this model? U. Graf (1941) also had a model made - where is it? 6.P. Geometrical Vanishing. 6.P.1. Paradoxical Dissections of the Chessboard Based on Fibonacci Numbers. I have a vague reference to W. Leybourn, Pleasure with Profit, 1694, but I could find nothing in it. Loyd, Cyclopedia, claims to have presented the 8 x 8 to 5 x 13, in 1858. The first publication appears to be that signed Schl. in Z. Math. Phys. 13 (1868) 162. Weaver (AMM 45 (1938) 234-236) attributes this to Schlomilch and this seems right since he was a co-editor at the time. Coxeter says Schlegel, apparently confusing this with a later article by Schlegel on the same problem in the same journal. Schlomilch gives no explanation, so the first known explanation is in Riecke (1873). The version where 8 x 8 is converted to an area of 63 squares is in Loyd's Cyclopedia, but Gardner, Math., Magic & Mystery, attributes it to Loyd Jr. S&B show a puzzle version of c1900 and it is in Boy's Own Paper of 14 Dec 1901. Escott gives it in 1907 and White gives an extension in 1908. Dudeney, "World's best puzzles" (1908), shows 5 x 5 to 3 x 8. 6.P.2. Other Types. The earliest is usually claimed to be Hooper, Rational Recreations (1774), but an accidental version appears in Serlio, Libro Primo d'Architettura, 1545. This was already noted by Cataneo in 1567. I have a vague reference to Ozanam (1723) (though I am not convinced there is such an ed. and I don't see it in the 1725 ed.). Get Off The Earth. Gardner (SA, Nov 1971) says this had predecessors, e.g. an 1880 premium by Wemple and Company, NY, called the Magical Eggs. I have seen an example from c1890, by R. March, London. Has anyone got an example of this or other early forms? I have found that Loyd has a US patent 563,778 - Transformation Picture (14 Jul 1896) but this is a simple version. Loyd published 'Get Off the Earth' in the Brooklyn Daily Eagle during April & May, 1896. Readers submitting clippings of of all the Chinamen received a coloured version of the puzzle. I would like to get an example of this. 6.Q. Knotting a Strip of Paper to Make a Pentagon. This is in Lucas, RM and in Tom Tit. Lucas (1895) and E. Fourrey (1924) attribute it to Urbano d'Aviso; Trattato della Sfera ..., (or Traite de la Sphere); Rome, 1682, NYS. Can anyone supply this?? 6.R.1&2. Geometric Fallacies. Rouse Ball, MRE (1st ed., 1892) shows every triangle is isosceles and an obtuse angle is acute and says he thinks this is their first publication, though the latter may be in Mittenzwey (1879), NYR. 6.R.3. Lines Approaching but Not Meeting. This is in Proclus (5C), van Etten et al. (1630), Ozanam-Montucla (1778). QUERIES - p.12

6.S. Tangrams. Needham refers to early 19C books in China. E. Scott had one with covers probably from 18C and one with Empire fashions. Nob Yoshigahara has sent a recent reprint of two booklets from Japan, attributed to Sei Shonagon (1742) & by anon. (1837), but the first is a different dissection than the tangram and the second seems to use this same dissection. The history and bibliography by Jan van der Waals in Joost Elffers' Tangram is the most thorough that I have seen. Has anyone got copies of the early works mentioned - have any been reissued? I have a facsimile of the 1881 Japanese translation of an 1803 Chinese book recognised as the earliest known Tangram book. Van der Waals refers to a woodcut of Utamaro (1780) which I haven't seen. In Dudeney's books of bound articles (in the Strens Collection at Calgary) are two letters from J. Murray about the word 'Tangram' (excerpted in Dudeney's AM). He says that 'tan' is not a Chinese syllable and that 'tangram' appears to be a mid 19C word. Was it invented by Loyd? 6.S.1. Loculus of Archimedes. Dijksterhuis' Archimedes has a thorough survey of the material. 6.T. No Three in a Line Problem. Ahrens, MUS, says he had it in a letter from Escott in 1909. It appears in Sam Loyd's Puzzle Magazine (Jan 1908). Dudeney gives it in The Tribune (7 Nov 1906) NX (at Colindale). Can anyone supply copies? 6.U.2. Packing Bricks in Boxes. Many people have discovered the conditions for an a x b brick to fill a p x q box and it was folklore by about 1965. But I don't know any published proof nor the originator of the idea. Greenblatt, Mathematical Entertainments (1968 or 1965?) asserts that filling a 6 x 6 x 6 with 1 x 2 x 4 was invented by R. Milburn of Tufts Univ. De Bruijn seems to have discovered the basic necessary condition for higher dimensions, c1969. 6.V. Silhouette and Viewing Puzzles. Van Etten (1653 ed.) has four silhouette versions. Ozanam (1725) appears to have copied van Etten and added an extra figure. Alberti copied the latter. None are the classic circle, triangle, square problem, which first(?) appears in Catel's 1790 catalogue. What is the origin of the puzzles where three or two orthogonal views are shown? 6.W. Burr Puzzles. Slocum says Wyatt (1928) is the first to use the word 'burr'. The Crambrook catalogue of 1843 mentions several items which may be these puzzles, but pictures of the objects are not known?? 6.W.1. Three Piece Burr. The earliest I know is Hoffmann (1893), but Crambrook (1843) may have the same puzzle. 6.W.2. Six Piece Burr = Chinese Cross. This appears in an 1790 catalogue of the Berlin firm of Catel, called the Small Devil's Hoof. Minguet é Irol (1822) diagrams the pieces. The Magician's Own Book (1857) shows a different set of pieces. The Illustrated Boy's Own Treasury (1860) diagrams the pieces, shows how to assemble it and calls it 'The Chinese Cross'. Is there any Chinese connection?? 6.W.3. Three Piece Burr With Identical Pieces. This appears in SA (1 Apr 1899), but Crambrook (1843) looks like it has it. 6.W.4. Diagonal Six Piece Burr = Trick Star. A spherical version with a key piece was patented in 1904. A version with six identical pieces was patented in 1905, but it doesn't show the 3 + 3 assembly. Other versions go back to 1875 and possibly to Crambrook (1843) 6.W.5. Six Piece Burr With Identical Pieces. A version with 6 U-shaped cards appears in 1891, and there may be a version in Crambrook (1843). The Bonbon nut is in Hoffmann (1893). 6.X. Rotating Rings of Polyhedra. Rouse Ball, MRE (11th ed., 1939) attributes the tetrahedral idea to J. M. Andreas and R. M. Stalker. Coxeter says neither ever published on the subject. Pedersen tells me that it appears in Fedorov, late 19C, NYS. Bruckner (1900) considers rings of tetrahedra. What about the cubical versions? The 'Shinsei Mystery' version is attributed to Naoki Yoshimoto, 1972, but may be in Slothouber & Graatsma, Cubics, 1970, NYS. Was Engel the first to consider using the 'Jacob's ladder' hinge? QUERIES - p.13

6.Y. Rope Round the Earth. That is, if it is moved out 1 m all around, it gains 6.28 m in length. The geometric idea is clear in Lucca 1754 (c1390); Muscarello (1478); Pacioli's Summa (1494). "A Lover of the Mathematicks"; A Mathematical Miscellany; Dublin, 1735; describes travellers whose heads go 12 yards more than their feet. Dudeney gives it in Strand Mag. (1909). 6.Z. Langley's Adventitious Angles. (Let ABC be an isosceles triangle with  A = 20o, and draw BD, CE making angles 50o and 60o with the base. Then  CED = 30o.) This appeared a few years ago in MG and they cite Langley, MG (1922) and 13 solutions in 1923. The latter cites the Peterhouse and Sidney entrance scholarship examination, Jan 1916. JRM 15 (1982-83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. Can anyone supply copies of these? 6.AA. Nets of Polyhedra. (I.e. unfoldings into a planar figure.) Dürer, 1525, gives a net for each regular polyhedron and some Archimedean ones. Panofsky's biography of Dürer asserts that Dürer invented the concept of a net. Cardan shows the nets of the regular polyhedra in 1557. Dudeney, MP, gives all 11 nets of the cube. Perelman [1920s?] gives the problem and finds 10 nets. The 11 nets of the octahedron are in Math. Teaching 40 (Aut 1967) 48-52 (where 13 are given), corrected Ibid. 41 (Wint 1967) 29. I believe the dodecahedron and the icosahedron were unsolved until my student Peter Light and I found (1984) that there are 43380 nets on these. Gardner mentions the 4-cube in his Foreword to Steinhaus's One Hundred Problems (1964). He posed it again in SA (Nov 1966) and the Addendum to this in Carnival says that he received several answers, no two agreeing! Peter Turney, JRM 17 (1984/5) finds 261 nets on the 4-cube. 6.AB. Self-Rising Polyhedra. Kac (AMM 81 (1974)) says the dodecahedron is due to Steinhaus. I have recently obtained a 1939 ed. of Steinhaus's Mathematical Snapshots with the original example in the pocket at the back. I have recently seen several pop-up octahedrons. Does anyone know the source of the other versions? 6.AC. Conway's Life. Gardner, SA (Oct 1970), seems to be the first printed material. 6.AD.1. Largest Parcel One Can Post. I have this from 1883, complete with the cylindrical solution. 6.AE. 6" Hole Through a Sphere Leaves Constant Volume. Gardner cites S. I. Jones, Mathematical Nuts, 1932, NYS. But the two-dimensional version is Holditch's theorem of 1858. 6.AF. What Colour Was the Bear? I find this in Leopold's At Ease! (1943); Northrop's Riddles in Mathematics (1944) and in Moulton (AMM 51 (1944) 216 & 220). Simpler North Pole problems go back to 1735. Moulton (1944) is the first to consider all solutions of the three-sided problem. Perelman (1920s?) is the first to consider a closed circuit, but he has a square. 6.AG. Moving Around a Corner. I don‘t know the origin of this. My earliest version with a table is AMM 47 (1940) 569. Abraham (1933) gives the problem of taking a ladder from one street to another, of different widths. Licks (1917) gives an equivalent problem phrased as getting a stick into a circular hole in the ceiling. 6.AH. Tethered Goat. M. Fraser (MM 56 (1983) 123) cites the Ladies Diary for 1748, NYS. Dudeney (1898) has a version with two pieces of field which avoids the usual transcendental equation. 6.AI. Trick Joints. The common square version was patented in 1888. S&B cite The Woodworker of 1902 as the earliest reference, but it occurs in Tom Tit (1892). The hexagonal and triangular versions are in Abraham, 1933. I have found a pentagonal version, 1940s?, but it splits and reglues the wood so it cannot come apart. I have a double dovetail T-joint from SA (25 Apr 1896) 267. Wyatt, Wonders in Wood, cites A. B. Cutler in Industrial Arts and Vocational Education (Jan 1930) for a triple QUERIES - p.14 dovetail, but I couldn't find it in vols. 1-40. 6.AJ. Geometrical Illusions. This is a large field. Principally I have considered the cases below, but I have started to compile origins of other illusions. 6.AJ.1. The Two Pronged Trident. R. L. Gregory told me this was devised by an MIT draughtsman about 1950. D. H. Schuster (Amer. J. Psychol. 77 (1964) 673) refers to an ad by California Technical Industries in Aviation Week and Space Technology 80:12 (23 Mar 1964) 5. A letter to the address in the ad was returned 'insufficient address'. Gardner, SA (May 1970), says it became known in 1964. Can anyone supply a copy of Mad Magazine (Mar 1965) which had it on the cover or of the 1964 issue reprinted in Mad Power? Oscar Reutersvård developed a similar pattern in the 1930s. 6.AJ.2. Tribar and Impossible Staircase. I have now seen Oscar Reutersvård's books of the early 1980s. There appear to have been earlier exhibition catalogues. I have a reference to 150 Omojliga Figurer, Malmo, 1981. Can anyone supply a copy? The books are not very clear as to when he first exhibited his tribar - it appears to have been in the early 1960s. Where is L. S. Penrose's original model staircase? Roger Penrose thinks it may be in Manchester? 6.AK. Polygonal Path Covering N by N Lattice of Points, Queen's Tours, etc. Loyd has 3 x 3, 7 x 7 and 8 x 8 versions in his Cyclopedia. A. C. White, Sam Loyd and His Chess Problems, shows an 8 x 8 Queen's circuit in 14 moves, by Loyd in ?Le Sphinx (Mar 1867) (or 15 Nov 1866) (NYS) and Chess Strategy (1878) no. or p. 336 (NYS). Can anyone help with the Le Sphinx? White also shows non-crossing Rook's circuits in 16 segments from Chess Strategy. I have a trick version on the 6 x 6 from 1887. The 3 x 3 problem must be old, but my earliest references are an interview with Loyd in 1907 and Pearson (1907). 6.AO. Configuration Problems. (a,b,c) denotes a points in b rows of c each. Jackson, Rational Amusements for Winter Evenings, (1821) seems to be the source, but several authors, e.g. Dudeney, "World's best puzzles" (1908), say (9,10,3) is attributed to Newton. 6.AO.1. Place Four Points Equidistantly = Make Four Triangles with Six Matches. My earliest example is c1826. 6.AO.2. Place an Even Number on Each Line. My earliest examples are Mittenzwey (1879?) and Hoffmann (1893). 6.AP. Dissections of a Tetrahedron. The classic bisection of a tetrahedron into two congruent pieces was patented in 1940, though I have clear picture from 1887. I have references from 1946 and 1957, but I also have a 1962 mention which indicates it was new to the author. These pieces can be cut in half in a simple way or by another midplane of the tetrahedron to give four congruent pieces. I have an example of the first from about the 1940s, but I think the latter is more recent and my earliest source is a puzzle version of the 1970s?? There is also a trisection, made by Pussy and Tenyo. Any idea of the origins of any of these? 6.AQ. Dissection of a Cross, T or H. S&B show a 1857 dissection of a cross which trims to the common T dissection. They say that crosses date from early 19C and T's from 1903. A T puzzle has been found from 1898. Charles Babbage's MSS of c1820 show a five piece cross and I have another version of 1826? 6.AR. Quadrisection of a Square. I have a 1903 description of this and it may be in Crambrook. 6.AS.1. Twenty 1, 2, 5 Triangles Make a Square or Five Equal Squares to a Square. The 10-piece version is in Les Amusemens (1749). The Boy's Own Book (1828) has the 20-piece version. 6.AS.2. Dissect Two Adjacent Squares to a Square. This must be ancient, as it is a QUERIES - p.15 proof of the Theorem of Pythagoras, but my earliest reference is Les Amusemens (1749), but I have recently been told that a version occurs in Arabic, c1100. 6.AT. Polyhedra and Tessellations. 6.AT.2 Star and Stellated Polyhedra. There is a mosaic attributed to Uccello on the floor of San Marco, Venice, which shows a 'spiky' dodecahedron. Judith Field says it is not a proper stellated dodecahedron and is much later than Uccello. I'd like a good picture of it. I've now found two pictures, one very good and one not good!! Jamnitzer (1568) shows a great dodecahedron. The stella octangula is generally attributed to Kepler. Coxeter [The Fifty-Nine Icosahedra] cites Harmonices Mundi, II, prop. XXVI, but this describes a sort of cube with ears whose faces are octagrams. I have not found anyone who can give the origin of the stella octangula. 6.AT.8. Dürer's Octahedron. New topic - I'd appreciate further references. 6.AU. Dead Dogs and Trick Ponies. I have 1849, 1857, c1859 & 1860. I have just found an early version, supposedly from Ulm, 1470, with a rotating centre section. There are many paintings showing mixtures of bodies and heads - I have c1600 Persian and several Japanese version, c1800. 6.AV. Cutting Up in Fewest Cuts. My early examples are in Perelman, 1920s?? 6.AW.1. Mitre Puzzle. This is in Hanky Panky (1872) and Hoffmann (1893), but Pearson (1907) calls it Loyd's. Dudeney (1908) says he has traced it back to 1835, but gives no details. It is in Babbage's MSS of the 1820s and in Endless Amusement (1826?). 6.AW.3. Dividing a Square into Congruent Parts. Is there any way to divide a square into three congruent, connected parts, other than rectangles? I have heard that this has been proven. 6.AX. The Packer's Secret. My earliest example is Tissandier, 1888. 6.AY. Dissect 3A x 2B to Make 2A x 3B, etc. (Using a staircase cut.) This is in Cardan (1557) and a Japanese booklet of 1727. Loyd (1914) claimed that any rectangle could be converted to a square in this way, but Dudeney pointed out this error. 6.AZ. Ball Pyramid Puzzles. Len Gordon has been studying these. He & Jerry Slocum can only trace them back to the early 1970's. The first example is Pyramystery by Piet Hein, 1970, but Hein produced two versions with the same name in 1970! 6.BA. Cutting a Card so One Can Pass Through It. This is in Ozanam (1725). 6.BB. Doubling a Square without Changing Its Height or Width. This is in The Sociable (1858). 6.BC. Hoffman's Cube. Dean Hoffman presented this at a conference at Miami Univ. in 1978. Was there any publication of this Conference? 6.BD. Bridge a Moat with Planks. This is in Mittenzwey (1879?), Lucas (1883), Hoffmann (1893). The first example with a circular moat seems to be Always (1969). 6.BE. Reverse a Triangular Array of Ten Circles. My earliest example is 1939. 6.BF.4. Rail Buckling. My earliest examples are 1943 and 1956. 6.BG. Quadrisect a Paper Square with One Cut. I reinvented this. Gardner says it is well known, but I have only found it once in print. 6.BH. Moiré Patterns. I haven't found any really good history of this. 6.BL. Tan-1 ⅓ + Tan-1 ½ = Tan-1 1, etc. This problem is usually presented with three squares in a row with lines drawn from one corner to the opposite corners of the squares. Euler has a general version, but the particular puzzle version seems to be mid-20C. 6.BN. Round Peg in Square Hole or Vice Versa. I am collecting quotations based on this simile. The fact that the round peg fits better was the basis of my first paper in 1964, but the 2-D case had already been done in 1944. 6.BR. What is a General Triangle? More references would be useful. 6.BS. Form Six Coins into a Hexagon. My earliest example is 1963. 6.BT. Placing Objects in Contact. My earliest example is 1926 putting five coins in QUERIES - p.16 contact. I know there is more recent work, but I don't know where.

7. ARITHMETICAL RECREATIONS

7.A. Fibonacci Numbers. D. Bernoulli knew the Binet formula for Fn in 1732. This is about a century before Binet and is much clearer! 7.B. Josephus or Survivor Problem. Josephus is very vague, though the Slavonic text implies that he definitely cheated. Ahrens, MUS II, cites Codex Einsidelensis No. 326 (early 10C) and other items. Murphy's article in Béaloideas - The Journal of the Folklore of Ireland Society 12 (1942) 3-28 describes early versions and variants from northern Europe and Ireland - he believes it originates in c800 Ireland! Apparently Cardan (1539) is the first to associate it with Josephus. Most of the early forms involve 15 & 15 counted out by 9 or 10, but Meermanische Codex (10C), Munich Codex 14836 (11C) and Chuquet give rules for counting out by other values. Murphy and some 19C articles associate the problem with St. Peter. Pacioli, De Viribus, describes other numbers, e.g. 2 Christians and 30 Jews. Ahrens is not clear as to the origin of the idea of counting out till one is left. Any information on early versions would be appreciated. The Japanese variant of the problem, with 15 & 15 counted by 10s, goes back at least to Muramatsu (1665) and Ahrens believes it may have an independent origin as early as 11C?, though I find this hard to believe. Can anyone supply copies of the early Japanese versions? 7.C. Egyptian Fractions. Was Sylvester the first to show that an expansion terminated? 7.E. Monkey and Coconuts Problems. Mahavira gives several versions, including two with indeterminate final result. Versions with final result specified occur in Chiu Chang Suan Ching, Zhang Qiujian, Ananias of Shirak, Bakhshali MS, etc. There are coconuts versions in Pearson (1907) and there are other indeterminate versions in Ozanam (1725) and Dudeney (1903). I have a reference to al-Tabari - can anyone supply details? 7.E.1. Versions with All Getting the Same. These have the i-th person getting a*i + b plus r of the rest or r of the remainder plus a*i + b more and all getting the same. These occur in Fibonacci, Maximus Planudes, Gherardi, dell'Abbaco, etc. 7.F. Illegal Operations Giving Correct Result. I have an article by Witting (1910) which gives some of the examples. Ahrens, A&N, gives it in 1918. Lietzmann, Lustiges und Murkwürdiges uber Zahlen und Figuren, gives 16/64 = 1/4 in 1922?? (my edition is 1923), citing Witting and Ahrens. The illegal cancellation of a - b into a2 - b2 is in SM 12 (1946) 111. 7.G. Inheritance Problems. 7.G.1. Half + Third + Ninth, Etc. It is in Dudeney, MP, and Loyd, Cyclopedia. The French ed. of MRE says it is Arabic. Kraitchik, Math. des Jeux, says it is a Hindu problem. Dudeney and Loyd both give it in a rephrased manner that makes the 18th camel unnecessary. Sanford, Short History of Math., says Tartaglia was one of the first to suggest the 18th camel, but I haven't found it yet. Tartaglia does discuss the problem and says people claim the problem is impossible or illegal, but he simply divides proportionally. Hanky Panky (1872) and Cassell's Book ... (1881) give it clearly. H&S says it is a modern problem in that previously it was considered as a proportion - 1/2 : 1/3 : 1/9 = 9 : 6 : 2. Ahrens, A&N, gives it clearly and cites the French edition of MRE - he adds that the problem has been in German oral tradition for a long time. Dell'Abbaco has division into half and third. I recall a paper which found all solutions of 1/a + 1/b + 1/c = n/(n+1), but I can't find it. 7.G.2. Posthumous Twins, Etc. I have references for the posthumous twins problem to the Lex Falcidia (-1C), Juventius Celsus (1C), and others, but don't yet know if they are easily found. QUERIES - p.17

7.H. Division and Sharing Problems - Cistern Problems. This is generally attributed to Hero(n), but it is in a dubious work and his solutions are very confused and wrong. (The dating of Hero seems to have recently changed - I have c150 - what is the new date??) Smith, History II 538, quotes a cistern problem from Bachet's Diophantos, but it is in a section taken from The Greek Anthology. H&S says a cistern problem is in Alcuin but I can only find a simple problem (8) which mentions a cistern, but otherwise is unrelated. Datta & Singh cite Brahmagupta, but it is actually in Prthudakasvami's commentary of 860. The Chiu Chang Suan Ching (c-150) gives a five pipe cistern problem, which Vogel says is the first example. However, the Chiu Chang Suan Ching also gives a number of 'assembly problems' which lead to the identical mathematics as cistern problems, but all the rates are inverted. Eleanor Robson has told me that there are several of these problems in Old Babylonian and I now have some details of these. The Bakhshali MS (c7C??) has several variants of the cistern problem, including one with seven rates. 7.H.2. Division of Casks. Alcuin divides 10 full, 10 half-full and 10 empty casks among three people. Abbot Albert divides 9 casks containing 1, 2, ..., 9 among 3 people. 7.H.3. Sharing Unequal Resources - Problem of the Pandects. E.g. one man has 5 loaves and another has 3 which they share with a third. He pays them. How do they split the money? It is in Fibonacci, 1202, and in Kazwini's Cosmographia, 1262. Kraitchik gives a Roman(?) version, taken from Unterrichtsblatter fur Math. & Naturw. 11, pp. 81-85, (NYS), and this is my only source for the term 'Pandects'. 7.H.4. Each Doubles Other's Money to Make All Equal. Diophantos gives general formulations for 3 and 4 people and it is in Mahavira and Fibonacci. 7.H.5. Sharing Cost of Stairs, Etc. Mahavira and Sridhara have versions where carriers are paid from the goods carried. This really leads to an exponential function and is a bit like the Explorer's problem (5.N). Mahavira and Sridhara also have problems of sharing payment among carriers who carry for different parts of the journey or spectators who watch different parts of a performance. Dell'Abbaco has a house being shared. 7.H.6. Sharing a Grindstone. I have just added this and my only example is 1928. 7.H.7. Digging Part of a Well. Qazwini and dell'Abbaco have versions where a well is being dug. 7.I. Four Fours. The earliest reference to four fours is a pseudonymous letter to Knowledge (1881), but Dilworth, c1744, asks for 12 in four identical figures and for 34 as four threes. I have now two 18C references and a number of examples of the idea before 1880. 7.I.1. Largest Number Using Four Ones, etc. My earliest version is Perelman, 1920s? 7.J. Salary Puzzle. That is, it is better to get one quarter of the raise twice as often. This is in Ball (MRE, 3rd ed., 1896), Cunnington (1904) and the Daily Mail (30 Jan 1905). The puzzle seems to be a fairly direct evolution from more straightforward salary problems. 7.K.1. Casting Out Nines. This is mentioned by St. Hippolytus, Philosphumena, c200, NYS. A special case is in Iamblichus. Al-Khowarizmi, c820, describes it. A fairly general use of 9s is in Aryabhata II's Mahasiddhanta, c950, and Narayana's Ganita-kaumudi, c1356, allows any modulus. Have either of these ever been translated into a western language?? There are also Arabic references from 952/953, c1000 and c1020 (Avicenna, who attributes it to the Hindus). 7.L.1. Geometric Progressions. I am looking for the source of the story of Sessa and the chessboard, i.e. 1 + 2 + 4 + ...+ 263. Murray feels it is of Indian origin, but his earliest version is Al-Yaqubi, c875. Mas'udi's Prairies d'Or (10C) refers to the summation without reference to Sessa. Al-Biruni computed 264 by repeated squaring but doesn't cite Sessa. It appears in Fibonacci, etc.. J. Wallis gives an Arabic and Latin version. The earliest horseshoe nails version seems to be AR (c1450). I am interested in occurrences of (1 + )7 + 49 + 343 + ... which appears in Papyrus QUERIES - p.18

Rhind and in Fibonacci and in Munich 14684. Buddha is said to have been asked to compute 717. (Cajori, History of Mathematics, p. 90.) Are there earlier nursery-rhyme versions than c1730? Are there other examples going up by 9s? Fibonacci also gives the use of binary weights (1, 2, 4, 8, 15) to get to 30 and Bachet's weights (1, 3, 9, 27, ...). Al-Tabari is said to have used the latter - can anyone supply details? 7.M. Binary System and Binary Recreations. I have just added material on the origins of the binary system, but the history seems contorted. Binary coding is present in the Chinese arrangement of the I-Ching hexagrams, c1060. But it is already implicit in Egyptian multiplication and the Chinese rings. Binary arithmetic seems due to Harriot (unpublished, c1604) and Napier (Rabdologia, 1617), before Leibniz (1679). 7.M.1. Chinese Rings. Cardan's 1550 description is brief! Ch'ung-En Yu's Ingenious Ring Puzzle Book calls it the Nine-Interlocked-Rings Puzzle and says it was well known in the Sung Dynasty (960-1279). Jerry Slocum and Needham give older Chinese references, but these are pretty vague. Culin, Games of the Orient, gives a legend that it was invented by Hung Ming (181-234). There is a Chinese musical drama, The Stratagem of Interlocking Rings, c1300 - but I have no information about it. Gardner says there are 17C Japanese haiku about the puzzle and it occurs in Japanese heraldry - can anyone supply details? Afriat says he had a copy of Gros's 1872 pamphlet obtained by the Radcliffe Science Library at Oxford, but they could not find it for me. Can anyone help with this? 7.M.2. Tower of Hanoi. I have now seen photocopies of the original versions deposited at the Conservatoire National des Arts et Métiers by Lucas himself, with his inscriptions, including the date of invention - Nov 1883 - and the assertion that he invented it. The commercial version had the fancy cover and the instructions are dated 1883. Edward Hordern has an example with the same instructions, but in a different box. The earliest article seems to be: G. de Longchamps; "Variétés"; Journal de Mathématiques Spéciales (2) 2 (1883) 286-287, but the British Library says there are no copies in the UK - can any French reader send a copy? De Parville has a note on the puzzle on 27 Dec 1883. There is an 1889 booklet (Brochure) by Lucas on the Tower of Hanoi, NYS (not in British Library or Bibliothèque Nationale catalogues and A. Hinz has not located any example). But Lucas' article in La Nature (1889) indicates that this last may refer to the game itself or its instructions. Dudeney, World's best puzzles, dates it to 1883. The original cover says 'Brevete', but the Conservatoire could not locate a patent in 1880-1890. 7.M.2.a. Tower of Hanoi with More Pegs. Lucas' La Nature article of 1889 mentions the use of 4 or 5 pegs and illustrates such a game. Dudeney gives a 4 peg version in London Magazine (May 1902), a 4 peg version in Weekly Dispatch (25 May 1902) and a 5 peg version in Weekly Dispatch (15 Mar 1903). The general version is unsolved. 7.M.3. Gray Code. Baudot used this c1878, but I have no contemporary reference other than Annales Télégraphiques of 1879. Was there a patent? 7.M.4. Binary Divination. Pacioli's De Viribus has a clear version for finding one thing from 16. 7.M.6. Binary Button Games. I have recently added this. The earliest material I have is Berlekamp's switching game at Bell Labs, c1970, and my analysis of the XL25 in 1985. 7.N. Magic Squares. This is getting sorted out?. I am back to c-650 in the Shu Ching and c-5C in the Confucian Analects, though these are cryptic references which may be irrelevant or later interpretations. Theon of Smyrna is often cited but his square is not magic. Ho Peng Yoke says Xu Yiu (Hsu Yo), c190, is the first to give the order 3 square, but there is doubt about the date and authorship - some say it was written by Zhen Luan in c570. Cammann, Lam and Hayashi refer to the Ta Tai Li Chi, c80, as the earliest example. Ahrens' survey in Der Islam (1917) has clarified most of the Arabic references. Jabir ibn Hayyan and Tabit ibn Korra give the first Arabic magic squares. The relevant works of Korra and ibn al-Haitham seem to have perished. QUERIES - p.19

I have vague references to Ts'ai Yuan-Ting (c1160) and Najm al-Din al-Lubudi (c1250). The 1C Hindu version remains mysterious - I have only a brief reference to this - A. N. Singh, "Hist. of magic squares in India", Proc. ICM, 1936. The 4 x 4 magic square seems likely to have originated in India. I'm not sure if the square at Khajuraho is the same as the one at Dudhai, Jhansi?? Narayana Pandita (1356) gives oddly even squares and the editor cites some earlier Hindu sources. (I believe someone has studied Narayana's work on magic squares??) Can anyone help with these? 7.N.1. Magic Cubes. Fermat seems to be the first to construct one. However, it fails along 8 of the 24 2-agonals as well as along all 4 3-agonals. Benson & Jacoby cite G. Frankenstein, in the Commercial Traveller (Cincinnati) (11 Mar 1875), for a perfect 83 (NYS). Maxey Brooke cites Joseph Sauvier (1710) for the first magic cube (NYS) and Schlegel (1892) for the first magic 34 (NYS). In W. S. Andrews' Magic Squares and Cubes, C. Planck cites a book of his, Theory of Path Nasiks, NYS, which seems to be the first approach to a theory. Gardner, SA (Feb 1976) says there is an unpublished MS of Rosser & Walker at Cornell and that there are reports by Schroeppel and Beeler on this. Can anyone help find these? 7.N.2. Magic Triangles. Frenicle (1640) and Scheffler (1882) give versions. 7.N.3. Anti-Magic Squares and Triangles. When do these begin? Loyd Jr (1928) gives an antimagic 3 x 3 square. Gardner mentions anti-magic squares in SA (Jan 1961), citing MM, 1951. Fults' book asserts Trigg considered these in 1951, but he gives no reference and I see this is a corruption of Gardner's reference. 7.N.4. Magic Knight's Tour. New section. Beverley, 1848, seems to be the first to consider this. De Jaenisch has a lot on it. 7.N.5. Other Magic Shapes. I have a magic cross from 1893 & 1907 and a number of odd magical shapes from late 19C. 7.O. Magic Hexagons. Trigg, RMM 14 (1964) 40-43 reports that Adams saw the idea in an issue of The Pathfinder, c1910. Can anyone find this? Gardner, Puzzles from Other Worlds (1984), reports that there are 1896 patents in the UK and the US by W. Radcliffe. Heinrich Hemme has kindly sent material on the first version by von Haselberg, 1888-89. 7.O.1. Other Magic Hexagons. Frenicle gives one form in 1640. Another form with constant 26 on a Star of David appears in 1895, 1908 and 1918. 7.P.1. Hundred Fowls and Other Linear Problems. This seems due to Zhang Qiujian (Chang Chhiu-Chin) (c475). The Indian Bakhshali MS also has it - this was formerly dated c4C, but is more recently dated c7C. (The BLLD doesn't have G. R. Kaye's Indian Mathematics, Calcutta & Simla, 1915.) Libbrecht quotes a number of nonsensical commentaries on Chang. I have examples from Alcuin, Mahavira, Sridhara, Abu Kamil, Bhaskara II, Fibonacci, etc. 7.P.5. Selling Different Amounts 'At Same Prices' Yielding the Same. Mahavira (c850) has 6 examples, but the results are not clearly found. Sridhara (c900) and Bhaskara II (c1150) have comprehensible examples. These all specify the relative amounts, by giving the capitals, and one price. The European versions specify the amounts, and sometimes the yield. The first European versions are Fibonacci (1202), Munich 14684 & dell'Abbaco. Ozanam (1694?) seems to be the first to give a general approach?? Ozanam-Riddle (1840) finds all 10 integral solutions to the case where the amounts are 10, 25, 30. I have analysed both the Indian and the western versions and found a new simple rule for the number of solutions in the western case. 7.P.7. Robbing and Restoring. First appears in al-Karkhi, c1010. This is said to be problem 5 in the contest between Fibonacci and John of Palermo in 1225. I haven't seen a description of this contest, but the problem is extensively treated in Fibonacci. 7.Q. Blind Abbess and Her Nuns. That is, how to arrange objects along edges of a QUERIES - p.20

3 x 3 square so edge sums are constant. This is in van Etten (1653). A simpler type of problem is in De Viribus. Murray's History of Chess says this appears in At-Tilimsani, c1370 - can anyone provide details? 7.Q.1. Rearrangement on a Cross. This is in De Viribus, c1500, then in Les Amusemens, 1749, then numerous 19C examples. I have seen only one example of rearrangement on a Y, from 1912. 7.Q.2. Rearrange a Cross of Six to Make Two Lines of Four. New section - my examples are 1749, 1921, 1930s?, 1939. 7.R. "If I Had One from You, I'd Have Twice You". This is in Heiberg & Menge's edition of Euclidis Opera. It is also in the Greek Anthology (c510). Diophantos (c250) gives a general formulation of this and a version for 3 and 4 people. Fibonacci gives many versions, including one which is inconsistent which he says has no solutions. 7.R.1. Men Find a Purse and 'Bloom' of Thymarides. Diophantos has similar problems. Iamblichus and Mahavira have several examples. Fibonacci has many examples, including some with negative solutions. He is the first to consider the case where the i-th says he'd have ai times the i+1st person. 7.R.2. "If I Had 1/3 of Your Money, I could buy the Horse". There are several variants of this in the Chiu Chang Suan Ching. Diophantos gives a general formulation for 3 and 4 people. 7.R.3. Sisters and Brothers. I have this only back to 1924? 7.R.4. "If I Sold Your Eggs at My Price, I'd Get ...." I find this in Simpson's Algebra, 1745, then in 1940. 7.S. Dilution and Mixing Problems. I have a reference to Recorde, The Grounde of Artes, 1579 ed., but it is not in my facsimile of the 1542 ed., though perhaps this was done from an imperfect version?? 7.S.1. Dishonest Butler Drinking Some and Replacing with Water. There is a simple version in the Rhind Papyrus. There is a version in the Bakhshali MS. Cardan (1539) has a version, as do Tartaglia, Quesiti, ..., (1546); Buteo (1559) and Trenchant (1566, NYS). 7.S.2. Water in Wine versus Wine in Water. What is the origin of this problem? It was a favourite of Lewis Carroll. I've found it in Ball (MRE, 3rd ed., 1896) and Pearson (1907). I also find more direct related versions c1900. 7.T. Four Number Game. That is: (a, b, c, d) goes to (a-b, b-c, c-d, d-a). The first reference seems to be Ciamberlini and Marengoni (1937). 7.U. Frobenius's Postage Stamp Problem. That is, find the largest value not obtainable from a set of values. I can't locate anything that relates this to Frobenius until 1962. Various authors cite Sylvester (1884), NYS, for the two value case, but Dickson is surprisingly obscure on this subject. The second reference on the subject appears to be a two part paper of A. Brauer in 1942 & 1954. 7.V. xy = yx and Iterated Exponentials. Archibald (AMM 28 (1921) 141-143) and Knoebel (AMM 88 (1981) 235-252) survey the history. Any other sources than those cited? 7.W. Card Piling Over a Cliff. This seems to first appear as Problem 3009, AMM 30 (1923) 76 and the Otto Dunkel Problem Book lists this as unsolved in 1957 and I believe it is still unsolved, though at least one author has claimed that the harmonic series gives the maximum overhang. Ramsey's Statics (1934, pp 47-48, NYS - I have seen the 2nd ed. of 1941) gives a related problem referred to a Tripos. The problem appears several times in the 1950s. Gamow & Stern give it in 1958. MG (Oct 1980) discusses the problem, but their earliest reference is Barnard's column in The Observer (1962). 7.X. How Old Is Ann? (Or Mary??). The name varies, though the problem is usually the same. Loyd refers to How Old is Ann and gives a How Old is Mary in the Cyclopedia. A. C. White, Sam Loyd and His Chess Problems (1913), says Loyd invented How Old Was Mary. G. G. Bain, "An Interview with Sam Loyd" (1907), refers to How Old Was Mary. Dudeney, AM, prob. 51, attributes it to Loyd. But Clark Kinnaird (1946) says it QUERIES - p.21 is due to Loyd Jr., and made him famous. He goes on to say that such problems go back to 1789, but gives no references. W. R. Ransom says it is much older than the early 1900s. I suspect these are referring to the following type of problem: X is now a times as old as Y; after b years, X is c times as old as Y, which I have traced back to 1745.. 7.Y. Combining Amounts and Prices Incorrectly. Also called the appleseller's problem. This is in Alcuin, ibn Ezra, Fibonacci, etc. 7.Y.1. Reversal of Averages Paradox. Sometimes known as Simpson's Paradox. I have an example from 1944 and a reference to 1934, then Simpson's paper of 1951. 7.Y.2. Unfair Division. Farmer is to give 2/5 of his yield to the landlord, but the farmer uses 45 bushels of the harvest before they can divide it. He then proposes to give 18 bushels to the landlord and then divide up the rest. Is this correct? I have three essentially identical versions from 1857-1860. Does this problem ever occur anywhere else? 7.Z. Missing Dollar Paradox. This is the one where one counts in different ways and comes out with $27 and $28. The earliest I have is 1939, but a related form confusing the amount withdrawn from a bank goes back to 1933 and there is an ancestral from in 1751. 7.AC.1. Cryptarithms: SEND + MORE = MONEY, etc. Shortz reports an example from 1864, NYR. Versions are in Loyd's Cyclopedia. SEND + ... is in Dudeney's column in Strand Mag. (1924). 7.AC.2. Skeleton Arithmetic: Solitary Seven, Etc. Berwick's seven sevens division appeared in 1906. Ackermann says Berwick was inspired by similar problems in Workman's Tutorial Arithmetic. Fred Schuh (Nieuw Tijds. voor Wisk. 8 (1920-21) 64) gives a skeleton division with no figures shown, but there is no answer in this or the succeeding volumes. The problem and answer do appear in AMM (1921 & 1922) and in Schuh's book (1943). In 1992, an anonymous postcard to The Science Correspondent, "The Glasgow Herald", 8 May 1963, was found in Prof. Lenihan's copy of Gardner's More Mathematical Puzzles and Diversions giving the full skeleton of 1062 / 16 = 66.375 with no digits specified - the solution is unique. I thought this might be from Tom O'Beirne, but Mrs O'Beirne says it is not in his handwriting. 7.AC.3. Pan-digital Sums. The earliest versions are tricks: 15 + 36 + 47 = 98 + 2 = 100. These start in 1857. Dudeney (1897-98) has examples like 235 + 746 = 981. 7.AC.4. Pan-Digital Products. Loyd (1897) has examples like 3907 * 4 = 15628. Dudeney (1902) has 2 * 78 = 4 * 39 = 156. 7.AC.5. Pan-Digital Fractions. I have versions from Pearson (1907). 7.AD. Selling, Buying and Selling the Same Item. A version is in Loyd's Cyclopedia, but he avoids giving an answer. 7.AD.1. Pawning Money. Carroll used to give this. 7.AE. Use of Counterfeit Bill. Versions begin in 1857. 7.AG. 2592. That is, 2592 = 25 * 92. This is in Dudeney, AM (1917), Phillips (1937) and RMM (1962). Greenblatt, Mathematical Entertainments (1968), asserts that this was discovered by his officemate. 7.AH. Multiplying by Reversing. I have just noted a curious connection of this with 1089, because 9 * 1089 = 9801; 4 * 2178 = 8712 and 2178 = 2 * 1089!! This follows since 1089 = 1100 - 11 so that k * 1089 = kk00 - kk = k,k-1,9-k,10-k in base 10. From this we see that k*1089 is the reverse of (10-k)*1089. Now 10-k is a multiple of k for k = 1, 2, but we get some new types of solution for k = 3, 4, namely: 7 * 3267 = 3 * 7623; 6 * 4356 = 4 * 6534. I cannot see a proof that this gives all solutions of this problem. 7.AI. Impossible Exchange Rates. Phillips, Week-End, 1932, describes two currencies, each valued at 90% of the other and cites New Statesman and Nation, late 1931, NYS. The Week-End Book, 1924 [no relation to the Phillips book], says the US & Mexico each valued the other's currency at 29/30. 7.AJ. Multiplying by Shifting. E.g. find abc...mn such that nabc...m is twice as QUERIES - p.22 big. I've recently seen this when the second was 3/2 times the first, given by Bronowski in 1949. Dickson's first entry for any form of the problem is Hausted, 1878, but special cases occur in Babbage's MSS, c1820, and in 1854. A version is in Hoffmann (1893). 7.AK. Lazy Worker. This is in Al-Buzajani (NYS), al-Karkhi, At-Tabari (NYS), Fibonacci. 7.AL. If A is B, What is C? Fibonacci discusses the simplest form of this in some detail. More complex forms appear in Hoffmann (1893), Pearson (1907), etc. 7.AM. Crossnumber Puzzles. When do these originate? Dudeney gives examples in 1926 and 1932. I also have a 1927 version. Hubert Phillips gives three simple ones in Brush (1936). The 'Dog's Mead Puzzle' or 'Little Pigley' or 'Little Pigsby' has various dates involved in it - I have seen 1935, 1936, 1939 and an attribution to Michael N. Dorey, but my earliest source is 1940 and I have no reference to an original location. 7.AN. Three Odds Make an Even, etc. Alcuin describes this as a fable. Numerous forms appear from 1694 onward 7.AP. Knowing Sum vs Knowing Product. I have a version from 1940, but then 1987. 7.AQ. Numbers in Alphabetic Order. When does this first appear? I learned it in college, c1957. 7.AR. 1089. The earliest version seems to be 1890, but all examples used money until 1898. 7.AS. Cigarette Butts. New section - I have just realised that if b butts are needed to make a smoke, then B butts produce [(B-1)/(b-1)] smokes. 7.AT. Bookworm's Distance. New section - me earliest version is 1914. 7.AV. How Long to Strike Twelve? New section - my earliest version is 1927.

8. PROBABILITY RECREATIONS

8.B. Birthday Paradox. Feller cites von Mises (1938-39), but von Mises gets the expected number of repetitions, not the usual result. Ball, MRE (11th ed., 1939) cites Davenport, but Coxeter says that Davenport did not publish anything on it and others, including Mrs Davenport, say that Davenport explicitly denied originality for it. However, George Tyson, who was a student at Manchester a year ahead of Davenport, says Davenport showed it to him in 1927 and with an attitude that made Tyson believe it was due to Davenport. 8.C. Probability That a Triangle is Acute. Lewis Carroll gives a version in Pillow Problems (1893). Sylvester (1865) attributes a version to Woolhouse, NYS. 8.D. Attempts to Modify Boy/Girl Ratio. Laplace discusses this in his Essai Philosophique, c1819. 8.E. St. Petersburg Paradox. Euler says this originates in a letter from N. Bernoulli to Montmort on 9 Sep 1713. I have a reference to Montmort's publishing it in his book, but I had trouble finding it. Tissandier (1880?) and Dan Pedoe give vague references to D. Bernoulli publishing it in the Transactions [sic] of the St. Petersburg Academy or other unknown journals and to Buffon playing the game 2048 times. I recall hearing something on this in one of Tom Stoppard's plays, but I can't find it in Rosencrantz & Guildenstern are Dead nor in Jumpers? 8.F. Problem of Points. This is in Pacioli's Summa and Tartaglia. Ore says he has seen it in Italian MSS back to 1380 and believes it may be Arabic. 8.G. Probability that Three Lengths Form a Triangle. Fourrey, Curiosities Geom. (1907), cites Lemoine (1872-73). There were a number of following papers, some of which I have seen. 8.H. Probability Paradoxes. I have examples deriving from Bertrand (1889), QUERIES - p.23 including vague reference to Weaver and Whitehead. 8.I. Taking the Next Train. My first example is 1940.

9. LOGICAL RECREATIONS

9.A. All Cretans are Liars, Etc. Bochenski's Ancient Formal Logic and History of Formal Logic give details of the classical versions. Cervantes gives the 'I will be hanged' version in Don Quixote (1605). What are the origins of other versions, e.g. the barber? Whitehead and Russell discuss several of the paradoxes in Principia Math, vol. 1, pp. 63-64. I have a reference to Descremps' Les Petits Aventures de Jerome Sharp, Brussels, 1769 for the sentinel paradox. Can anyone help with this? An oriental version is the following: The Chief Warden swallows an elixir of immortality which he was supposed to convey to the Prince. The Prince orders the Warden's execution, but the Warden argues that if the execution succeeds, then the elixir was false and he is innocent of crime. The Prince pardons him. There are two versions which may be -2C. 9.B. Smith-Jones-Robinson. This is in Dudeney's Puzzles and Curious Problems, 1932. Gardner's note in 536 says Dudeney invented it. Schaaf recalls it appearing about 1926, so it probably was in Dudeney's Strand column. 9.C. Forty Unfaithful Wives. This occurs in Gamow and Stern, 'communicated by V. Ambarzuminian'. I now realise that it is a form of the 'Spots on foreheads' problem (9.D). 9.D. Spots on Foreheads. This appears in Kraitchik's Mathematical Recreations, but not in his earlier Math. des Jeux. There is a predecessor of the spots problem which has several persons in a railway carriage getting smudged in a tunnel and then laughing at each other, until they realise. I find this in the AMM 42 (1935) 256 and the solution says it was originated by 'Dr. Church of Princeton' and refers to a three person version in SSM 35 (1935) 429. Phillips, Week-End, 1932, gives the problem of two boys, one getting smudged and the other going to wash his own face. There is an ancestral version from 1903 (or 1883). Phillips, Brush, 1936, gives the three spots problem. A book of Arabic stories gives the three spots problem with the simpler solution based on symmetry. The author says he heard it in Palestine before 1948. Littlewood (1953) seems to be the first to note that the problem can be extended to n people, i.e. section 9.C. 9.E. Strange Families. Widower and son marry widow and daughter; widows and sons marry; are in the Bede version of Alcuin. Ripley's Believe It Or Not, 6th Series, reports a widower and son marrying a girl and her widowed mother. Bombaugh (1905) reports two widowers marrying each other's daughter. More real examples would be welcome. 9.E.1. That Man's Father is My Father's Son. New section - my earliest example is 1883, but similar problems occur back to 1844. 9.F. Unexpected Hanging. N. Falletta (1983) relates that this occurred as a Swedish wartime radio announcement and the Lennart Ekbom noticed its paradoxical nature and discussed it with students. Did he ever publish anything on it?? 9.G. Truthtellers and Liars. This is in Kraitchik's Mathematical Recreations (1942), but not in Math. des Jeux (1930). Phillips gives problems with truthtellers, liars and alternators in 1937 and 1945. In the mid to late 19C, there were complex problems with several persons with known probabilities of lying. 9.H. Prisoner's Dilemma. This is attributed to Merrill M. Flood and Melvin Dresher, c1950, but I have no reference. When was it first published? A. W. Tucker is said to have formalised it or invented it. I find anticipations of it in Babbage (1832) and F. W. Taylor (1911). 9.J. Fallen Signpost. I have this from 1927, where it is considered one of the best QUERIES - p.24

100 puzzles.

10. PHYSICAL RECREATIONS

10.A. Overtaking and Meeting Problems. There are many versions of these. Smith, History, II, gives some. There are several in the Chiu Chang Suan Shu (c-1C), some of considerable complexity. Several versions occur in the Bakhshali MS. 10.A.2. Number of Buses Met. New section. Mittenzwey, 1879, has a version. Proctor does it in 1883. Carroll (1885) gives some complex examples. A version is in Pearson (1907). 10.A.4. The Early Commuter. This is in a collection of best puzzles in 1943. 10.B. Fly Between Trains. This appears, as moderately well known, in a letter of G. H. Hardy in 1924. It appears in Dudeney, PCP (1932), and he has several versions in Strand Mag. for 1919, 1924? and 1925. In the MAA film John von Neumann (1966), Eugene Wigner relates this as being posed by Max Born to von Neumann, involving a swallow flying between two bicyclists. Wigner says it was popular in the 1920s. H&S (1927) says it is 'modern'. However, I have found it in Laisant (1906). 10.D. Mirror Reversal Paradox. That is, why does it reverse right and left but not up and down? This interested Carroll, who used to pose it to his girl friends. This is in Dudeney, PCP (1932). Gardner, Ambidextrous Universe, doesn't give an origin. 10.E.2. One Wheel Rolling Around Another. My earliest puzzle version in 1907, but there must be earlier examples since the phenomenon has been known in astronomy for many centuries. 10.E.3. Hunter and Squirrel. I have material from 1883. 10.F. Floating Body Problems. Gamow & Stern give the barge full of iron in a lock. Dudeney gives it in Strand Mag. (1931). What happens to a glass full of ice water as the ice melts - I have found this in the Daily Mail (3 Feb 1905). Ackermann has a barge going over a canal bridge - is there more weight on the bridge? I may add the question of the heavy lorry load of birds - can anyone supply early references? Perelman has a version: which is heavier, a bucket full of water or a bucket full to the same level with a piece of wood floating in it? Again, I don't know the original dates of Perelman's works. 10.G. Motion in a Current or Wind. Loyd's Cyclopedia has an against and with the wind problem. Sanford quotes Nicholas Pike (1788) and says it is the earliest version of the problem. There are many versions of this problem and I have few references. What is the origin of up & down a stream versus across & back? Did it first occur as a recreational or arithmetic problem or did it first arise in studying light and the ether? 10.H. Snail Climbing Out of Well. Kraitchik, Math. des Jeux, says this is a Hindu problem, but the Hindu versions don't really involve alternation of rates nor the end effect. Mahavira has a snake crawling into a hole while growing and other problems of multiple rates, but doesn't deal with the end effect due to alternation. Dell'Abbaco and the Columbia Algorism have the first proper versions. Chuquet has several proper versions. It is in Calandri's Arithmetic of 1491, but he doesn't note the end effect. 10.I. Two Men and a Bike, etc. Loyd gives a version with three men and a tandem. Dudeney, 1932, has 12 soldiers and a taxi which can take 4 of them. I have a 1939 version of two men and a bike. I have now found this in Laisant, 1906. I have developed a symmetric approach which solves the problem neatly. 10.J. Resistor Networks. AMM solution E2620 (1978) says Coxeter, Regular Polytopes, was first to give the cubical case, but I can't find it in the 2nd ed. This problem, QUERIES - p.25

E2620, appears to be the first to consider the regular polyhedra, though it only does between furthest points. It also says Gardner (SA Dec 1958 = 2nd Book, p. 22) gives the other polyhedral cases, but Gardner only considers the cube and cites Trigg, MM (Nov-Dec 1960) who cites Brooks & Poyser, Magnetism and Electricity, 1920, NYS. Any idea of the source of the infinite square lattice problem? It appears in Eureka (1951) where the resistances from (0,0) to (0,1) and (1,1) are wanted. Two solvers of SIAM Review prob. 79-16, 21 (1979) 559, obtained the resistance between points distance i apart in the n-dimensional cubical lattice, in terms of an n-fold integral. Van der Pol & Bremmer's book on the Laplace transform obtains a general solution. 10.K. Problem of the Date Line. Oresme proposed this c1355. This is in an almanac of 1493, then in Cardan (1539). 10.L. Falling Down a Hole Through the Earth. Ozanam (1725) discusses this. Ozanam-Riddle (1840) computes the time to reach the centre. Gardner's Annotated Alice says this problem interested Plutarch, Bacon and Voltaire (references??) and that it was resolved by Galileo. I have recently seen an argument that g increases with depth for a while due to the non-uniform density of the earth and this is said to be a late 19C result. 10.M. Celts = Rattlebacks. I have just seen a paper by G. T. Walker (Proc. Camb. Phil. Soc. 8 (1892/5) 305-306) and he refers to 'celts' as though the term and the object were well known. Celt was a common term for a hand-axe or chisel by early 19C, but when did it get associated with the rattleback? 10.M.1. Tippee Tops. This was patented in Germany in 1891, but appears to have been a fad in 1953 - can anyone provide information on this? 10.N. Ship's Ladder in Rising Tide. Phillips gives this in Brush, 1936. 10.O. Erroneous Averaging of Rates. My earliest example is 1941, but this is closely related to the older 7.Y. 10.P. False Balance. Cardan has a version, 1550, and there are versions from 1582, 1624, 1694. Determining the true weight by weighing on each side is in Ozanam-Hutton (1803) and Jackson, 1821.. 10.Q. Push a Bicycle Pedal. My first example is Pearson 1907). 10.R. Clock Problems. There are many 20C versions. My earliest version is Ozanam (1725), who asks when the hands meet. 10.W. Puzzle Vessels. I'd like details of ancient examples. 10.X. How Far Does a Phonograph Needle Travel? New Section - I only have a 1984 example.

11. TOPOLOGICAL RECREATIONS

11.A,B,C,D. Scissors on String, Two People Joined at Wrists, String Puzzles. These appear in Ozanam (1725), though the 1692 ed. and the 1708 English ed. do not have these plates. Alberti (1747) copies the plates. A number of these puzzles appear in the Catel and Bestelmeier Catalogues of 1790 & 1801-1803. 11.E. Loyd's Trick Pencil (on a short loop of string). Any idea of the date of this? Gardner (SA (Nov 1971) = Wheels, Chap. 12) says Loyd invented it for the New York Life Insurance Company. Loyd's account of this is given in the Delineator (April 1911). I have a 1910 description. 11.G. Trick Purses. There are examples in van Etten and Ozanam (1725). 11.H. Removing Waistcoat without Removing Coat. This is in Cassell's Book, ..., 1881. 11.H.1. Removing Loop from Arm. I have this from 1850, then 1930s? 11.I. Loop Puzzles. Cardan has two versions of the Alliance or Victoria puzzle. 11.J. Möbius Strip. I haven't yet got references to Möbius and Listing. My earliest QUERIES - p.26 usage as an amusement or puzzle is Tissandier (1888). I am interested in modern applications - in ink ribbon cartridges; as a low inductance resistor; as molecules. Can anyone provide details? 11.K. Wire Puzzles. The first books that I have seen which show many of these are Hoffmann (1893) and The Boy's Own Book of Indoor Games and Recreations, c1912. These are supposed to have been popular in China, but I have no specific sources. 11.K.4. Puzzle Rings. Any information on the origins would be interesting. 11.L. Jacob's Ladder and Other Hinging Devices. See also 6.X. I have a picture of a c1850 example 11.M. Puzzle Boxes. I have examples from the Catel & Bestelmeier catalogues, 1790 & 1801.