A bibliography of recreational mathematics (5 vols.)

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Ives Problem. Herodotus attributed them to the Lydians in the reign of Atys. Date uncertain - cf Cf The earliest MS versions are 9C. Verse mnemonics already exist by mid 12C. The word 'mate' is recorded in Latin before However, it is evident that these were the 'domino' cards still in use in China. By , the game has spread over most of Europe, reaching as far as Iceland, the Baltic and Bohemia. They are also recorded in 12C Arabia I've forgotten this source - it may refer to the following facts.

References and Resources

There is a fragment of a 12 or 13 C card and an almost complete early 15C deck from Egypt which show that the 52 card deck came to Europe from Egypt or thereabouts. John Scarne says there is an 11C card from Chinese Turkestan. Various works, including a poem about chess. It was one of the first books published by Caxton in Nicholai attrib. Possibly 13C. But I have a source that says cards were mentioned in , that they are mentioned in German MSS of to and were used in Itlay in Within a short time, they are widespread in Europe, but they are not mentioned in several lists of games of the previous decade.

They are also not mentioned in the general literature before this time, even by authors such as Petrarch, Boccaccio and Chaucer with an interest in games. A Paris ordinance regulating gaming in makes no mention of cards, but the equivalent ordinance of mentions them. By , cards are recorded in Florence, Basel, Regensburg, Brabant, Paris and Barcelona, and several of the records describe cards as new or having arrived this year.

Calandri: Tractato d'Abbacho. This formulation is odd if there had been two different authors. For a complete bibliography of the editions by Jacques Ozanam and the English translations, see David Singmaster, The bibliography, pp. The correct reference is Nicolas Hunt, Iudiciary exercises, or Practicall conclusions Whereby any one of meane capacitie, may readily and infallibly finde out the Christian names of men and women, their titles of honour, ages, offices, trades or callings of life, places of birth, houses of residence appertaining to scholars, either in the vniversities of Oxford or Cambridge, or the Innes of Court and Chauncerie.

With many other things both pleasant and profitable.

By Nicolas Hunt Master of Arts. London : Printed by Aug. Math[hewes] for Luke Faune and are to be sold at the great noth [sic] doore of Saint Pauls, There is no indication in the book who was responsible for the translation. The addition of this part is repeated on the front page which gives the false impression that William Oughtred was responsible for the translation. However, this is very unlikely as Hall has persuasively shown Malthus indeed is the most plausible candidate for the translation.

My belief is based on some details in the third part on fireworks. In Malthus published a work on fireworks, with some basic arithmetic, geometry and a part on fortifications, in the same year in French in Paris and English in London The strongest support for Malthus as the translator is the fact that he added some copper plate illustrations from his own book which are not in the French source The Dutch editions have been translated by Wynant van Westen, organ player and mathematician from the city of Nijmegen.

A bibliographic search located seven editions published between and in Arnhem and later Amsterdam In fauour of mathematicall students. Reprinted in , , and Me Chyrurg[ie]n. More about this in section 5. I compared Robert Norton, The gunner shevving the vvhole practise of artillerie: vvith all the appurtenances therevnto belonging. Together with the making of extra-ordinary artificiall fireworkes, as well for pleasure and triumphes, as for warre and seruice. VVritten by Robert Norton, one of his Maiesties gunners and enginiers. London: Printed by A[ugustine] M[athewes] for Humphrey Robinson, and are to be sold at the three Pidgeons in Paules-Churchyard, with Hanzelet and Thybourel, , but could not find any common illustrations.

The and editions have added to the confusion with different years of publication on the front pages of the different parts. The is not listed in William L. A Supplement to the Lists in William L.


Schaaf's A Bibliography of Recreational Mathematics. Collected by William L. Schaaf; typed and annotated by David Singmaster. Van Westen reproduces the woodcuts from the French edition including the original numbering, which does not correspond with his own. An in depth comparison of the illustrations suggest that the first Dutch translation is based on the Lyon edition.

Subsequent editions from include also the second and third part. The Latin translation, known as Thaumaturgus mathematicus, is from Caspar Ens and was published in Munich in and , to be followed by a Venice edition in Problems from the second part are included but not the third part on fireworks. Almost problems can be matched between the two works.

As usual in such cases it is very hard to trace the original source for the attribution. Singmaster mistakenly lists Arnhem as the first edition, and gives two entries for the same book of weduw' Loots-Man, Amsterdam , while there is only one. Leipzig, Teubner, , Gaspar Schott in was responsible for this claim, as cited in Singmaster, The bibliography, p. Alegambe took responsibility for a new edition of the Bibliotheca Scriptorum Societatus Jesu, originally started by Pedro de Ribadeneira in He provides no further explanation and does not mention van Etten.

This is the only evidence we have about the claimed authorship of Leurechon. Illustrium scriptorum religionis Societatis Jesu catalogus. Auctore P. Petro Ribadeneira, Antwerp: apud J. Moretum, , cited in Hall, Old Conjuring Books, p. Opus inchoatum a R. Petro Ribadeneira Eiusdem Societatis Theologo, anno salutis Continuatum a R. Philippo Alegambe A Nathanaele Sotuello Rome: ex typographia Iacobi Antonij de Lazzaris Varesij, XXXVI, pp From till Leurechon was part of the Missio Castrensis in Brussels, which indicates he was then an army chaplain.

This is described in the Catalogus primus ac secundus personorem at the Belgian state archives Rijksarchief. He is further listed in the Admissi in Societatem ante Divisionem Provinciae Beligiciae, a manuscript stored at the Collegium Maximum in Louvain, and the Album novitiorum vol. II, 69, at the Royal Library in Brussels. To substantiate this claim he had to establish that van Etten was not just a pseudonym, as often stated, but a person that really existed and was related to Lambert Verreycken. Hall succeeded to some degree as he was able to trace a relationship between the van Etten and Verreycken families to a point of finding evidence of the marriage between Christophe van Etten and a daughter of Louis Verreycken More details can now be added.

As shown in figure 2 and 3, the arms used on the title page of the first editions are indeed from Verreycken. Lambert was the latest child of Louis Verrycken in a family of ten. He must have been born after and is known to have died in as a captain in the siege of s'Hertogenbosch in French: Bois-le-Duc His oldest sister Marie born Dec.

I did not find any evidence to corroborate the name Hendrik or Henri, as a son of Christophe. In Jacques Voignier published the results of his study on the book in an obscure periodical called The Perennial Mystics. He challenges the claim of Hall that van Etten is the real author and bases his argument on the Jesuit 46 Hall, Old Conjuring Books, p. Bruxelles: A. Bieleveld, , Vol. Gent: Gyselynck, , , Vol II, p. Guillaume Le Blond, The military engineer: or, a treatise on the attack and defence of all kinds of fortified places London : printed for J.

No printer would dare to use the jesuit emblem if not asked to do so by a jesuit. This argument can be contested. One needs to show that at least one of them is not from a Jesuit. But there is an easier way: the copy, which Voignier has not seen, does not include the emblem! Only the edition used the IHS- emblem as shown in figure 4. Alegambe lists precisely the edition which does not contain any reference to Jesuits and still atributes the work to Leurechon.

Emblem edition, Fol. But there is more: a copper plate used for a sun dial, currently at the Lorraine Museum in Nancy, is signed by Hanzelet and uses the IHS emblem An iconographical explanation for the signs is that Jesus symbolizes the sun, and Maria the day. His first engravings to be published in a book appeared in In he published a book together with Thybourel on military machines and fireworks His license is retracted and Hanzelet is fined 50 francs for printing a book by Hordal without his consent The fifth word is as follows I have used copper plate engravings for the most needed illustrations to clarify some propositions.

I have done so rather than using the more expedient woodcuts where they could have been put on their proper place; nonetheless, using numbering to overcome this minor inconvenience. This is clearly the word of the master engraver Jean Appier Hanzelet and not of a Jesuit mathematician or his year old student. Indeed subsequent editions, starting with Lyon edition had the woodcut illustrations inside the text.

Given his credentials there is little doubt that the first five parts on artillery and fireworks are by Hanzelet.

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The surgeon Thybourel could have contributed with the sixth and seventh book on secret writing. The separate title pages list Thybourel first, but the frontispiece of the book puts Hanzelet first. Bernard, Authore Joanne hordal I. Almae Universitatis Pontimussanae ordinario professore Pontimussi: apud J. Appier Hanzel, This foreword is not translated into Latin, or Dutch, but does appear in the English edition with the exception of the quoted citation.

The reason that this important quote has been overlooked by many, might be because it only appears in these early editions. This rather technical defense for the use of copper plates instead of wood cuts is likely to be one from the printer engraver himself. Finally, there is the expanded edition of , adding two parts to an already succesful work. In the third book he describes several contrivances to lift a heavy canon by one or two persons While the inclusion of a treaty on fireworks in a book on recreational mathematics might seem a little odd to some, the fact that Hanzelet is the author makes it fit together.

The 15 chapters in between have the same title and mostly the same text as the Receuil. All the copper plate illustrations from are included as wood cuts, with the addition of a new one in the last chapter. Why would Hanzelet allow his book and precious illustrations from to be used if he was not involved?

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If he still would have held his printing license in he very likely would have printed the book himself, but now he had to make an agreement with Charles Osmont to get the book on the market. Expanding an already succesful work with a section, using previous work available, makes perfect sense. With a few exceptions, the direct source of all these problem can be attributed with poise to Problemes plaisants by Claude Bachet Aiiiv; Rouen, F. Several commentators claim that the author does not acknowledge his sources. Those have clearly not seen the early editions.

The English translator leaves out this passage and replaces the genuine sources by a preposterous long list of Greek philosophers, medieval and renaissance authors to end with ironically so Tyberill [Thybourel]. The fifth edition of was reprinted in and More significantly, there is textual evidence. Puis autre, et derechef multiplier, ou diviser multiplier le quotient par quelque autre, et par un autre, et ainsi tant que tu derechef multiplier, ou diviser par un voudras. Also Bachet misled his readers about the originality of his problems as he rarely mentions any of his sources Both Bachet and Leurechon relied heavily on the traditions of mercantile arithmetic and practical geometry which spread from Italy to France from the fifteenth century.

By the end of the nineteenth century several authors have shed light on the origins of the problems contained in these seventeenth-century works. Some years later Wilhelm Ahrens did make the attempt and was the first to give alternative sources for several problems of Bachet His Mathematische Unterhaltungen und Spiele could be considered as the first work on the history of recreational mathematics which took considerable care in mentioning the sources of previously published problems.

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Par D. References to Bachet on p. The foreword was written by Montucla. Also for [AU05] see p. Both references to Tartaglia were deleted in the following editions. Of monumental value for the history of many problems is JohannesTropfke, who gives both a chronology and a classification of recreational problems Johannes Tropfke, Geschichte der Elementarmathematik in systematischer Darstellung.

IV : Ebene Geometrie, Bd. VI : Analyse, analytische Geometrie, Bd. I : Rechnen, Bd. II : Algemeine Arithmetik, Bd. But also Bachet relied heavily on previous writers. These sources provide evidence for a long-lasting tradition of recreational mathematics through the Renaissance and Middle Ages, back to Arab, Hindu and Babylonian sources. The Columbia Algorismus [c. Although it is unlikely that Bachet had access to this manuscript or a related copy, the correspondance with many of the problems is significant. These are most of the number and permutation divinations, cistern problems, the two crossing problems, the ring game, the weights problems and the problems involving geometric progressions With the exception of the cistern problems and the geometric progressions, they also appear in Bachet.

Almost all of the other texts deal with recreational problems. Some include only a few problems as illustrations of arithmetical or algebraic rules, others exclusively list recreational problems. In any case they provide evidence of a continuous tradition of recreational mathematics throughout the Middle Ages. Lutetiae Parisiorum: Sumptibus Sebastiani Cramoisy, An early study appeared in Elizabeth B.

Fiske ed.

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A complete transcription was made by Kurt Vogel, Ein italienisches Rechenbuch aus dem Problems will be referred to by their number in the English editions, between square brackets. Chuquet displays a profound knowledge of arithmetic and algebra which made this manuscript something of a rarity given the place and time in which it was conceived.

Only recently it became recognized that Chuquet connects with the Italian abacus tradition through Provencial intermediaries. Chuquet himself gives little cues, but at some point criticizes a certain Bartholomy de Romans, an unknown Dominican monk. Several other Provencial authors active during that period are listed by van Egmond The notebook of Francesco Bartoli, an Italian business man travelling between Italy and the south of France at the beginning of the fifteenth century, provides rare evidence of the transmission of recreational problems throughout Europe.

In addition to arithmetic tools such as exchange and multiplication tables, the itinerary from Florence to Avignon and price lists, it contains a collection of thirty problems of the recreational sort We can assume that Bartoli was only one of the many links in the trade routes by which the tradition of recreational mathematics was passed from Italy to France and the Low Countries.

Marre discovered that a printed work of by Estienne de la Roche, contained large fragments that were literally copied from the manuscript Large parts of the manuscript are translated into English by G. Flegg, C. Hay and B. Moss eds. The correspondence of problems from Bachet and Chuquet is conspicuously high. The provenance of the manuscript was traced by Marre. After de la Roche it came into the hands of the Italian Leonardo de Villa, only to return to France, in the library of Jean-Baptiste Colbert, long after the publications of Bachet As these problems also appear in many sixteenth-century arithmetic books, Bachet seems to have used several written sources, or was acquainted with these problems through oral dissemination.

Luca Pacioli [] Pacioli is famous for his Summa de arithmetica published in Venice in This book is one of the first printed books on mathematics. This encyclopedic work deals with many problems of arithmetic, geometry, trigonometry and algebra, which took Pacioli 20 years to collect. The work has clearly been written for practical purposes. Composed in Italian, it deals with practical problems of exchange, conversion of measures and double-entry bookkeeping.

The Summa is filled with large sections unscrupulously copied from manuscripts from earlier writers After working on the Summa, between and , Pacioli compiled a text with the title De Viribus Quantitatis. This three-part manuscript of pages is now preserved at the University Library of Bologna and has only recently been published. The first part, Delle forze numerali cioe de Arithmetica, contains recreational problems Folkerts and J.

Hogendijk, Vestigia mathematica, , Amsterdam: Rodopi, , p. Trascrizione di Maria Garlaschi Peirani dal Codice n. Dario Uri has pictures of the complete manuscript on-line and is in the process of doing the transcription. Pacioli writes in De Viribus that the sort of problems he considers are alike the kind that were discussed in public schools during that time.

It is possible that many of the problems in the first part of De Viribus were communicated within the oral tradition. Pacioli himself admits that he borrowed from Euclid, Boethius, Sacrobosco and Fibonacci. The major part of the problems that can be attributed to Pacioli are also treated in printed books that were accessible to Bachet: Cardan89, Tartaglia90 and Trenchant So the influence could have been indirect. Both problems have the character of parlour tricks and could have been part of the oral tradition.

The importance of the oral tradition became clear to me when going through the card tricks of Bachet together with someone who had surprised me before with his skills in this discipline. He could show me several variations and explain the arithmetic behind them. However, he was taught all this by friends and relatives and did not learn them from books.

So, the tradition in which recreational problems and parlour tricks were communicated before the age of printing continues to be part of our culture. Francesco di Leonardo Ghaligai [? The lessons were divided into seven parts or muta, covering subjects typically found in the arithmetic books of the early sixteenth century. Hoepli, , pp.

Milan: Imprimebat impensis Bernardini Calusci, Ensemble vn petit discours des changes. Auec l'art de calculer aux getons. Lyon, Jean Pillehotte, , reprinted in , , , and The most comprehensive historical overview of recreational problems is David Singmaster, Sources in Recreational Mathematics, An Annotated Bibliography. Eighth Preliminary Edition, unpublished, electronic copy from the author, However Singmaster does not mention any sources for this type of problem between and in section 7. Renouard et cle, , vol. II, p. Swetz, Capitalism and arithmetic : the new math of the 15th century, including the full text of the Treviso arithmetic of , translated by David Eugene Smith, La Salle: Open Court, , pp.

Solving recreational problems by arithmetical and algebraic methods were an integral part of the curriculum. Ghaligai published his Summa de arithmetica in in Florence. This book, written in Italian, is now very rare. It was reprinted in and as Pratica d'arithmetica. The book is divided into 13 chapters, treating practical subjects such as money exchange, the rules of fellowship and barter, as well as algebra and the works of Fibonacci and Euclid.

Chapters 9 and 13 contain several recreational problems. Eight problems from Bachet correspond with those by Ghaligai As there are alternative sources for each of these problems, it is unlikely that Ghaligai was a direct source for Bachet or Leurechon. He was a brilliant, critical, unconventional and controversial man. In his younger years he played chess, dice and card games for money, a practice that earned him good living thanks to his knowledge of probabilities.

He studied and practiced medicine but did not succeed in obtaining a post lecturing medicine until From onwards he wrote many books on mathematics, the most famous being Ars Magna in However, it is his first book, the Practica arithmetice et mensurandi singularis of , that is of most interest to us. Submit Search. Successfully reported this slideshow. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads.

You can change your ad preferences anytime. Upcoming SlideShare. Like this document? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Published in: Education , Technology. Full Name Comment goes here. Are you sure you want to Yes No. Alexis Lawson. Nurhidayah Mahmud.

No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Fraenkel, Aviezri. Fraenkel weizmann. The issues are published in the exact moments of the equinox. Contents Page Editorial. Occasionally, some papers that do not require any mathematical background will be published. Examples of topics to be addressed include: games and puzzles, problems, mathmagic, mathematics and arts, history of mathematics, math and fun with algorithms, reviews and news. Recreational mathematics focuses on insight, imagination and beauty.

His- torically, some areas of mathematics are strongly linked to recreational mathematics - probability, graph theory, number theory, etc. Thus, re- creational mathematics can also be very serious. While there are conferences related to the subject as the amazing Gathering for Gardner, and high-quality magazines that accept recreational papers as the American Mathematical Monthly, the number of initiatives related to this important subject is not large.

This context led us to launch this magazine. We will seek to provide a qua- lity publication. Ideas are at the core of mathematics, therefore, we will try to bring in focus amazing mathematical ideas. We seek sophistication, imagination and awe. Key-words: impossibility proofs, Marriage Theorem, matching problems.

Puzzle enthusiasts know that a really good puzzle is more than just a problem to solve. The very best problems and puzzles can provide insights that go beyond the original setting. Sometimes even classic puzzles can turn up something new and interesting. The way I present it here is slightly unusual - bear with me for a moment. So consider a chess board, and a set of dominos, each of which can cover exactly two squares. Figure 1: Standard Chessboard. Recreational Mathematics Magazine, Number 1, pp. The classic problem then asks - is it possible to cover the board when two op- posite corners are removed?

Insight strikes when if! Why is this important? The reason is simple. As we cover pairs of squares, we must cover the same number of blacks as whites. Figure 2: Mutilated Chessboard. If you remove any single black and single white and try to cover the remain- der it seems always to be possible.

Indeed, in mathematician Ralph E. Gomory [3] showed with a beautifully simple and elegant argument that it is Recreational Mathematics Magazine, Number 1, pp. What about if two whites and two blacks are removed? Can it always be covered? And so we continue. What if three whites and three blacks are removed still leaving the board connected. Can it then always be covered? Well, that was quick. There is a hint later [X]. So consider. The original setting is this. Suppose we have a collection of men and a collection of women, and each woman is acquainted with some of the men.

Under what circumstances is this possible? In another formulation, suppose we have a collection of food critics. Naturally enough, they all hate each other, and refuse to be in the same room as each other. In one case we are matching women with suitable men, in another we are matching critics to restaurants. So when can we do this? If we do successfully create such an ar- rangement then we know that every woman must know at least one man, namely, her husband. But we can go further. If a matching is possible, then any collection of, say, k women will collectively know their husbands and possibly more.

Later, Marshall Hall Jr. So how does this solve our problem with the mutilated chess board? A covering of dominos matches each white square with a black square, so we have two collections that we are trying to match. If some collection of white squares collectively are not attached to enough black squares, a covering is im- possible.

Therefore: To show that a mutilated chess board is coverable - cover it. We can now use this result to create a minimal, connected, uncoverable set of squares. If we used just one black square then it would have to be connected to no squares at all, and so it would be disconnected, so we must use at least two black squares. They must then be attached to just one white square. That means we need another black square. We now have a white square with three black neighbours, and so we need two more white squares to balance the numbers. Figure 4: Uncoverable. This is the unique shape.

It also lets us answer the question above about the mutilated chess board with three of each colour missing. Some pile will have an ace, possibly more than one.

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  • Some pile will have at least one deuce, and so on. So remove those cards to leave thirteen piles of three. Now we can do it all a second time. And a third time. Alternatively, deal the cards into four piles of thirteen. What may surprise you is that you can do it again, and again, and again, and so on, right through to the end. Thank you! References [1] Martin Gardner. My Best Mathematical and Logic Puzzles. Dover Publica- tions, Mathematical Gems I, Mathematical. Mathematical Asso- ciation of America, There are actually two distinct types. Gardner had managed to trace such puzzles back to Hooper in In , I was visiting Leipzig and reading Schwenter which referred to an error of Serlio, in his book of I will describe the history and some other versions of the idea.

    Key-words: Sam Loyd, vanishing area puzzles. Martin Gardner discussed this extensively in [7] and [8]. Recrational Mathematics Magazine, Number 1, pp.

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    These are considerably older than vanishing object puzzles. This article is primarily concerned with the early history of such puzzles. Figure 2: Schlomilch. In , I was visiting Leipzig and looked at some obscure books in their library and discovered references going back to The history of this particular version is obscure. David Singmaster 13 Figure 4: LoydCyc For a long time, I tended to ignore this as it seemed smudged.

    But when I went to scan Fig. Figure 6: AWGL. David Singmaster 15 The oldest known version of this is an actual puzzle, dated c, [1], shown in [18] and there is a publication [5]. Loyd would have been 17 at the time. If this is true, he is ten years before any other appearance of the area 65 puzzle and about 42 years before any other appearance of the area 63 puzzle. I am dubious about this as Loyd did not claim this as his invention in other places where he was describing his accomplishments. In , Sam Loyd Jr. For example, on p. In , Coxeter [3] said it was V. Schlegel, but he apparently con- fused this with another article on the problem by Schlegel.

    Taking a smaller example based on the numbers 2, 3, 5 makes the trickery clear. Figure 7: Nine to Ten. However, there are other versions of vanishing area or object puzzles. Since Recrational Mathematics Magazine, Number 1, pp. Figure 8: CreditSqueeze. Lennart Green uses a version of this in his magic shows, but he manages to reassemble it three times, getting an extra piece out each time!

    Needless to say, this involves further trickery. A version of this is available on the Internet. But there are earlier examples. Gardner and others tracked the idea back to Hooper [9] in , as seen in Fig. This was corrected in the second edition of and this version occurs fairly regularly in the century following Hooper. Sebastiano Serlio was born in Bologna and worked in Rome in with the architect Peruzzi. This was studied by Wallis, leading to a system of 25 linear equations for the Sheldonian roof.

    Figure 9: SerlioRot. Since that time, I have found two other late 18C examples, possibly predating Hooper. The Plan in what Manner the Plank must be cut and applied to the Table is required? Vyse is clearly unaware that area has been lost. By dividing all lengths by 3, one gets a version where one unit of area is lost. Note that 4, 8, 9 is almost a Pythagorean triple. I have not Recrational Mathematics Magazine, Number 1, pp. Like Serlio, the author is unaware that some area has vanished!

    The edition of Ozanam by Montucla [12] has an improvement on Hooper. Figure Ozanam. The image is Fig. Here just one unit of area is gained, instead of two units as in Hooper. He remarks that M. In conclusion, we have found that vanishing area puzzles are at least two hundred years older than Gardner had found. We have also found a number of new forms of the puzzle. Who knows what may turn up as we continue to examine old texts?

    I think Martin would have enjoyed these results. Described and shown in [10], p. Aldus, Venice, pp. Scripta Math, 19, pp. Messenger of Mathematics, 6, p.