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View This Volume's Front and Back Matter John McCleary John EXERCISES IN (Mathematical) STYLE 10.1090/nml/050 Exercises in (Mathematical) Style Stories of Binomial Coefficients Originally published by The Mathematical Association of America, 2017. ISBN: 978-1-4704-4783-0 LCCN: 2017939372 Copyright c 2017, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2018 The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 23 22 21 20 19 18 Exercises in (Mathematical) Style Stories of Binomial Coefficients John McCleary Vassar College Providence, Rhode Island Council on Publications and Communications Jennifer J. Quinn, Chair Anneli Lax New Mathematical Library Editorial Board Karen Saxe, Editor Timothy G. Feeman John H. McCleary Katharine Ott Katherine Socha James S. Tanton Jennifer M. Wilson ANNELI LAX NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F. Beckenbach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinoff 5. The Contest Problem Book I Annual High School Mathematics Examinations 1950–1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by P. J. Davis 7. Uses of Infinity by Leo Zippin 8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. Olds 10. Replaced by NML-34 11. Hungarian Problem Books I and II, Based on the Eötvös Competitions 12. 1894–1905 and 1906–1928, translated by E. Rapaport 13.o Episodes from the Early History of Mathematics by A. Aaboe 14. Groups and Their Graphs by E. Grossman and W. Magnus 15. The Mathematics of Choice by Ivan Niven 16. From Pythagoras to Einstein by K. O. Friedrichs 17. The Contest Problem Book II Annual High School Mathematics Examinations 1961–1965. Compiled and with solutions by Charles T. Salkind 18. First Concepts of Topology by W. G. Chinn and N. E. Steenrod 19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations II by I. M. Yaglom, translated by A. Shields 22. Elementary Cryptanalysis by Abraham Sinkov, revised and updated by Todd Feil 23. Ingenuity in Mathematics by Ross Honsberger 24. Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book III Annual High School Mathematics Examinations 1966–1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26. Mathematical Methods in Science by George Pólya 27. International Mathematical Olympiads—1959–1977. Compiled and with solutions by S. L. Greitzer 28. The Mathematics of Games and Gambling, Second Edition by Edward W. Packel 29. The Contest Problem Book IV Annual High School Mathematics Examinations 1973–1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione, and N. Shell 30. The Role of Mathematics in Science by M. M. Schiffer and L. Bowden 31. International Mathematical Olympiads 1978–1985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin 32. Riddles of the Sphinx by Martin Gardner 33. U.S.A. Mathematical Olympiads 1972–1986. Compiled and with solutions by Murray S. Klamkin 34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson 35. Exploring Mathematics with Your Computer by Arthur Engel 36. Game Theory and Strategy by Philip D. Straffin, Jr. 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger 38. The Contest Problem Book V American High School Mathematics Examinations and American Invitational Mathematics Examinations 1983–1988. Compiled and augmented by George Berzsenyi and Stephen B. Maurer 39. Over and Over Again by Gengzhe Chang and Thomas W. Sederberg 40. The Contest Problem Book VI American High School Mathematics Examinations 1989–1994. Compiled and augmented by Leo J. Schneider 41. The Geometry of Numbers by C. D. Olds, Anneli Lax, and Giuliana P. Davidoff 42. Hungarian Problem Book III, Based on the Eötvös Competitions 1929–1943, translated by Andy Liu 43. Mathematical Miniatures by Svetoslav Savchev and Titu Andreescu 44. Geometric Transformations IV by I. M. Yaglom, translated by A. Shenitzer 45. When Life is Linear: from computer graphics to bracketology by Tim Chartier 46. The Riemann Hypothesis: A Million Dollar Problem by Roland van der Veen and Jan van de Craats 47. Portal through Mathematics: Journey to Advanced Thinking by Oleg A. Ivanov. Translated by Robert G. Burns. 48. Exercises in (Mathematical) Style: Stories of Binomial Coefficients by John McCleary Other titles in preparation. Preface . it is extraordinary how fertile in properties the triangle is. Every- one can try his hand. Treatise on the Arithmetical Triangle, Blaise Pascal At the age of twenty-one, he wrote a treatise upon the binomial theorem, which has had a European vogue. On the strength of it he won the mathematical chair at one of our smaller universities, and had, to all appearances, a most brilliant career before him. (Sherlock Holmes about James Moriarty) The Final Problem, Sir Arthur Conan Doyle Who has anything new to say about the binomial theorem at this late date? At any rate, I am certainly not the man to know. (Moriarty to Watson) The Seven-Per-Cent Solution, Nicholas Meyer . every style of reasoning introduces a great many novelties, in- cluding new types of (i) objects, (ii) evidence, (iii) sentences, new ways of being a candidate for truth or falsehood, (iv) laws, or at any rate modalities, (v) possibilities. One will also notice, on occasion, new types of classification, and new types of explanations. Ian Hacking According to the Oxford English Dictionary, a pastiche is a work that in- corporates several different styles, drawn from a variety of sources, in the style of someone else. In this book I am consciously imitating the work Exercices de Style (Éditions Gallimard, Paris, 1947) of Raymond Queneau (1903–1976). In a page, Queneau introduces a banal story of a man on a bus that he then tells in 99 different ways. The manners of speech include philosophical, metaphorical, onomatopoeic, telegraphic; also the forms of an advertisement, a short play, a libretto, a tanka; in passive voice, imperfect tense; and so on. He celebrates the potential of language, and his experiments in style, even the most demanding ones to read, seek to delight the reader with new forms. vii viii Preface Queneau was no stranger to mathematics, having published a note in the Comptes Rendus and a paper in the Journal of Combinatorial Theory (both on s-additive sequences). He also published Cent mille milliards de poèmes (100,000,000,000,000 Poems1), consisting of a set of ten sonnets with a uni- fied rhyme scheme from which a poem is constructed by making a choice of each line from among the corresponding lines in the ten sonnets. This scheme produces the 1014 distinct poems. The poet estimated that it would require more than 190 million years to recite them all. Queneau was also the cofounder in 1960 of the experimental writing group OuLiPo (Ouvroir de Littérature Potentielle). From their earliest meetings the OuLiPo discussed the use of mathematical structures in literary endeavors. When strict rules (constraints) are chosen and followed, a writer needs to find new modes of expression. By focusing on new operations applied to literary works or generating new works, writing can become freer and more thorough, a labo- ratory of invention. The canonical example is La Disparation by George Perec (1936–1982), Denoël, Paris, 1969, a novel in which the vowel e has vanished. Such a text is called a lipogram. Mathematicians and authors well known to the mathematical community have been members of the OuLiPo since its origin, including cofounder François Le Lionnais, Claude Berge, and Michèle Audin. So what have I tried to do? I have chosen a (not at all) banal object in mathematics—the binomial coefficients—and written 99 short notes in which these numbers (or their cousins) play some role. Certain particular properties of these numbers appear in different exercises, seen from different viewpoints. The behavior of the binomial coefficients provides a framework that suggests properties of related families of numbers. The binomial coefficients even appear as a sort of character in a mathematical story, or they structure the choices in a piece. The temptation to write a mathematical pastiche of Exercices de Style has been on my mind for a long time, and it has also been taken up by others: Ludmilla Duchêne and Agnès Leblanc have written the delightful Rationnel mon Q (Hermann, Paris, 2010) in which 65 exercises explore the irrationality of the square root of 2. What is style? According to Wikipedia, style is a manner of doing or pre- senting something. In mathematics, there may be an algebraic way to under- stand certain results, and there also may be a combinatorial way to understand the same fact. Thus viewpoint contributes to style. Other aspects of style might include the method of a proof, be it induction, recursion, or algorithm; or by contradiction, by the introduction of a fancy zero, or a conscious imitation of the proof of a different fact. On the other hand, the presentation of a mathemat- ical proof might take a particular form of discourse. 1 An English translation of Cent mille milliards de poèmes can be found in Oulipo Compendium, compiled by Harry Mathews and Alastair Brotchie.
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