Mathematical Circles (Russian Experience) PDF
Preview Mathematical Circles (Russian Experience)
Mathematical World • Volume 1 Mathematical Circles (Russian Experience) Dmitri Fomin Sergey Genkin Ilia ltenberg American Mathematical Society Selected Titles in This Series Dmitri Fomin, Sergey Genkin, and Ilia ltenberg, Mathematical circles (Russian experience), 1996 David W. Farmer and Theodore B. Stanford, Knots and surfaces: A guide to discovering mathematics, 1996 David W. Farmer, Groups and symmetry: A guide to discovering mathematics, 1996 V. V. Prasolov, Intuitive topology, 1995 L. E. Sadovskil' and A. L. SadovskiX, Mathematics and sports, 1993 Yu. A. Shashkin, Fixed points, 1991 V. M. Tikhomirov, Stories about maxima and minima, 1990 Mathematical World • Volume 7 Mathematical Circles (Russian Experience) Dmitri Fomin Sergey Genkin Ilia ltenberg Translated from the Russian by Mark Saul American Mathematical Society C.A.rEHKHH,H.B.HTEHBEP~ll.B.~OMHH MATEMATHqECKHH KPYmO:K CAHKT-IIETEPBYPr 1992, 1993 Translated from the' Russian by Mark Saul 2000 Mathematics Subject Classification. Primary OOAOB; Secondary OOA07. ABSTRACT. This book is intended for students and teachers who love mathematics and want to study its various branches beyond the limits of school curriculum. It is also a book of mathematica1 recreations, and at the same time a book containing vast theoretical and problem material in some areas of what authors consider to be 11extracurricular mathematics". The book is based on an experience gained by several generations of Russian educators and scholars. Library of Congress Cataloging-in-Publication Data Genkin, S. A. (Sergei Aleksandrovich) !Matematicheskii kruzhok. English) Mathematical circles : (Russian experience) / Dmitri Fomin, Sergey Genkin, Ilia Itenberg; translated from the Russian by Mark Saul. p. cm. - (Mathematical world, ISSN 1055-9426; v. 7) On Russian ed., Genkin's name appears first on t.p. Includes bibliographical references (p. - ). ISBN 0.8216-0430.8 (alk. paper) 1. Mathematical recreations. I. Fomin, D. V. (Dmitri! Vladimirovich) II. Itenberg, I. V. (Il'fa. Vladimirovich) III. Title. IV. Series. QA95.G3813 1996 510'. 76-<lc20 96-17683 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionClams, org. © Copyright 1996 by the American Mathematical Society. Printed in the United States of America. 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 URL: http://vvw.ams.org/ 109876543 070605040302 Contents Foreword vii Preface to the Russian Edition ix Part I. The First Year of Education Chapter 0. Chapter Zero Chapter l. Parity 5 Chapter 2. Combinatorics-1 11 Chapter 3. Divisibility and Remainders 19 Chapter 4. The Pigeon Hole Principle 31 Chapter 5. Graphs-I 39 Chapter 6. The Triangle Inequality 51 Chapter 7. Games 57 Chapter 8. Problems for the First Year 65 Part II. The Second Year of Education Chapter 9. Induction (by I. S. Rubanov) 77 Chapter 10. Divisibility-2: Congruence and Diophantine Equations 95 Chapter 11. Combinatorics-2 107 Chapter 12. Invaxiants 123 Chapter 13. Graphs-2 135 Chapter 14. Geometry 153 Chapter 15. Number Bases 167 Chapter 16. Inequalities 175 Chapter 17. Problems for the Second Year 187 vi CONTENTS Appendix A. Mathematical Contests 201 Appendix B. Answers, Hints, Solutions 211 Appendix C. References 269 Foreword This is not a textbook. It is not a contest booklet. It is not a set of lessons for classroom instruction. It does not give a series of projects for students, nor does it offer a development of parts of mathematics for self-instruction. So what kind of book is this? It is a book produced by a remarkable cul tural circumstance, which fostered the creation of groups of students, teachers, and mathematicians, called mathematical circles, in the former Soviet Union. It is pred icated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport, without necessarily being competitive. Thus it is more like a book of mathematical recreations-except that it is more serious. Written by research mathematicians holding university appointments, it is the result of these same mathematicians' years of experience with groups of high school students. The sequences of problems are structured so that virtually any student can tackle the first few examples. Yet the same principles of problem solving developed in the early stages make possible the solution of extremely challenging problems later on. In between, there are problems for every level of interest or ability. The mathematical circles of the former Soviet Union, and particularly of Lenin grad (now St. Petersburg, where these problems were developed) are quite different from most math clubs in the United States. Typically, they were run not by teach ers, but by graduate students or faculty members at a university, who considered it part of their professional duty to show younger students the joys of mathemat ics. Students often met far into the night, and went on weekend trips or summer 4 retreats together, achieving a closeness and mutual support usually reserved in our country for members of athletic teams. We are fortunate to be living in a time when Russians and Americans can easily communicate and share their cultures. The development of mathematics education is an aspect of Russian culture from which we have much to learn. It is still very rare to find research mathematicians in America willing to devote time, energy, and thought to the development of materials for high school students. So we must borrow from our Russian colleagues. The present book is the result of such borrowings. Some chapters, such as the one on the triangle inequality, can be used directly in American classrooms, to supplement the development in the usual textbooks. Others, such as the discussion of graph theory, stretch the curriculum with gems of mathematics which are not usually touched on in the classroom. Still others, such as the chapter on games, offer a rich source of extra-curricular materials with more structure and meaning than many. Each chapter gives examples of mathematical methods in some of their barest forms. A game of nim, which can be enjoyed and even analyzed by a third grader, vii viii FOREWORD turns out to be the same as a game played with a single pawn on a chessboard. This becomes a lesson for seventh graders in restating problems, then offers an introduction to the nature of isomorphism for the high school student. The Pigeon Hole Principle, among the simplest yet most profound mathematics has to offer, becomes a tool for proof in number theory and geometry. Yet the tone of the work remains light. The chapter on combinatorics does not require an understanding of generating functions or mathematical induction. The problems in graph theory, too, remain on the surface of this important branch of mathematics. The approach to each topic lends itself to mind play, not weighty reflection. And yet the work manages to strike some deep notes. It is this quality of the work which the mathematicians of the former Soviet Union developed to a high art. The exposition of mathematics, and not just its development, became a part of the Russian mathematician's work. This book is thus part of a literary genre which remains largely undeveloped in the English language. Mark Saul, Ph.D. Bronxville Schools Bronxville, New York
The list of books you might like
