Counting and Configurations: Problems in Combinatorics, Arithmetic, and Geometry (CMS Books in Mathematics) PDF
Preview Counting and Configurations: Problems in Combinatorics, Arithmetic, and Geometry (CMS Books in Mathematics)
Jiri Herman Radan Kucera Jaromir Simsa Translated by Karl Dilcher Counting and Configurations Problems in Combinatorics, Arithmetic, and Geometry A ba at ms "a" ea . Pp Y jj: Canadian Mathematical Society Société mathématiaue du Canada Canadian Mathematical Society Société mathématique du Canada Editors-in-Chief Rédacteurs-en-chef Jonathan Borwein Peter Borwein Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo CMS Books in Mathematics Ouvrages de mathématiques de la SMC 1 HERMAN/KUCERA/SIMSA_ Equations and Inequalities 2 ARNOLD Abelian Groups and Representations of Finite Partially Ordered Sets Ww BORWEIN/LEWIS Convex Analysis and Nonlinear Optimization DD LEviNLUBINSKY Orthogonal Polynomials for Exponential Weights HH I KANE Reflection Groups and Invariant Theory DH PHILLIPS Two Millennia of Mathematics O DEUTSCH Best Approximation in Inner Product Spaces FABIAN ET AL. Functional Analysis and Infinite-Dimensional Geometry oO KRiZEK/LUCA/SOMER 17 Lectures on Fermat Numbers 10 BORWEIN Computational Excursions in Analysis and Number Theory Pt REED/SALES Recent Advances in Algorithms and Combinatorics 12 HERMAN/KUCERA/SIMSA_ Counting and Configurations 13 NAZARETH Differentiable Optimization and Equation Solving 14 PHILLIPS Interpolation and Approximation by Polynomials Jiri Herman Radan Kuéera Jaromir SimSa Counting and Configurations Problems in Combinatorics, Arithmetic, and Geometry Translated by Karl Dilcher With 111 Figures 6 Springer isk Jiti Herman Radan Kuéera Brno Grammar School Department of Mathematics tf. Kpt. Jarode 14 Faculty of Science 658 70 Brno Masaryk University Czech Republic Jandtkovo nfm. 2a 662 95 Brno Czech Republic Jaromir Sim3a Translator Mathematical Institute Karl Dilcher Academy of Sciences of the Department of Mathematics Czech Republic and Statistics Zizkova 22 Dalhousie University 616 62 Brno Halifax, Nova Scotia B3H 3J5 Czech Republic Canada Editors-in-Chief Reédacteurs-en-chef Jonathan Borwein Peter Borwein Centre for Experimental and Constructive Mathematics Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 186 Canada [email protected] Mathematics Subject Classification (2000): 05-01, 11-01, 51-01, 97D50 Library of Congress Cataloging-in-Publication Data Herman, Jifi. [Metody resenim atematickych Woh n. English] Counting and i i and geometry /Jizi He rman, Radan Kutera, “Joro mir Sima; translated by Karl Dilcher. p.cm, (CMS books in mathematics; 12) Includes bibliographical references and index. ISBN 0-387-95552-6 (alk. paper) 1. Combinatorial geometry—Problems, exercises, etc. I. Kutera, Radan. I Sima, Jaromir. IU. Title. IV. Series. QA167 -H47 2002 511’.6—de21 2002026655 ISBN 0-387-95552-6 Printed on acid-free paper. © 2003 Springer-Verlag New York, Inc. Translated from the Czech Afetody fedeni matematickych toh I, by J. Herman, R. Kutera, and J. SimSa. Masaryk University: Brno, 1997. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc.. 175 Fifth Avenue, New York, NY 10010, USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 SPIN 10887276 Typesetting: Pages created by the authors using IAT5X 2.09. www springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelarnannSpringer Science+Business Media GmbH Preface This book can be seen as a continuation of Equations and Inequalities: El- ementary Problems and Theorems in Algebra and Number Theory by the same authors, and published as the first volume in this book series. How- ever, it can be independently read or used as a textbook in its own right. This book is intended as a text for a problem-solving course at the first- or second-year university level, as a text for enrichment classes for talented high-school students, or for mathematics competition training. It can also be used as a source of supplementary material for any course dealing with combinatorics, graph theory, number theory, or geometry, or for any of the discrete mathematics courses that are offered at most American and Canadian universities. The underlying “philosophy” of this book is the same as that of Equations and Inequalities. The following paragraphs are therefore taken from the preface of that book. There are already many excellent books on the market that can be used for a problem-solving course. However, some are merely collections of prob- lems from a variety of fields and lack cohesion. Others present problems according to topic, but provide little or no theoretical background. Most problem books have a limited number of rather challenging problems. While these problems tend to be quite beautiful, they can appear forbidding and discouraging to a beginning student, even with well-motivated and carefully written solutions. As a consequence, students may decide that problem solving is only for the few high performers in their class, and abandon this important part of their mathematical, and indeed overall, education. vi Preface One of the reasons why problem solving is often found to be difficult is the fact that in recent decades tbere has been less emphasis on technical skills in North American high-school mathematics. Furthermore, such skills are rarely taught at a university, where most courses are quite theoretical or structure-oriented. As a result, a lack of “mathematical fluency” is often evident even in upper years at a university; this reduces the enjoyment of the subject and impairs progress and success. A second reason is that most students are not used to more complex or multilayered problems. It would certainly be wrong to give the impres- sion that all mathematical problems should succumb to a straightforward approach. Indeed, much of the attractiveness of mathematics lies in the satisfaction derived from solving difficult problems after much effort and several futile attempts. On the other hand, being “stuck” too often and for too long can be very discouraging and counterproductive, and is ulti- mately a waste of time that could be better spent learning and practicing new techniques. This book, as well as the earlier Equations and Inequalities, attempts to address these issues and offers a partial remedy. This is done by emphasizing basic combinatorial ideas and techniques that are reinforced in numerous examples and exercises. However, even the easiest ones require a small twist. We therefore hope that this process of practicing does not become purely rote, but will retain the reader’s interest, raise his or her level of confidence, and encourage attempts to solve some of the more challenging problems. Another aim of both our books is to familiarize the reader with methods for solving problems in elementary mathematics, accessible to beginning university and advanced high-school students. This can be done in different ways; for instance, the authors of some books introduce several general methods (e.g., induction, analogy, or the pigeonhole principle) and illustrate each one with concrete problems from different areas of mathematics and with varying degrees of difficulty. Our approach, however, is different. We present a relatively self-contained overview of some parts of elementary mathematics that do not receive much attention in high-school and university education. We give only enough theoretical background to make these topics self-contained and rigorous, and concentrate on solving particular problems. The chapters of our books are fairly independent of one another, with only a limited number of cross references. Within each chapter, clusters of sections and subsections are tied together either by topic or by the methods needed to solve the examples and exercises. The problem-book character of this text is underlined by the large number of exercises; they can be solved by using a method or methods previously introduced. We suggest that the reader first carefully study all relevant examples before attempting to solve any of the exercises. Preface vil The individual problems in the present book (divided into approximately 310 examples and 650 exercises) are of varying degrees of difficulty, from completely straightforward, where the use of a method under consideration will immediately lead to a solution, to much more difficult problems whose solutions will sometimes require considerable effort. The more demanding exercises are marked with an asterisk (*). Answers to all exercises can be found in the final chapter, where additional hints and instructions to the more difficult ones are given as well. The problems were taken from a variety of mainly Eastern European sources, such as Mathematical Olympiads and other competitions. Many of them are therefore not otherwise easily accessible to the English-speaking reader. An important criterion for the selection of problems was that their solutions should, in principle, be accessible to high-school students. We believe that even with this limitation one can successfully stimulate creative work in mathematics and illustrate its diversity and richness. One of our objectives has been to stress alternative approaches to solving a given problem. Sometimes we provide several different solutions in one place, while at other times we return to a problem in later parts of the book. This book is a translation of the second Czech edition of Metody regent matematickych tloh II (Methods for solving mathematical problems II) pub- lished in 1997 at Masaryk University in Brno. Apart from the correction of some minor misprints, the main material has been left unchanged. Three short sections with “competition-type” problems at the ends of the chap- ters were deleted, and an alphabetical index was added. All the figures from the Czech editions were redrawn for this translation by Karel Hordk, we thank him for this important contribution. The Czech editions of this book have been used by the authors in spe- cia] enrichment classes for secondary-school students and for Mathematical Olympiad training in the Czech Republic. It has also been used on a regu- lar basis at Masaryk University in courses for future mathematics teachers at secondary schools. We hope that a course partly or entirely based on this book will work well even in a class with a wide range of mathematical backgrounds and abilities. The book’s structure, the worked examples, and the range in level of difficulty of the exercises make it particularly well suited as a source for assigned readings and homework exercises. Brno, Czech Republic Jiti Herman Radan Kuéera Jaromir Simsa Halifax, Nova Scotia, Canada Karl Dilcher, Translator Contents Preface Symbols 1 Combinatorics 1 Fundamental Rules ........°°; 2. .-fe 0ee n 2 Standard Concepts. ..........-.- 3 Problems with Boundary Conditions ...., 4 Distributions into Bins .- ............ 5 Proving Identities ... 2.2.222. .ee .ee. ee e 6 The Inclusion-Exclusion Principle.. ... ......- 7 Basics of Pélya's Theory of Enumeration ....... 8 Recursive Methods . . 2 Combinatorial Arithmetic 1 Arrangements. 6. ee ee ee 2 Sequences ..... ..... a oe 3 Arrays. 0 wee ee eee = a ae 4 Unordered Configurations.... ...... 5 Iterations ........ a 192 3 Combinatorial Geometry 217 1 Systems of Points and Curves... -..--. 219 2 Systems of Curves and Regions .......- 241
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