Table Of ContentJiri Herman Radan Kucera
Jaromir Simsa
Translated by Karl Dilcher
Counting and
Configurations
Problems in
Combinatorics,
Arithmetic, and Geometry
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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
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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.
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Springer-Verlag New York Berlin Heidelberg
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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
