Table Of ContentMathematical 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
