Selected Title s i n Thi s Serie s

7 Dmitr i Fomin, Sergey Genkin, and Ilia Itenberg, Mathematica l circle s (Russia n experience) , 1996

6 Davi d W. Farme r and Theodor e B . Stanford , Knot s an d surfaces : A guid e t o discoverin g

mathematics, 199 6

5 Davi d W. Farmer , Group s an d symmetry : A guid e t o discoverin g mathematics , 199 6

4 V . V . Prasolov , Intuitiv e topology , 199 5

3 L . E. Sadovski ï and A. L . Sadovskiï , Mathematic s an d sports , 199 3

2 Yu . A . Shashkin , Fixe d points , 199 1

1 V . M . Tikhomirov , Storie s abou t maxim a an d minima , 199 0

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Mathematical World • Volume 7

Mathematical Circles (Russian Experience)

http://dx.doi.org/10.1090/mawrld/007

C. A . T E H K H H , H . B . H T E H B E P r , ü . B . # O M H H

MATEMATH^ECKMH KPY>KO K C A H K T - n E T E P B y p r 1992 , 199 3

Transla ted fro m th e Russia n b y Mar k Sau l

2000 Mathematics Subject Classification. Primar y 00A08 ; Secondar y 00A07 .

ABSTRACT. Thi s boo k i s intende d fo r student s an d teacher s wh o lov e mathematic s an d wan t t o study it s various branches beyon d th e limit s of school curriculum. I t i s also a book of mathematica l recreations, an d a t th e same time a book containin g vas t theoretica l an d proble m materia l i n som e areas o f wha t author s conside r t o b e "extracurricula r mathematics" . Th e boo k i s base d o n a n experience gaine d b y severa l generation s o f Russia n educator s an d scholars .

L i b r a r y o f C o n g r e s s Ca ta log ing - in -Pub l i ca t io n D a t a Genkin, S . A . (Serge ï Aleksandrovich )

[Matematicheskiï kruzhok . English ] Mathematical circle s : (Russia n experience ) / Dmitr i Fomin , Serge y Genkin , Ili a Itenberg ;

translated fro m th e Russia n b y Mar k Saul . p. cm . — (Mathematica l world , ISS N 1055-9426 ; v . 7 )

On Russia n ed. , Genkin' s nam e appear s first o n t.p . Includes bibliographica l reference s (p . - ) . ISBN 0-8218-0430- 8 (alk . paper ) 1. Mathematica l recreations . I . Fomin , D . V . (Dmitri ï Vladimirovich ) II . Itenberg , I . V .

(Il'ia Vladimirovich ) III . Title . IV . Series . QA95.G3813 199 6 510'.76—<lc20 96-1768 3

CIP

Copy ing a n d r e p r i n t i n g . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-penaissionOams.org .

© Copyrigh t 199 6 b y th e America n Mathematica l Society . Printed i n th e Unite d State s o f America .

The America n Mathematica l Societ y retain s ai i right s except thos e grante d t o th e Unite d State s Government .

© Th e pape r use d i n thi s boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http:/ /www.ams.org /

10 9 8 7 6 5 4 1 5 1 4 1 3 1 2 1 1

Contents

Foreword vi i

Preface t o th e Russia n Editio n i x

Part I . Th e Firs t Yea r o f Educatio n

Chapter 0 . Chapte r Zer o 1

Chapter 1 . Parit y 5

Chapter 2 . Combinatorics- 1 1 1

Chapter 3 . Divisibilit y an d Remainder s 1 9

Chapter 4 . Th e Pigeo n Hol e Principl e 3 1

Chapter 5 . Graphs- 1 3 9

Chapter 6 . Th e Triangl e Inequalit y 5 1

Chapter 7 . Game s 5 7

Chapter 8 . Problem s fo r th e Firs t Yea r 6 5

Part II . Th e Secon d Yea r o f Educatio n

Chapter 9 . Inductio n (b y I . S . Rubanov) 7 7

Chapter 10 . Divisibility-2 : Congruenc e an d Diophantin e Equation s 9 5

Chapter 11 . Combinatorics- 2 10 7

Chapter 12 . Invariant s 12 3

Chapter 13 . Graphs- 2 13 5

Chapter 14 . Geometr y 15 3

Chapter 15 . Numbe r Base s 16 7

Chapter 16 . Inequalitie s 17 5

Chapter 17 . Problem s fo r th e Secon d Yea r 18 7

CONTENTS

Appendix A . Mathematica l Contest s 20 1

Appendix B . Answers , Hints , Solution s 21 1

Appendix C . Reference s 26 9

Foreword

This i s not a textbook . I t i s not a contes t booklet . I t i s not a se t o f lessons fo r classroom instruction . I t doe s no t giv e a serie s of projects fo r students , no r doe s i t offer a developmen t o f part s o f mathematics fo r self-instruction .

So wha t kin d o f boo k i s this ? I t i s a boo k produce d b y a remarkabl e cul -tural eireumstanee , whic h fostered th e creation of groups of students, teachers , an d mathematicians, calle d mathematical circles , in the former Sovie t Union. I t i s pred-icated o n the idea that studyin g mathematic s ca n generate the sam e enthusiasm a s playing a team sport , withou t necessaril y bein g competitive .

Thus i t i s more like a book of mathematical recreations—excep t tha t i t i s more serious. Writte n b y research mathematician s holdin g universit y appointments , i t i s the resul t o f thes e sam e mathematicians ' year s o f experienc e wit h group s o f hig h school students . Th e sequence s o f problem s ar e structure d s o tha t virtuall y an y student ca n tackle the first fe w examples. Ye t the same principles of problem solving developed i n th e earl y stage s mak e possibl e th e solutio n o f extremel y challengin g problems late r on . I n between , ther e ar e problem s fo r ever y leve l o f interes t o r ability.

The mathematical circle s of the former Sovie t Union , an d particularly o f Lenin-grad (no w St. Petersburg , wher e these problems were developed) ar e quite differen t from mos t mat h club s i n the Unite d States . Typically , the y wer e run no t b y teach -ers, bu t b y graduat e student s o r facult y member s a t a university , wh o considere d it par t o f thei r professiona l dut y t o sho w younge r student s th e joys o f mathemat -ics. Student s ofte n me t fa r int o th e night , an d wen t o n weeken d trip s o r summe r retreats together , achievin g a closeness an d mutua l suppor t usuall y reserve d i n ou r country fo r member s o f athleti c teams .

We are fortunate t o be living in a time when Russians and Americans can easily communicate an d shar e thei r cultures . Th e developmen t o f mathematics educatio n is a n aspec t o f Russia n cultur e fro m whic h w e have muc h t o learn . I t i s stil l ver y rare to find research mathematicians i n America willing to devote time, energy, an d thought t o th e developmen t o f material s fo r hig h schoo l students .

So we must borro w fro m ou r Russian colleagues . Th e present boo k i s the resul t of such borrowings. Som e chapters, such as the one on the triangle inequality, can be used directl y i n America n classrooms , t o supplemen t th e developmen t i n the usua l textbooks. Others , suc h a s th e discussio n o f grap h theory , stretc h th e curriculu m with gem s of mathematics whic h ar e not usuall y touche d o n in the classroom. Stil l others, such as the chapter on games, offer a rich source of extra-curricular material s with mor e structur e an d meanin g tha n many .

Each chapte r give s examples o f mathematica l method s i n som e o f thei r bares t forms. A game o f nim, whic h ca n b e enjoyed an d eve n analyze d b y a third grader ,

vii

viii F O R E W O R D

turns ou t t o b e th e sam e a s a gam e playe d wit h a singl e paw n o n a chessboard . This become s a lesso n fo r sevent h grader s i n restatin g problems , the n offer s a n introduction t o th e nature o f isomorphism fo r th e hig h schoo l student . Th e Pigeo n Hole Principle , amon g th e simples t ye t mos t profoun d mathematic s ha s t o offer , becomes a too i fo r proo f i n number theor y an d geometry .

Yet the tone o f the wor k remains light . Th e chapte r o n combinatorics doe s no t require a n understandin g o f generatin g function s o r mathematica l induction . Th e problems i n grap h theory , too , remai n o n th e surfac e o f thi s importan t branc h o f mathematics. Th e approac h t o eac h topi c lend s itsel f t o min d play , no t weight y reflection. An d ye t th e wor k manage s t o strik e som e dee p notes .

It i s thi s qualit y o f th e wor k whic h th e mathematician s o f th e forme r Sovie t Union develope d t o a hig h art . Th e expositio n o f mathematics , an d no t jus t it s development, becam e a par t o f th e Russia n mathematician' s work . Thi s boo k i s thus par t o f a literar y genr e whic h remain s largel y undevelope d i n th e Englis h language.

Mark Saul , Ph.D . Bronxville School s

Bronxville, Ne w Yor k

Preface t o th e Russia n Editio n

§1. Introductio n

This boo k wa s originall y writte n t o hel p peopl e i n th e forme r Sovie t Unio n who dealt wit h extracurricula r mathematica l education : schoo l teachers, universit y professors participatin g i n mathematica l educatio n programs , variou s enthusiast s running mathematica l circles , o r peopl e wh o just wante d t o rea d somethin g bot h mathematical an d recreational . And , certainly , student s ca n als o us e thi s boo k independently.

Another reaso n fo r writin g thi s boo k wa s tha t w e considere d i t necessar y t o record th e rol e playe d b y th e tradition s o f mathematica l educatio n i n Leningra d (now St. Petersburg) ove r the last 6 0 years. Thoug h our city was, indeed, the cradl e of th e olympia d movemen t i n th e USS R (havin g see n th e ver y firs t mathematica l seminars fo r student s i n 1931-32 , an d th e first cit y olympia d i n 1934) , an d stil l remains on e o f th e leader s i n thi s particula r area , it s hug e educationa l experienc e has no t bee n adequatel y recorde d fo r th e intereste d readers .

* * *

In spit e o f th e stylisti c variet y o f thi s book' s material , i t i s methodologicall y homogeneous. Her e we have, we believe, all the basic topics for sessions of a mathe-matical circle for the first tw o years of extracurricular educatio n (approximately , fo r students of age 12-14) . Ou r main objective wa s to make the preparation o f sessions and th e gatherin g o f problem s easie r fo r th e teache r (o r an y enthusias t willin g t o spend tim e wit h children , teachin g the m non-standar d mathematics) . W e wante d to talk abou t mathematica l idea s which are important fo r students , an d abou t ho w to dra w th e students ' atten t ion to thes e ideas .

We mus t emphasiz e tha t th e wor k o f preparin g an d leadin g a sessio n i s itsel f a creativ e process . Therefore , i t woul d b e unwis e t o follo w ou r recommendation s blindly. However , w e hop e tha t you r wor k wit h thi s boo k wil l provid e yo u wit h material fo r mos t o f you r sessions . Th e followin g us e o f thi s boo k seem s t o b e natural: whil e working on a specific topi c the teacher read s an d analyze s a chapte r from th e book, an d afte r tha t begin s to construct a sketch of the session. Certainly , some adjustment s will hav e t o b e mad e becaus e o f th e leve l o f a give n grou p o f students. A s supplementar y source s o f problem s w e recommend [13 , 16 , 24 , 31 , 33], an d [40] .

* * *

We woul d lik e t o mentio n tw o significan t point s o f th e Leningra d traditio n o f extracurricular mathematica l educationa l activity :

ix

x PREFAC E T O T H E RUSSIA N EDITIO N

(1) Session s featur e vivid , spontaneou s communicatio n betwee n student s an d teachers, i n which eac h studen t i s treated individually , i f possible .

(2) The proces s begins a t a rather earl y age : usuall y durin g the 6th grad e (ag e 11-12), an d sometime s eve n earlier .

This boo k wa s written a s a guide especiall y fo r secondar y schoo l student s an d for thei r teachers . Th e ag e o f th e student s wil l undoubtedl y influenc e th e styl e o f the sessions . Thus , a fe w suggestions :

A) We consider i t wrong to hold a long session for younge r student s devote d t o only one topic . W e believe tha t i t i s helpful t o chang e th e directio n o f the activit y even withi n on e session .

B) I t i s necessar y t o kee p goin g bac k t o materia l alread y covered . On e ca n do thi s b y usin g problem s fro m olympiad s an d othe r mathematica l contest s (se e Appendix A) .

C) I n discussin g a topic , tr y t o emphasiz e a fe w o f th e mos t basi c landmark s and obtai n a complet e understandin g (no t jus t memorization! ) o f these fact s an d ideas.

D) W e recommen d constan t us e o f non-standar d an d "gamelike " activitie s i n the sessions , with complet e discussio n o f solutions an d proofs . I t i s important als o to us e recreational problem s an d mathematica l jokes. Thes e ca n be found i n [5—7, 16-18, 26-30] .

We must mentio n her e our predecessors—those wh o have tried earlie r to creat e a sor t o f anthology fo r Leningra d mathematica l circles . Thei r book s [32 ] and [43] , unfortunately, di d no t reac h a larg e numbe r o f reader s intereste d i n mathematic s education i n secondar y school .

In 1990-91 the original version of the first par t o f our book was published by the Academy of Pedagogical Science s of USSR as a collection o f articles [21 ] written b y a number o f authors. W e would lik e to thank al l our colleague s whose materials we used whe n workin g o n th e preparatio n o f the presen t book : Deni s G . Benua , Igo r B. Zhukov , Ole g A . Ivanov , Alexe y L . Kirichenko , Konstanti n P . Kokhas , Nikit a Yu. Netsvetaev , an d Ann a G . Prolova .

We also expres s ou r sincer e gratitud e t o Igo r S . Rubanov , whos e pape r o n in -duction written especially for the second part o f the book [21 ] (but never published , unfortunately) i s included her e a s the chapte r "Induction" .

Our specia l thank s g o t o Alexe y Kirichenk o whos e hel p i n th e earl y stage s o f writing thi s boo k canno t b e overestimated . W e woul d als o lik e t o than k Ann a Nikolaeva fo r drawin g th e figures.

§2. S t ruc tu r e o f t h e boo k

The boo k consist s o f this preface , tw o main parts , Appendi x A "Mathematica l Contests", Appendix B "Answers , Hints, Solutions", and Appendix C "References" .

The firs t par t ("Th e Firs t Yea r of Education") begin s with Chapte r Zero , eon-sisting o f tes t question s intende d mostl y fo r student s o f age s 10-11 . Th e problem s of this chapte r hav e virtually n o mathematical content , an d thei r mai n objectiv e i s to reveal the abilitie s of the students i n mathematics an d logic . Th e rest o f the first part i s divide d int o 8 chapters . Th e first seve n o f thes e ar e devote d t o particula r topics, an d th e eight h ("Problem s fo r th e first year" ) i s simpl y a compilatio n o f problems o n a variet y o f themes .

P R E F A C E T O T H E RUSSIA N EDITIO N x i

The second par t ("Th e Secon d Year of Education") consist s of 9 chapters, som e of whic h jus t continu e th e discussio n i n th e firs t par t (fo r example , th e chapter s "Graphs-2" an d "Combinatorics-2") . Othe r chapter s ar e comprise d o f materia l considered t o b e too complicated fo r th e firs t year : "Invariants" , "Induction" , "In -equalities".

Appendix A tell s abou t five mai n type s o f mathematica l contest s popula r i n the forme r Sovie t Union . Thes e contest s ca n b e hel d a t session s o f mathematica l circles o r use d t o organiz e contest s betwee n differen t circle s o r eve n schools .

Advice t o th e teache r i s usually give n unde r th e remar k labelle d "Fo r teach -ers" . Rar e occasion s o f "Methodologiea l remarks " contai n mostl y reeommen -dations about th e methodology of problem solving: the y draw attention to the basi c patterns o f proofs o r method s o f recognizing an d classifyin g problems .

§3. Technicalitie s an d legen d

(1) Th e mos t difflcul t problem s ar e marke d wit h a n asteris k (*) . (2) Almos t al l o f th e problem s ar e commente d o n i n Appendi x B : eithe r a

full solutio n o r a t leas t a hin t an d answer . I f a proble m i s computational , the n we usuall y provid e onl y a n answer . W e d o no t giv e th e solution s t o problem s fo r independent solutio n (this , i n particular , goe s fo r al l th e problem s fro m Chapter s 8 an d 17) .

(3) All the references can be found a t the end of the book in the list of references . The book s w e recommend mos t ar e marke d wit h a n asterisk .

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