DOCUM7NTPFSUM7 SE 007 731 03", C90 rtri Vf +h a?patics. 'T,TmT77, ',,=kna4" ior711 t-, N MathPmaics,:, Tnc., -r 1T C IT Tr'"r, TTa-oral roueit cff mPach,Prs 79,71,ina+nn, r 7) ri r 7), 1) "I 771 1r7n. ouncil oc mPacher of mathematics,1221 T,VA-TT,A.,v7-rT)()%1 r QtrPot, 7.W.rWashington, 7).C.2223'5 * from 71r)q. T7 r CZ 71T) 77 C T7 ur "of kvailab1=-?. T) *A,nri-a+r,r1 *LitPraturP c7(' 11)r 'DT() (7 T,it:sratu/-P royiews,*;6ithematicalPnrichmeilt, *mathe,matics 77-ancation,rPfPrPnce nooks rT litoraturn auiTh is ahil-liograrooy o' books, concccrneya withmathPmatical recreations. arfirlPc;, an-3 n0ro'rat than 2,n20 ml-is is tl-n +hirl itior of a hookwhich, con+ainPd mar,. Snpnlr,mPnts have henad1P1 to bring onfriPg it inal Plition. snide is intondndfor VIP, the }-11-,linaranT,y u interested in nronssional mat},Pmatician and tiqe amateurwho ma,thPmafics as a1-ohhv. 'or thiF: reasonboth popular articles ,.4ions are, irclud0'1. Tn many cases,entries are' tPchnical discus auth6',- poin-ls out as anail to t'lP user ofthis hook. Trhe annotaf to look for sourceratPrials and fl-at this guie can serve PF-.a place loolrig for nrojPctmaterial advanced will ImP hPinf,fl to (741-nPnts technical articlPs aPri in research.41so, -,c, many non- stulPrts enqa fn 7-1thematics, as a will proyia0PnlovmPnt for +.hPlayman intereste(7, recrPation.(T) 1 PHOCESSwailMICROFICHE AND PUBLISHER'S PRICES. MICROFICHE, REPRODUCTION ONLY. William L. Schaaf U -S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT. POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS g OMMI EIPM WA LW*444 4 0 0 0-0740 CD 4. 1..CN PROCESS WITH MICROFICHE Lris AND PUBLISHER'S PRICES. rrN MICROFICHE REPRODUCTION ONLY. Rev:4404mgal Nadestatied 4 c. s'estse,,nosr s s",as --ar - as, 41 rits1AS* f).5 ve - S Ti sai i la j. :-..4 ...' 4_ N'1 r i.=.,_ - ,::-. -1., =4.,....G.,,...a , 4. 4 4.41k :=1'IA ig 4I fi4 -41 ,s , sa -- -,..: ese R'emeareed#141 .4 A Guide to the Literature WILLIAM L. S?1.1AAF Brooklyn College, Brooklyn, N. Y. 1 yr NATIONAL COUNCIL OF SI TEACHERS OF MATHEMATICS It 1201 Sixteenth Street, N.W. , Washington, D.C. 20036 0 "PERMISSION TO REPRODUCE THIS COPYRIGHTED. MATERIAL BY MICROFICHE ONLY HAS BEEN GRANTED BY Na TP h Ma th TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE U. S. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER." Copyright 0 1955, 1958; 1963, by THE NATIONAL COUNCIL OFTEACHERS OF MATHEMATICS, INC. All rights reserved THIRD EDITION (1963) Second printing, May 1965 Third printing, February 1967 Pt.: mission to reproduce this copyrighted work Wasbeen gi'anted to the Educational Resources InformationCenter (ERIC) and to they organization operating undercontract with the OfficeofEducation to reproduce documents in- c 1 uded in the ERIC system by means of microfiche only, but this right ianot conferred to any userofthe micro- fiche received from the ERIC DocumentReproduction Service.Further reproduction of any part requiresper- mission of the copyright owner. PRINTED IN THE UNITED STATES OF AMERICA It N, tr,,,evt 4 PREFACE The late G. H. Hardy once observed that there are few more"popular" subjects than mathematics. His contention isamply borne out by the uni- versal interest manifested in mathematical recreationsfor over 2000 -years, ranging from the loculus of Archimedesand the talismen magic squares of the early Chinese to the cryptanaLysis andtopological recreations of -modern times. One need only re.,-;a11 how testamentproblems, ferrying problems, coin problems, problems of pursuit andproblems of arrangements 4ave come down throughthe ages, ever dressed anew, yet always the sameold friends.Labyrinths, dissections, *acrostics, tangrams,palindromes, and so on, are likewise virtuallyageless. Hence it should occasion little surprise that an enormous body of literature has arisen inthe last 300 years. It has been my purpose to gather aconsiderable part of this material between the covers of one book for the convenienceof students and teach- 4 J. ers, as well aslaymen and specialists. The more than 2000 entriesby no means represent acomplete or exhaustive compilation. But enoughhas been given, I hope, to be of real help.I have tried to meet the needs of almost any readerthebeginner, the, dilettante, the professional scholar.--Hence I ip have deliberately included some "popular"articles along with erudite and technical discussions; many contemporary and recentpublications, as well as some of an earlierperiod; some that are readily accessible, andothers that are to be found only in importantlibraries; most of them in English, some in French, German,and Italian; most of them significant, a few, some- what superficial. In this way, it is hoped,both the neophyte and the sophis- ticated authority will find what they need. -The task of organizing this materialyielded a more or less arbitrary classification of mathematical recreations.Occasionally, where helpful, en- - tries have been annotated; to have commented uponeach item seemed quite unnecessary, and would in any eventhave been prohibitive. It would scarcely seem necessary to suggest'how this guide may be used. To be sure, a number of entries listedunder each of the more than 50 head- ings will not be available to the reader unlessthe facilities of a large library are at hand; yetthere will almost surely be some that areaccessible. In most instances the reader. willhave little difficulty in selecting items per- taining to a given topic: he should beguided by the title of the book or article; by the annotation, if any; by the sortof periodical, whether scholarly, popular, professional, newsy, and so on;and, to some extent, by the length of an article. Naturally, the reader's purpose, aswell as his familiarity with the subject, will loom large as factors inhelping him select items to be (4 =t, ti 0 consulted. Nor should he be deterred by references in a foreign language; after all, the mathematical symbols ad geometric figures are essentially the same, so that even a moderate facility inFrench Or German often suffices. This guide will serve as a place to begin to look for source materials.It will help the student pursuing his mathematical stuies in high sdhool or college; the mathematics club looking for program and project material; the teacher gathering human interest or motivation material; thmore, ad- winced student engaged in research; the amateur mathematician othe proverbial layman happily engaged in that most delectable of all activities a hobby or a recreation. May following these trails afford the reader as much pleasure as it has been for me to map them out for him. W. L. S. July 1954. PREFACE TO THE SECOND EDITION In preparing thenew edition, certain topics of general interesthave been included: mathemat'cal models, mathematical instruments, the abacus., mathe- matics and philatelymathematics in nature.In 'all, more than 500 references 6 have been added, soe in nearly every section, thusfilling in gaps'and bring- ing the bibliographyup to date.To make room for the new material, some old, inaccessible, oress appropriate items had tobe omitted: many 19th century works, itemson number mysticism andnumerology, Latin squares, "mathematical thinkig and invention," "machinesthat think."It is hoped that this edition willrove more useful to an evenwider group of readers. W. L. S. July 1957 PREFACE TO THE TylIRD EDITION A Supplement has been added, pages 144-56, to bring this bibliography up-to-date. It contains a partial listing of books, pamphlets, and periodical references that have appeared during the years 1957-62 and which are par- ticularly significant and readily accessible. W.L.S. January 1963 Contents Chapter 1.General Works 1 2 1.1Early. Twentieth CenturyBooks-1900-1924 4 1.2Contemporary Books-From 1925On 12 1.3Periodical Literature L4 Mathematics ClubPrograms; Plays 16 18 1.5Mathematics and Philately 19 1.6Mathematical Contests 20 1.7Mathematical Models 23 1.8Mathematical Instruments 24 1.9The Abacus 26 Chapter 2.Ariththetical and Algebraic.Recreations 26 2.1General Arithmetical Recreations 29 2.2Specific Problems and Puzzles 34 2.3Number Pleasantries 39 2.4Calculating Prodigies 42 2.5Theory of Numbers-Factorizations-----Primes 45 2.6Perfect Numbers-Mersenne'sNumbers 47 2.7Fermat's Last Theorem 49. 2.8Fibonacci Numbers and Series 4 51 Chapter 3.Geometric Recreations 51 3.1General Geometric Problemsand Puzzles 54 3.2Geometric Fallacies-OpticalIllusions.) 55 3.3Geometric Dissections-Tangranis 57 3.4Regular Polygons andPolyhedrons 60 3.5Geometric Constructions 63 3.6Mascheroni Constructions 64 3.7Linkages-ThVantograph 3.8Mechanical Construction ofCurves 66 Cha\pter 4.Assorted Recreations 68 4.1Boss Puzzle 68 4.2Card Tricks-Manipulative Puzzles 69 4.3Chessboard Problems 71 4.4Topological Questions 71 4.5String Figures-Theory of Knots 73. 4.6The MObius Strip 74 4.7Map-Coloringlb Problems 4.8Paper Folding 76 4.9.1.Tnicursal Problems-Labyrinths Chaptei 5.Magic Squares 79 5.1Books--1900-1924 79 '5.2Contemporary Books-from 1925 On 81 5.3Periodical Literature 83 Chapter 6.The Pythagorean Relationship 89 6.1The Theorem of Pythagoras 89 6.2Pythagorean Numbers-Rational Right Triangles 93 6.3Special Triangles--Heronian Triangles 96 6.4Miscellaneous Pythagorean Recreations 98 Chapter 7.Famous Problems. of Antiquity 100 7.1Classical Constructions 100 7.2Trisecting an Angle 102 7.3Duplicating a Cubet 105 7.4Squaring a Circle 106 7.5History and Value of Pi (71-) 109 7.6Zeno's Paradoxes 112 Chapter8.Mathematical Miscellanies 114 8.1Mathematics in Nature 114 8.2Machines That Think 116 .