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Intendd for Both A DOCUMENT RESUME ED 040 874 SE 008 968 AUTHOR Schaaf, WilliamL. TITLE A Bibli6graphy of RecreationalMathematics, Volume INSTITUTION National Council 2. of Teachers ofMathematics, Inc., Washington, D.C. PUB DATE 70 NOTE 20ap. AVAILABLE FROM National Council of Teachers ofMathematics:, 1201 16th St., N.W., Washington, D.C.20036 ($4.00) EDRS PRICE EDRS Price ME-$1.00 HC Not DESCRIPTORS Available fromEDRS. *Annotated Bibliographies,*Literature Guides, Literature Reviews,*Mathematical Enrichment, *Mathematics Education,Reference Books ABSTRACT This book isa partially annotated books, articles bibliography of and periodicalsconcerned with puzzles, tricks, mathematicalgames, amusements, andparadoxes. Volume2 follows original monographwhich has an gone through threeeditions. Thepresent volume not onlybrings theliterature up to material which date but alsoincludes was omitted in Volume1. The book is the professionaland amateur intendd forboth mathematician. Thisguide canserve as a place to lookfor sourcematerials and will engaged in research. be helpful tostudents Many non-technicalreferences the laymaninterested in are included for mathematicsas a hobby. Oneuseful improvementover Volume 1 is that the number ofsubheadings has more than doubled. (FL) been 113, DEPARTMENT 01 KWH.EDUCATION & WELFARE OffICE 01 EDUCATION N- IN'S DOCUMENT HAS BEEN REPRODUCED EXACILY AS RECEIVEDFROM THE CO PERSON OR ORGANIZATION ORIGINATING IT POINTS Of VIEW OR OPINIONS STATED DO NOT NECESSARILY CD REPRESENT OFFICIAL OFFICE OfEDUCATION INt POSITION OR POLICY. C, C) W A BIBLIOGRAPHY OF recreational mathematics volume 2 Vicature- ligifitt.t. confiling of RECREATIONS F DIVERS KIND S7 VIZ. Numerical, 1Afironomical,I f Antomatical, GeometricallHorometrical, Mechanical,i1Cryptographical, i and Statical, Magnetical, [Htlorical. Publifhed to RecreateIngenious Spirits;andto induce them to make fartherlcruciny into tilde( and the like) Suut.tm2. SCIENCES. AND To divert them from following filchVices, to whichYouci ( in this Age)are fo much Inclin'd. By WILLIAM 1.,1(.110LIKN3Philomathes,' To this Work isalto Annext, A 11REATISE of According,. ALGEBRA) to the late Improvements,applied tri klions and Geiinutyr; with a NEW Niuniericai Mots opP as_alf6 a CONN SERIES for thefpeedy ExtrA ner of epatiow. ER.GING SERIESfor ailman- By R: SAut. t, Mafler of the Court; in Matbmatick Schoolin jd.rm's. Ero2dibtft, near thelicy.s1 Ex:b.tnge, L 0 N Db N. ,/ - .7f- 7. 0 N :00 N: Cof Print forilialarb Zathimin, S Arms in WarmckLine and on IDuntim i near digOXfOrgi : And at the R41.Pta in thePoalgrg. :1 *OR 169+, A BIBLIOGRAPHYOF recreationalmathematics VOLUME 2 William L. Schaaf Professor Emeritus Brooklyn College The City Universityo I New York NATIONAL COUNCIL OFTEACHERS OF MATHEMATICS 1201 Sixteenth Street, Northwest,Washington, D.C. 20036 i i TO REPRODUCETHIS COPYRIGHTED MATEY3! BY itcRE.°TEEVSKE12HI PARf TO ERIC AND ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THEU. S. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ER ICSYSTEM REQUIRES PERMISSIONOF THE COPYRIGHT OWNER." Copyright © 1970 by theNational Council of Teachers of Mathematics,Inc. All rights reserved. r Printed in the United States ofAmerica. II-/, eV t Preface Popular interest in recreational mathematics wouldseem to be at an all-time high. The last few decades have witnessed theappearance of literally scores of books and hundreds of articleseloquent testimonyto the wide appeal of mathematical games and puzzles, tricks, pastimes,amusements, brainteasers, and paradoxes. In addition to this embarrassment of richesthere is to be noted a shift in emphasis in recentyears. Whereas formerly much attention was given to numerical curiosities, magic squares, the fourth dimension, tangrams, dissections, angl'. trisection and circle squaring,contemporary writers tend to devote more attention to suchmatters as acrostics, alphameties, colored cubes, combinatorics, electroniccomputersI primes, chess playing, etc.), hexahedrons, game strategy, inferential and logicproblems, modular arithmetic, networks and graphs, packing problems, tessellations,polyominoes, probability (queuing, etc.), scales of notation, the Soma cube,topological recreations. In this connection one should mentionsome of the outstanding leaders in the field. The close of the nineteenth century and the early decadesof the present century saw the work of W. Ahrens, W. W. Rouse Ball, Lewis Carroll, Henry Dudeney, F. Fitting, E. Fourrey, J. Ghersi, Bruno Kerst, W. Lietzmann, Sam Loyd, E. Lucas, L. Mittenzwey, Hubert Phillips, and Hermann Schubert. In more recent years the roster of notable contributors would include,among others, the following: A. R. Amir-Moez, Irving Adler, Leon Bankoff, Stephen Barr, L. G. Brandes, Maxey Brooke, Mannis Charosh, John Conway, Martyn Cundy, A. P. Domoryad, Martin Gardner, Michael Goldberg, Solomon Golomb, Philip Heaford, Piet Hein, J. A. H. Hunter, Murray Klamkin, Boris Kordem- sky, Sidney Kravitz, Harry Lindgren, Joseph Madachy, Karl Menninger, Leo Moser, T. H. O'Beirne, C. Stanley Ogilvy. the late Hans Rademacher, Howard Saar, Sidney Sackson, Fred Schuh, Hugo Steinhaus, Otto Toeplitz, Charles W. Trigg, and William Tutte. Some indication of the deep interest in recreational mathematics evinced by teachers is the fact that since this bibliography first appearedsome fifteen years ago it has passed through three editions and seven printings. Meanwhile, the literature has become so extensive that itwas felt a second volume would prove more serviceable than a revision of the original monograph. Thus the present monograph, volume 2, not only brings the literatureup to date but also fills in many gaps and omissions. It containsnone of the earlier material except vi PREFACE the supplementary periodicalreferences, which occupied the last eightpages of the third edition. Thesereferences have been incorporatedin the new classification in thepresent volume. The organization of thecontents of volume 2 is such that the majortopics parallel the original classificationfairly closely, making it relativelyeasy to consult both volumes on a given topic. There are several innovationswhich it is hoped will materiallyincrease the usefulness of the'monograph. To begin with, the number of subheadingshas beenmore than doubledand this should help the readerto track down a particular subject or special type of recrea- l tion. Secondly, in additionto periodical references,many references to "spe- cialized" booksare now includedbooks dealing solely witha single topic, such asmazes, or dissections, or tangrams,or inferential problems. Finally, there are now included alsomany references to specific topics in important general workson recreational mathematics, suchas books by Ball, Domoryad, Gardner, Madachy, O'Beirne, Schuh,and others. As in the case of volume 1, theclassification of mathematicalrecreations is at best an arbitrary affair, reflectingthe bibliographer's personaljudgment.I have again endeavored to make the coverage sufficientlycomprehensive to include comparatively simpleas well as highly sophisticated materialtopics usually thought ofas mathematical puzzles and pastimesas well as certain fringe topics suchas mathematical humor, mathematics insports, mathematics in art, literature, and music, andmathematics in nature.In this way it is hoped to reach a %vide spectrum of readers: students, teachers,laymen, amateurs, and professional mathematicians. I take this opportunity to acknowledgemy indebtedness over the pastyears to many friends and colleagues, including,among others, Professor H. S. M. Coxeter, Mr. J. A. H. Hunter,Mr. Joseph S. Madachy, andProfessor Adrian Struyk. I am indebted alsoto Mr. Leon Bankoff for dataon the arbelos and other geometric recreations; to Maxey Brooke for help inconnection with magic squares, Miibius bands,and other topological ideas;to Professor R. L. Morton in connection with perfectnumbers and amicable numbers;to Mr. Charles W. Trigg for assistancewith linkages, packing problems,and tessella- tions. My greatest debt isto Martin Gardner, not only for specifichelp, but for encouragement andinspiration. I am grateful alsoto the several publishers from whose works brief excerpts have been quoted; to the NationalCouncil of Teachers of Mathematics fortheir generous cooperation;and to my wife for her infinite patience inmany ways. William L. Schaaf Boca Raton, Florida September 1969 Contents Chapter 1.Algebraic andArithmetical Recreations 1.1The Abacus 1 1 1.2Acrostics; Crossword Puzzles 1 1.3Algebraic Puzzles and Pastimes 3 1.4Alphametics and Cryptarithms 09 1.5Calendar Problems 4 1.6Continued Fractions 6 1.7Cross-Number Puzzles 6 1.8Fallacies-Illegal Operations-Paradoxes 7 1.9Farey Sequences 8 1.10 Figurate Numbers-Polygonal Numbers 9 1.11 Four Fours 11 1.12 Number Bases-Binary Recreations 12 1.13 Number Giants 14 1.14 Number Mysticism and Numerology 15 1.15 Number Pleasantries and Curiosities 16 1.16 Pascal's Triangle 21 1.17 Repeating Decimals 22 1.18 Story Problems 23 Chapter 2. Number Theoryas Recreation 26 2.1Amicable Numbers 26 2.2Diophantine Puzzles 26 2.3Fermat Numbers 27 2.4.Fermat Theorems 28 2.5Fibonacci Numbers 29 2.6Mersenne Numbers ................ ,......... ,.. 32 2.7Number Theory: General 33 2.8Odd Perfect Numbers 34 2.9Perfect Number,. 35 2.10 Prime Numbers 37 2.11 Tests for Divisibility 39 Chapter 3. Geometric Recreations 41 3.1Apollonian Problem 41 3.2The Arbelos 4.1 vii viii CONTENTS 3.3The Butterfly Problem 42 3.4 The CrossedLadders 43 3.5Curves and Their Construction 44 3.6Curves of Constant
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