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A Bibliography of Recreational Mathematics, Volume 3. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C DOCUMENT RESUME ED 087 631 SE 017 302 AUTHOR Schaaf, William L. TITLE A Bibliography of Recreational Mathematics, Volume 3. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 73 NOTE 187p. AVAILABLE FROM National Council of Teachers of Mathematics, 1906 Association Drive, Reston, Virginia 22091 EDRS PRICE MF-$0.65 HC Not Available from EDRS. DESCRIPTORS *Annotated Bibliographies; Classroom Games; *Games; Geometric Concepts; Instructional Materials; *Literature Guides; Literature Reviews; *Mathematical Enrichment; *Mathematics Education; Number Concepts; Probability; Reference Books ABSTRACT This book is a partially annotated bibliography of books, articles and periodicals concerned with mathematical games, puzzles, tricks, amusements, and paradoxes. Because the literature in recreational mathematics has proliferated to amazing proportions since Volume 2 of this series (ED 040 874), Volume 3 is more than just an updating of the earlier monographs. The overall organization of the book has been retained, although there has been a notable rearrangement of subtopics in the interest of economy and clarity. Major changes include (1) the addition of two new sections on classroom games and recreational activities which will be quite useful for teachers; (2)a chronological synopsis of Martin Gardner's popular column in Scientific American; and (3)a glossary of terms related to recreational mathematics. The main topic headings are: Arithmetic Recreations; Number Theory as Recreation; Geometric Recreations; Topological Recreations; Magic Squares and Related Configurations; Pythagorean Recreations; Recreations in Antiquity; Combinatorial Recreations; Manipulative Recreations; and Mathematics in Related Fields. (JP) U 5 DEPARTMENT OF HEALTH, EDUCATION & WELFARE NATIONAL INSTITUTE OF EDUCATION THIS DOCUMENT HAS BEEN REPRO DUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGIN ATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRE SENT OFFICIAL NATIONAL INSTITUTE OF EDUCATION POSITION OR POLICY ; A BIBLIOGRAPHY OF recreational mathematics volume 3 A BIBLIOGRAPHY OF recreational mathematics VOLUME 3 William L. Schaaf Professor Emeritus Brooklyn College The City University of New York NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS 1906 Association Drive, Reston, Virginia 22091 "PERMISSION 10 REPRODUCE THIS COPYRIGHTED MATERIAL BY MICRO. FICHE ONLY HAS BEEN GRANTED BY TO ERIC AND ORGANIZATIONS OPERAT INC UNDER AGREEMENTS WITH.THE NA TIONAL INSTITUTE OF EDUCATION FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PERMIS SION DF THE COPYRIGHT OWNER Copyright 1973 by the National Council of Teachers of Mathematics, Inc. All rights reserved. Library of Congress Cataloging in Publication Data (Revised): Schaaf, William Leonard, 1898 A bibliography of recreational mathematics. Vols. 2 and 3 lack edition statement. First3d editions published under title: Recrea- tional mathematics. 1.Mathematical recreationsBibliography. I.Title. Z6654.M3S32 016.7937'4 73.12168 Printed in the United States of America Preface Since the appearance of volume 2 in 1970, general interest in mathematical recreations has not abated. On the contrary, not only has the literature be- come more sophisticated,it has proliferated to amazing proportions.As Maxey Brooke suggests, it is like.Alice: You have to keep running as hard as you can to stay in the same plaCe. "Thus the present volume is morc than just an updating of the earlier monographs. For the reader's convenience, the overall organization of the book has been retained, although there has been a notable rearrangement of subtopics in the interest of economy and greater clarity. Major changes include (1) the addi- tion of two new sections on classroom games and recreational activities, which will doubtless appeal to teachers;(2)a chronological synopsis of Martin Gardner's popular column in Scientific American; and (3) a glossary of terms related to recreational mathematics. As heretofore, some references have been listed under two different head- ings. Also, several hundred entries that were somehow previously overlooked have been belatedly included. Finally, a few items listed in the earlier vol- umes have been deliberately repeated here to enable the reader to retrieve related information concerning a given topic. To acknowledge my indebtedness to many friends and correspondents would entail a lengthy list of names, and so I hereby express my gratitude to them all. My thanks are especially due to Maxey Brooke, Mannis Charosh, Martin Gardner, James A. H. Hunter, and Charles W. Trigg, whose help and sugges- tions have been invaluable. My thanks are also extended to the National Council of Teachers of Mathematics for its customary cooperation, and to my wife for her unflagging encouragement and patience. William L. Schaaf Boca Raton, Florida April 1973 Contents Chapter 1.Arithmetical Recreations 1 1.1Calendar Problems 1 1.2Cryptarithms; Alphametics; Anagrams 1 1.3Determinants 3 1.4Fallacies; Illegal Operations; Paradoxes 3 1.5Four Fours; Four Digits; Related Problems 4 1.6Fractions; Farey Sequences 5 1.7Number Bases; Numeration 5 1.8Number Mysticism; Numerology 7 1.9Number Pleasantries; Digital Oddities 8 1.10 Number Tricks; Calculating Prodigies 11 1.11 Repeating Decimals 12 1.12 Story Problems 13 Chapter 2. Number Theory as Recreation 16 2.1Amicable Numbers 16 2.2Divisibility 17 2.3Fibonacci and Lucas Numbers 18 2.4Figurate Numbers 19 2.5Palindromes; Repunits and Repdigits 21 2.6Perfect, Deficient, and Abundant Numbers 22 2.7Prime Numbers 24 2.8Recursive Operations; Bracelets; Digital Inversions 26 2.9General Theory of Numbers 27 Chapter 3. Geometric Recreations 30 3.1Curves: Their Properties and Construction 30 3.2Curves of Constant Width 32 3.3The Fourth Dimension 32 3.4Geometric Constructions; Mascheroni Constructions 33 vii Viii CONTENTS 3.5Geometric Problems and Theorems 34 3.6Geometric Recreations and Puzzles 35 3.7Lattices; Taxicab Geometry; Geoboard 37 3.8Optical Illusions 38 Chapter 4. Topological Recreations 39 4.1Braids; Knots; Flexagons; Mobius Bands; String Figures 39 4.2Dissection Problems 39 4.3Graphs; Networks 41 4.4Mai) Coloring 43 4.5Mazes and Labyrinths 46 4.6Paper Folding; Origami 46- 4.7Polyhedrons; Platonic and Archimedean Solids 48 4.8Polytopes; Irregular Polyhedrons 50 4.9Polyominoes; Polyiamonds; Rep-tiles 51 4.10 Soma Cubes; Polycubes 52 4.11 Squared Rectangles; Squaring the Square 53 4.12 Tangrams 53 4.13 Tessellations; Packing Problems 53 4.14 Topological Amusements 56 Chapter 5. Magic Squares and Related Configurations... 57 5.1Magic Squares 57 5.2The Magic Knight's Tour 59 5.3Antimagic Squares; Heterosquares 60 5.4Magic Triangles and Other Plane Figures 60 5.5Magic and Antimagic Solids 60 5.6Latin Squares and Euler Squares 61 Chapter 6. Pythagorean Recreations 62 6.1The Pythagorean Theorem 62 6.2Pythagorean Triples 63 6.3Pythagorean Relationships 64 6.4Heronic Triangles; Figures with Integer Dimensions 65 Chapter 7. Recreations in Antiquity 67 7.1Computation of Pi (7r) 67 7.2History of Pi; e, 7r, and i 67 7.3Pi and Probability 68 7.4Trisection of an Angle 68 CONTENTS iX Chapter 8. Combinatorial Recreations 70 8.1 Permutations, Combinations, and Partitions; Factorials 70 8.2Pascal's Triangle 73 8.3Probability: Problems and Theory 73 8.4Games of Chance; Gaint."-; 75 Chapter 9. Manipulative Recreations 77 9.1Ticktacktoe 77 9.2The Fifteen Puzzle 77 9.3Binary Recreations; Nim; Wythoff's Game 77 9.4Board Games 78 9.5. Pencil and Paper Games. 79 9.6Manipulative Puzzles and Recreations 80 9.7Card Games; Card Tricks 82 9.8Colored Squares; Tiles and Cubes 83 9.9Games and Game Strategy 84 9.10 Domino Recreations 85 9.11 Chess Recreations 86 Chapter 10.Miscellaneous Recreations 88 10.1 Logic; Inferential Problems; Logical Paradoxes; Infinity 88 10.2 Cryptography; Cryptanalysis; Codes and Ciphers 89 10.3 Humor and Mathematics 90 10.4 Sports and Mathematics 92 10.5 Philately and Mathematics 92 10.6 Assorted Diversions and Amusements 92 Chapter 11.Mathematics in Related Fields 95 11.1 Ornament and Design; Art and Architecture 95 11.2 Dynamic Symmetry; Golden Section 96 11.3 Music and Mathematics 97 11.4 Music and Computers 98 11.5 Mathematics in Nature 99 Chapter 12.Recreations in the Classroom 100 12.1 Elementary School Activities 100 12.2 Secondary School Activities 104 12.3 Mathematics Clubs, Plays, Programs, Projects 108 12.4 Mathematics Contests, Competitions, Leagues 109 cONTENTS , Appendix A.Contemporary Works on Mathematical Recreations 111 Appendix B.Chronological Synopsis of Martin Gardner's Column in Scientific American 114 Glossary. A List of Terms Related to Recreational Mathematics 119 Problems Problems worthy of attack prove their worth by hitting back. PIET HEIN Grooks 1, p. 2 (1966). Last Things First Solutions to problems are easy to find: the problem's a great contribution. What is truly an art is to wring from your mind a problem to fit a solution. PIET HEIN Grooks 3, p. 15 (1970) Reprinted by permission of the Piet Hein International Information Center. 0 Aspila 1970, all rights reserved. Principal Abbreviations Used A.iW.i't. =American Mathematical Monthly A.T. ,arithmetic Teacher Fib.Q. = Fibonacci Quarterly J.R.M. = I ournal o f Recreational Mathematics M.Gaz. = Mathematical Gazette M.Mag. = Mathematics Magazine M.S.J. = Mathematics Student Journal M.T. = Mathematics Teacher M.Tchg. = Mathematics Teaching (England) NCTM = National Council of Teachers of Mathematics N.M.M. = National Mathematics Magazine P.M.E.J. = Pi Mu Epsilon Journal R.M.M. = Recreational Mathematics Magazine Sci.Am. = Scientific American Sci.Mo. = Scientific Monthly S.S.M. = School Science and Mathematics Scrip.M. = Scripta Mathematica Chapter 1 /11eastereetat Rematioaa 1.1 Calendar Problems Austin, A. K. A perpetual calendar. M.Tchg., no. 56, p. 18; Autumn 1971. Feser, Victor G. Annual sums. J.R.M. 5:252; Oct. 1972. Product dates. J.R.M. 5:251-52; Oct. 1972. Fredregill, Ernest J. 1000 Years: A Julian/Gregorian Perpetual Calendar. New York: Exposition Press. 1970. 38 pp. Brief history of calendars and methods of construction. Gardner, Martin. Birthday paradox. Sci.Am. Apr. 1957, p. 166. Calendar problems. Sci.Am., May 1967; Apr.
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