~1ATHEMATICAL PUZZLES FOR THE SECONDARY MATHE~1ATICS .-' TEACHER: A COLLECTION, CLASSIFICATION, AND EVALUATION A Thesis Presented to the Graduate Faculty of California State University, Hayward In Partial Fulfillment of the Requirements for the Degree Master of Science in Education By Randall R. Baumback \ November,\ 1980 Copyright @ 1980 By Randall R. Baumback ii ABSTRACT The purpose of this study is to provide secondary mathematics teachers with a collection of classified mathematical puzzles which when used in conjunction with the appropriate level and concept in the mathematics curricula, will serve to stimulate, illustrate, and enrich the desired objectives of study. As a basis for this study, three assumptions were made: (1) attitudes are fundamental to, and contribute to what students learn, (2) mathematical puzzles intrigue and motivate students, and (3) the use of puzzles as an integral part of the mathematics instruction is an effective way to teach concepts and skills. Teachers recognize that students' mathematical achievement and interest has suffered, and that recreational materials provide a welcomed relief from the rigorous mathematics curricula. However, a brief review of the literature will show that it does not provide the secondary mathematics teacher with a practical, effective classroom resource. From the available literature on mathematics recreations, the author made a final selection, edition, and classification of the 117 puzzles in Chapter 3 iii according to four criteria: (1) clarity and under­ standability, (2) time factor practicality, (3) motiva­ tional value, and (4) illustration of a mathematical concept. An evaluation of the puzzles was made by five secondary mathematics teachers. Each teacher rated each puzzle with respect to the four criteria on a four-point scale. An analysis of the data followed revealing that nearly all of the puzzles were rated very highly. Hence, the puzzle collection was judged to be an excellent resource for secondary mathematics teachers by those teachers who served on the evaluation panel. iv Y..)._THEl1ATICAL ?GZZI.ES FOR THE SECONDARY :1J:._THE:1ATICS TEACHER: A COLLECTION, CLASSIFICATION AND EVALUATION 3y ?,andall ?. 3a.umbac;·: APPROVED: DATE: Lk'.2. 7:> 1'180 ~e~:f::(/(~ 9 ///0 / v ACKNOWLEDGMENTS I am indebted to Dr. John Hancock for his immeasurable guidance in developing this work, and which is in large part a reflection of his professional concern for the student's and teacher's needs. Also, I wish to thank Dr. Albert Lepore for his crucial assistance, and Dr. Gerald Brown for his invaluable references. A special thanks to Cathy Vulpi, Vicki Reese, Bill Oliver, Rich Cordes, and Janice Carter for their participation, and a word of appreciation is due the James Logan High School Mathematics Department for their cooperation in this study. I am most grateful to my Mother and Father whose love, and encouragement has enabled me the fulfillment of undertakings such as this one. vi TABLE OF CONTENTS Page ABSTRACT .... iii ACKNOWLEDGMENTS vi LIST OF TABLES . xv Chapter 1. INTRODUCTION 1 PURPOSE OF THE STUDY . 1 BACKGROUND OF THE STUDY 1 Inaccessibility of Appropriate Puzzle Literature .... ... .. 2 ASSUMPTIONS 3 REVIEW OF THE LITERATURE .. 4 Use of Mathematical Puzzles to Illustrate a Concept .... 4 Use of Mathematical Puzzles to Stimulate Student Interest 11 2. METHOD OF STUDY 15 PROCEDURE FOR DEVELOPING PUZZLE COLLECTION 15 DISCUSSION OF CRITERIA . .'. 16 Clarity and Understandability 16 Time Factor Practicality 16 Motivational Value. .. 17 Illustration of a Mathematical Concept. 17 vii TABLE OF CONTENTS (Continued) Chapter Page DESCRIPTION OF POPULATION ... 17 INSTRUMENTATION ......... 18 PROCEDURE FOR COLLECTING DATA . 18 3. MATHEMATICAL PUZZLES .. .. 20 CONCEPTS COVERED ... 20 ARITHMETIC PUZZLES ... 24 1. A Snail in the Well .... 24 2. A Tennis Tournament. .. 25 3. Ten Million Bacter~a 25 4. How Many Squares? 25 5. The Bookworm ... 26 6. Squares. 26 7. A Cake of Soap ... 27 8. The Puzzled Driver ... 27 9. Thirty Flasks. .. .. 28 10. How Many Sandals? .... 28 11. From 1 to 1,000,000,000 28 12. Grandfather's Faded Bill .... 29 13. A Pile of Paper .. 29 14. The Deleted Checkerboard 30 15. Twenty-One Pigs 30 16. Me Pay You? .. 30 17. Ten Vacant Rooms 31 viii TABLE OF CONTENTS (Continued) Chapter Page 18. Three Men in a Hotel .. 32 19. Guess an Age .. 33 20. A Trick with Three Dice 34 21. A Right Answer ... 34 22. A Phonograph Record 35 Solutions to Arithmetic Puzzles . 35 ALGEBRA PUZZLES ..... 42 23. De Morgan's Birthday. 42 24. Chain Letter .... 42 25. If Half of Five is Three 43 26. A Bottle and a Cork 43 27. Brothers and Sisters. 43 28. How Much does the Brick Weigh? . 43 29. The Annual Pioneer Pancake Eating Contest ....... 43 30. Algebra Homework .. 44 31. How Many Students Did Mr. Einstein Have? ..... .. 44 32. A Basket of Eggs . 44 33. Jack and Jill 45 34. Phil Anthrope 45 35. More or Less? . .. 45 36. Cinch Around the Earth 46 ix TABLE OF CONTENTS (Continued) Chapter Page 37. Counting Sheep 46 38. Find the Flaw 46 39. Is 4 = 8? 47 40. What is the Error? . 47 4l. The Lucas Problem 47 42. A Mountain Climber's Journey. 48 43. A Card Trick ... 48 44. Crossing Tiles 49 45. The Refreshed Runner . 49 Solutions to Algebra Puzzles 50 GEOMETRY PUZZLES ... 61 46. Finding a Proof 61 47. Irish Friends ..... 61 48. The Game of "Stogey" 62 49. Toe-Tac-Tic . 62 50. Medians of a Triangle 62 51. A Paradox . 63 52. What is the Length? 64 53. To Find the Center of a Circle. 64 54. The Geometry Club's Badge 65 55. Remaining Metal 66 56. Yang, Ying and Yung 66 x TABLE OF CONTENTS (Continued) Chapter Page 57. A Grazing Goat 67 58. Separating the Sheep . 67 59. Koch's Triangles .. 68 60. Overlapping Figures 68 61. A Square Peg in a Round Hole . 69 62. The Carpenter's Rope .. 69 63. The Spider and the Fly 69 64. The Hunter's Dilemma. 70 65. The Fly and Honey 70 66. A Familiar Object 71 67. A Bottle's Volume 71 68. A Curious Sphere .. 72 69. A Geometrical "Vanish" 72 70. What's My Angle? . 73 71. Planting Trees . 73 Solutions to Geometry Puzzles . 73 INTERMEDIATE ALGEBRA PUZZLES 88 72. Three Daughters 88 73. How Many Pages? 88 74. The Evasive Engineer . 88 75. Watches 89 76. Two Candles 89 xi TABLE OF CONTENTS (Continued) Chapter Page 77. Two Horses and a Fly .. 89 78. Draining a Water Tank 90 79. The Physicist and the Escalator 90 80. Average Speed 90 8l. The Confused Teller 91 82. How Long was the Vacation? . 91 83. The Restaurant Bill 91 84. Scales .. 92 85. Bank Shot 92 86. An Infinitude of Twos 93 87. The Golden Ratio .. 93 88. A Swarm of Bees ... 93 89. An Algebra Error .. 94 90. An Elephant and a Mosquito 95 9l. Is 1 = -l? 96 92. Explain the Paradox 96 93. Another Paradox 96 Solutions to Intermediate Algebra Puzzles 97 ADVANCED MATHEMATICS PUZZLES 107 94. Two Trees with the Same Number of Leaves ... 107 95. Slicing a Cube. 108 xii TABLE OF CONTENTS (Continued) Chapter Page 96. Tower of Hanoi ... 108 97. A Deceitful Proof by Mathematical Induction .. ... 109 98. Glasses 110 99. Faulty Scale. 110 100. Time to Trisect any Angle 110 101. The Triangle is Equilateral 110 102. Four Bugs 110 103. A House Number III 104. A Calculus Paradox. 111 105. How Many Handshakes? . 112 106. Triangles 112 107. Thirty-one Flavors. 112 108. How Many Routes? .. 113 109. The Prize Contest 113 110. A Fast Deal 114 111. Four Letters 114 112. The Pentagon Building 115 113. The Same Birthday 115 114. Three Darts 115 115. Professor of Ancient History 115 116. Are You Certain? .. 116 xiii TABLE OF CONTENTS (Continued) Chapter Page 117. A Chance for Survival . 116 Solutions to Advanced Mathematics Puzzles 116 PUZZLE SOURCES . 128 4. RESULTS .... 130 5 . ANALYSIS, SUMMARY, AND RECOMMENDATIONS 139 ANALYSIS OF EVALUATIONS 139 SUMMARY ........ 140 RECOMMENDATIONS . 141 BIBLIOGRAPHY ...... 142 APPENDICES I. DATA SHEET ............... 148 II. RATING FORM AND PUZZLE EVALUATIONS 152 xiv LIST OF TABLES Table Page 1. Distribution of Ratings of Criteria for Puzzles ....... 154 2. Puzzle Ratings: Summarized Ratings by Respondents . 132 3. Summary of Ratings of Puzzles .. 138 xv Chapter 1 INTRODUCTION PURPOSE OF THE STUDY The purpose of this study is to provide secondary mathematics teachers with a collection of classified mathematical puzzles which, when used in conjunction with the appropriate level and concept in the mathematics curricula, will serve to stimulate, illustrate, and enrich the desired objectives of study. BACKGROUND OF THE STUDY There are several strategies available for teaching most secondary mathematical concepts~ however, rarely is an amusing mathematical puzzle used to stimulate the interest or attention required to demonstrate the desired topic. Although most experienced mathematics teachers know a few such puzzles, well organized collections of mathematical puzzles are scarce at this level. At the same time, teachers recognize that student achievement and interest has suffered froITl the most abstract of all the secondary subjects, and that recreational materials provide a welcomed relief from the rigor of the 1 2 mathematics curricula. This study is designed to partially free that gap--to promote both learning and enthusiasm for students and teachers alike of secondary mathematics. Inaccessibility of Appropriate Puz£IeLT"terature- ---- The pleasure and stimulation found in grappling with mathematical puzzles is not new, attested by the many books in the field. However, a brief review of the literature will show that it does not provide the secondary mathematics teacher with a practical, effective classroom resource. In general, the available books suffer from one or more serious pedagogical defects. Many books offer a collection of puzzles with no deliberate classification or continuity.