**Problem Books**

Volume: 36;
2020;
286 pp;
Softcover

MSC: Primary 00;

**Print ISBN: 978-1-4704-4759-5
Product Code: PRB/36**

List Price: $55.00

AMS Member Price: $41.25

MAA Member Price: $41.25

**Electronic ISBN: 978-1-4704-5702-0
Product Code: PRB/36.E**

List Price: $55.00

AMS Member Price: $41.25

MAA Member Price: $41.25

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#### Supplemental Materials

# Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles

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*Daniel J. Velleman; Stan Wagon*

MAA Press: An Imprint of the American Mathematical Society

Bicycle or Unicycle? is a collection
of 105 mathematical puzzles whose defining characteristic is the
surprise encountered in their solutions. Solvers will be surprised,
even occasionally shocked, at those solutions. The problems unfold
into levels of depth and generality very unusual in the types of
problems seen in contests. In contrast to contest problems, these are
problems meant to be savored; many solutions, all beautifully
explained, lead to unanswered research questions. At the same time,
the mathematics necessary to understand the problems and their
solutions is all at the undergraduate level. The puzzles will,
nonetheless, appeal to professionals as well as to students and, in
fact, to anyone who finds delight in an unexpected discovery.

These problems were selected from the Macalester College Problem of
the Week archive. The Macalester tradition of a weekly problem was
started by Joseph Konhauser in 1968. In 1993 Stan Wagon assumed
problem-generating duties. A previous book written by Wagon,
Konhauser, and Dan Velleman, Which Way Did the Bicycle Go?,
gathered problems from the first twenty-five years of the archive.
The title problem in that collection was inspired by an error in logic
made by Sherlock Holmes, who attempted to determine the direction of a
bicycle from the tracks of its wheels. Here the title problem asks
whether a bicycle track can always be distinguished from a unicycle
track. You'll be surprised by the answer.

#### Readership

Undergraduate and graduate students and researchers interested in problem solving.

#### Table of Contents

# Table of Contents

## Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles

Table of Contents pages: 1 2

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface xiii15
- Part 1 Problems 119
- Chapter 1. Can a Bicycle Simulate a Unicycle? 321
- Chapter 2. Geometry 523
- 2. Trisecting to Benefit Society 523
- 3. Ten Bottles of Wine 523
- 4. Into the Woods 523
- 5. The Attraction of the Golden Ratio 624
- 6. Reflect on This 624
- 7. Skewed Pizza 725
- 8. Four-Regular Squares 725
- 9. Venn Symmetry 826
- 10. A Crosscut Quadrilateral 927
- 11. The Legacy of H. G. Wells 927
- 12. Tiling Surprise 927
- 13. A Well-Balanced Clock 1129
- 14. On the Level 1129
- 15. A Polyhedral Puzzle 1129
- 16. Wiggle Room 1230
- 17. Drink Me 1230
- 18. Right-Angled Polygons 1230
- 19. A Rolling Parabola 1230
- 20. Rocket Science 1331
- 21. The Icing on the Cake 1432
- 22. More Cake 1432

- Chapter 3. Number Theory 1533
- 23. Power Matching 1533
- 24. Triplets 1533
- 25. The Mysterious Seventeenth Divisor 1533
- 26. True or False? 1533
- 27. An Exponential Diophantine Problem 1634
- 28. A Prime Characterization 1634
- 29. Two Sums and Many Differences 1634
- 30. Reciprocals to Squares 1634
- 31. Nondivisibility by 11 1634
- 32. Equally Powerful Splits 1634
- 33. Tripling in Two Steps 1735
- 34. Root Closure 1735
- 35. A Double Power Leads to a Sum of Two Squares 1735
- 36. Prime Subsets 1735
- 37. The Incredible Shrinking Superpowers 1735
- 38. Special Numbers 1937

- Chapter 4. Combinatorics 2139
- 39. Power up Your Radio 2139
- 40. A Parking Puzzle 2139
- 41. A Black and White Issue 2240
- 42. A Choice Problem 2240
- 43. Water Wave 2240
- 44. An Acceptable Committee 2240
- 45. A Compatible Committee 2341
- 46. Using Blocks and Chains to Unlock a Safe 2341
- 47. A Universal Set of Directions 2442
- 48. A Competition Problem About a Competition Problem 2442

- Chapter 5. Probability 2543
- 49. The Importance of Irrelevant Information 2543
- 50. Creeping Ants 2543
- 51. Roll the Dice 2543
- 52. Conditioned Throws of a Die 2644
- 53. Where Are the Rounded Powers of Two? 2644
- 54. The Holy Game of Poker 2644
- 55. A Shrinking Random Walk 2745
- 56. BINGO! 2745
- 57. How Much is a Penny Worth? 2745
- 58. A Historically Interesting Truth-Teller Problem 2846
- 59. Monty Hall Revisited 2846

- Chapter 6. Calculus 3149
- Chapter 7. Algorithms and Strategy 3351
- 69. Where’s Bob? 3351
- 70. It’s a Horse Race 3351
- 71. Your Two Best Shots 3351
- 72. Determine the Martian Majority 3452
- 73. Majority Rules 3452
- 74. Going for Gold 3452
- 75. The Mensa Correctional Institute 3553
- 76. Matching Hats 3553
- 77. One Hat Too Many 3553
- 78. The Prisoners Must Agree 3654
- 79. How to Use Irrelevant Information 3755
- 80. Flipping Pennies 3755
- 81. Battleship Destruction 3755
- 82. Real Battleship Destruction 3856
- 83. A Very Local Maximum 3856
- 84. Detecting a Black Hole 3856
- 85. Pablito’s Solitaire 3856
- 86. Are the Coins Authentic? 3957
- 87. Web Site Analysis 3957
- 88. Find the Car, and the Car Key 4058
- 89. Magic Coins 4058
- 90. Russian Cards 4058
- 91. The Generous Automated Teller Machine 4159

- Chapter 8. Miscellaneous 4361
- 92. A Self-Descriptive Crossword 4361
- 93. Serious Implications 4361
- 94. Hermione Granger and the Deadly Bottles 4361
- 95. Forbidden Polynomials 4563
- 96. A Polynomial Scandal 4563
- 97. Pascal’s Determinant 4664
- 98. Pains of Imperfect Glass 4664
- 99. On the Highway 4664
- 100. A Running Mystery 4765
- 101. The Star of This Problem is Addition 4765
- 102. Don’t Get Your Wires Crossed 4765
- 103. Weight Loss Through Juggling 4765
- 104. Cantor Set Arithmetic 4866
- 105. The Miracle of the Colliding Blocks 4967

- Part 2 Solutions 5169
- Chapter 1. Can a Bicycle Simulate a Unicycle? 5371
- Chapter 2. Geometry 6785
- 2. Trisecting to Benefit Society 6785
- 3. Ten Bottles of Wine 6785
- 4. Into the Woods 6886
- 5. The Attraction of the Golden Ratio 6987
- 6. Reflect on This 7088
- 7. Skewed Pizza 7290
- 8. Four-Regular Squares 7492
- 9. Venn Symmetry 7997
- 10. A Crosscut Quadrilateral 8098
- 11. The Legacy of H. G. Wells 8098
- 12. Tiling Surprise 8199
- 13. A Well-Balanced Clock 84102
- 14. On the Level 86104
- 15. A Polyhedral Puzzle 89107
- 16. Wiggle Room 89107
- 17. Drink Me 95113
- 18. Right-Angled Polygons 98116
- 19. A Rolling Parabola 101119
- 20. Rocket Science 102120
- 21. The Icing on the Cake 104122
- 22. More Cake 106124

- Chapter 3. Number Theory 109127
- 23. Power Matching 109127
- 24. Triplets 111129
- 25. The Mysterious Seventeenth Divisor 111129
- 26. True or False? 112130
- 27. An Exponential Diophantine Problem 113131
- 28. A Prime Characterization 114132
- 29. Two Sums and Many Differences 114132
- 30. Reciprocals to Squares 116134
- 31. Nondivisibility by 11 117135
- 32. Equally Powerful Splits 118136
- 33. Tripling in Two Steps 122140
- 34. Root Closure 123141
- 35. A Double Power Leads to a Sum of Two Squares 124142
- 36. Prime Subsets 124142
- 37. The Incredible Shrinking Superpowers 125143
- 38. Special Numbers 127145

- Chapter 4. Combinatorics 131149
- 39. Power up Your Radio 131149
- 40. A Parking Puzzle 135153
- 41. A Black and White Issue 138156
- 42. A Choice Problem 138156
- 43. Water Wave 139157
- 44. An Acceptable Committee 141159
- 45. A Compatible Committee 141159
- 46. Using Blocks and Chains to Unlock a Safe 142160
- 47. A Universal Set of Directions 147165
- 48. A Competition Problem About a Competition Problem 149167

- Chapter 5. Probability 151169
- 49. The Importance of Irrelevant Information 151169
- 50. Creeping Ants 154172
- 51. Roll the Dice 154172
- 52. Conditioned Throws of a Die 155173
- 53. Where Are the Rounded Powers of Two? 157175
- 54. The Holy Game of Poker 162180
- 55. A Shrinking Random Walk 163181
- 56. BINGO! 165183
- 57. How Much is a Penny Worth? 168186
- 58. A Historically Interesting Truth-Teller Problem 169187
- 59. Monty Hall Revisited 171189

- Chapter 6. Calculus 177195
- Chapter 7. Algorithms and Strategy 199217

Table of Contents pages: 1 2