Lovers of mathematics, young and old, professional and amateur, will enjoy this book. It is mathematics with fun: a collection of attractive problems that will delight and test readers. Many of the problems are drawn from the large number that have entertained and challenged students, guests and colleagues over the years during afternoon tea. The problems have their roots in many areas of mathematics. They vary greatly in difficulty: some are very easy, but most are far from trivial, and quite a few rather hard. Many provide substantial and surprising results that form the tip of an iceberg, providing an introduction to an important topic. To enjoy and appreciate the problems, readers should browse the book choosing one that looks particularly enticing, and think about it on and off for a while before resorting to the hint or the solution. Follow threads for an enjoyable and enriching journey through mathematics.
Author(s): Béla Bollobás
Edition: 1
Publisher: Cambridge University Press
Year: 2022
Language: English
Pages: 348
Tags: Mathematics; Puzzles; Problems
Contents
Preface
The Problems
The Hints
The Solutions
1. Real Sequences – An Interview Question
2. Vulgar Fractions – Sylvester’s Theorem
3. Rational and Irrational Sums
4. Ships in Fog
5. A Family of Intersections
6. The Basel Problem – Euler’s Solution
7. Reciprocals of Primes – Euler and Erdős
8. Reciprocals of Integers
9. Completing Matrices
10. Convex Polyhedra – Take One
11. Convex Polyhedra – Take Two
12. A Very Old Tripos Problem
13. Angle Bisectors – the Lehmus–Steiner Theorem
14. Langley’s Adventitious Angles
15. The Tantalus Problem – from The Washington Post
16. Pythagorean Triples
17. Fermat’s Theorem for Fourth Powers
18. Congruent Numbers – Fermat
19. A Rational Sum
20. A Quartic Equation
21. Regular Polygons
22. Flexible Polygons
23. Polygons of Maximal Area
24. Constructing √2 – Philon of Byzantium
25. Circumscribed Quadrilaterals – Newton
26. Partitions of Integers
27. Parts Divisible by m and 2m
28. Unequal vs Odd Partitions
29. Sparse Bases
30. Small Intersections – Sárközy and Szemerédi
31. The Diagonals of Zero–One Matrices
32. Tromino and Tetronimo Tilings
33. Tromino Tilings of Rectangles
34. Number of Matrices
35. Halving Circles
36. The Number of Halving Circles
37. A Basic Identity of Binomial Coefficients
38. Tepper’s Identity
39. Dixon’s Identity – Take One
40. Dixon’s Identity – Take Two
41. An Unusual Inequality
42. Hilbert’s Inequality
43. The Central Binomial Coefficient
44. Properties of the Central Binomial Coefficient
45. Products of Primes
46. The Erdős Proof of Bertrand’s Postulate
47. Powers of 2 and 3
48. Powers of 2 Just Less Than a Perfect Power
49. Powers of 2 Just Greater Than a Perfect Power
50. Powers of Primes Just Less Than a Perfect Power
51. Banach’s Matchbox Problem
52. Cayley’s Problem
53. Min vs Max
54. Sums of Squares
55. The Monkey and the Coconuts
56. Complex Polynomials
57. Gambler’s Ruin
58. Bertrand’s Box Paradox
59. The Monty Hall Problem
60. Divisibility in a Sequence of Integers
61. Moving Sofa Problem
62. Minimum Least Common Multiple
63. Vieta Jumping
64. Infinite Primitive Sequences
65. Primitive Sequences with a Small Term
66. Hypertrees
67. Subtrees
68. All in a Row
69. An American Story
70. Six Equal Parts
71. Products of Real Polynomials
72. Sums of Squares
73. Diagrams of Partitions
74. Euler’s Pentagonal Number Theorem
75. Partitions – Maximum and Parity
76. Periodic Cellular Automata
77. Meeting Set Systems
78. Dense Sets of Reals – An Application of the Baire Category Theorem
79. Partitions of Boxes
80. Distinct Representatives
81. Decomposing a Complete Graph: The Graham–Pollak Theorem – Take One
82. Matrices and Decompositions: The Graham–Pollak Theorem – Take Two
83. Patterns and Decompositions: The Graham–Pollak Theorem – Take Three
84. Six Concurrent Lines
85. Short Words – First Cases
86. Short Words – The General Case
87. The Number of Divisors
88. Common Neighbours
89. Squares in Sums
90. Extension of Bessel’s Inequality – Bombieri and Selberg
91. Equitable Colourings
92. Scattered Discs
93. East Model
94. Perfect Triangles
95. A Triangle Inequality
96. An Inequality for Two Triangles
97. Random Intersections
98. Disjoint Squares
99. Increasing Subsequences – Erdős and Szekeres
100. A Permutation Game
101. Ants on a Rod
102. Two Cyclists and a Swallow
103. Almost Disjoint Subsets of Natural Numbers
104. Primitive Sequences
105. The Time of Infection on a Grid
106. Areas of Triangles: Routh’s Theorem
107. Lines and Vectors – Euler and Sylvester
108. Feuerbach’s Remarkable Circle
109. Euler’s Ratio–Product–Sum Theorem
110. Bachet’s Weight Problem
111. Perfect Partitions
112. Countably Many Players
113. One Hundred Players
114. River Crossings: Alcuin of York – Take One
115. River Crossings: Alcuin of York – Take Two
116. Fibonacci and a Medieval Mathematics Tournament
117. Triangles and Quadrilaterals – Regiomontanus
118. The Cross-Ratios of Points and Lines
119. Hexagons in Circles: Pascal’s Hexagon Theorem – Take One
120. Hexagons in Circles: Pascal’s Theorem – Take Two
121. A Sequence in Zp
122. Elements of Prime Order
123. Flat Triangulations
124. Triangular Billiard Tables
125. Chords of an Ellipse: The Butterfly Theorem
126. Recurrence Relations for the Partition Function
127. The Growth of the Partition Function
128. Dense Orbits
