Book contents
- Frontmatter
- 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 Erdõs
- 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 3/2 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 – Sarkozy and Szemeredi
- 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 ShortWords – First Cases
- 86 ShortWords – 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 – Erdõs 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
117 - Triangles and Quadrilaterals – Regiomontanus
from The Solutions
Published online by Cambridge University Press: 08 July 2022
Book contents
- Frontmatter
- 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 Erdõs
- 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 3/2 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 – Sarkozy and Szemeredi
- 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 ShortWords – First Cases
- 86 ShortWords – 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 – Erdõs 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
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- The Art of Mathematics – Take TwoTea Time in Cambridge, pp. 305 - 307Publisher: Cambridge University PressPrint publication year: 2022