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The Art of Mathematics – Take Two : Tea Time in Cambridge
Constructing 3/2 Philon of Byzantium
monograph
Publication date:
June 30 2022
Publisher:
Cambridge University Press
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Book Chapter
Publication date:
June 30 2022
Pages
: 105-107
DOI:
10.1017/9781108973885.028
SO-VID:
ad4f0c4e-6d2d-4ea6-aff5-9a61f6e1b225
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Book chapters
pp. 1
The Problems
pp. 47
Real Sequences – An Interview Question
pp. 49
Vulgar Fractions – Sylvester’s Theorem
pp. 53
Rational and Irrational Sums
pp. 55
Ships in Fog
pp. 56
A Family of Intersections
pp. 58
The Basel Problem – Euler’s Solution
pp. 61
Reciprocals of Primes – Euler and Erdõs
pp. 65
Reciprocals of Integers
pp. 66
Completing Matrices
pp. 68
Convex Polyhedra – Take One
pp. 70
Convex Polyhedra – Take Two
pp. 72
A Very Old Tripos Problem
pp. 74
Angle Bisectors – the Lehmus–Steiner Theorem
pp. 76
Langley’s Adventitious Angles
pp. 79
The Tantalus Problem – from The Washington Post
pp. 81
Pythagorean Triples
pp. 84
Fermat’s Theorem for Fourth Powers
pp. 86
Congruent Numbers – Fermat
pp. 89
A Rational Sum
pp. 92
A Quartic Equation
pp. 95
Regular Polygons
pp. 99
Flexible Polygons
pp. 100
Polygons of Maximal Area
pp. 105
Constructing 3/2 Philon of Byzantium
pp. 108
Circumscribed Quadrilaterals – Newton
pp. 111
Partitions of Integers
pp. 114
Parts Divisible by m and 2m
pp. 115
Unequal vs Odd Partitions
pp. 117
Sparse Bases
pp. 119
Small Intersections – Sarkozy and Szemeredi
pp. 122
The Diagonals of Zero–One Matrices
pp. 123
Tromino and Tetronimo Tilings
pp. 126
Tromino Tilings of Rectangles
pp. 33
The Hints
pp. 128
Number of Matrices
pp. 129
Halving Circles
pp. 131
The Number of Halving Circles
pp. 134
A Basic Identity of Binomial Coefficients
pp. 136
Tepper’s Identity
pp. 138
Dixon’s Identity – Take One
pp. 140
Dixon’s Identity – Take Two
pp. 143
An Unusual Inequality
pp. 145
Hilbert’s Inequality
pp. 147
The Central Binomial Coefficient
pp. 149
Properties of the Central Binomial Coefficient
pp. 151
Products of Primes
pp. 153
The Erdős Proof of Bertrand’s Postulate
pp. 155
Powers of 2 and 3
pp. 156
Powers of 2 Just Less Than a Perfect Power
pp. 158
Powers of 2 Just Greater Than a Perfect Power
pp. 159
Powers of Primes Just Less Than a Perfect Power
pp. 164
Banach’s Matchbox Problem
pp. 166
Cayley’s Problem
pp. 168
Min vs Max
pp. 170
Sums of Squares
pp. 172
The Monkey and the Coconuts
pp. 174
Complex Polynomials
pp. 175
Gambler’s Ruin
pp. 179
Bertrand’s Box Paradox
pp. 181
The Monty Hall Problem
pp. 185
Divisibility in a Sequence of Integers
pp. 187
Moving Sofa Problem
pp. 191
Minimum Least Common Multiple
pp. 193
Vieta Jumping
pp. 195
Infinite Primitive Sequences
pp. 197
Primitive Sequences with a Small Term
pp. 199
Hypertrees
pp. 200
Subtrees
pp. 201
All in a Row
pp. 203
An American Story
pp. 205
Six Equal Parts
pp. 208
Products of Real Polynomials
pp. 210
Sums of Squares
pp. 212
Diagrams of Partitions
pp. 214
Euler’s Pentagonal Number Theorem
pp. 218
Partitions – Maximum and Parity
pp. 220
Periodic Cellular Automata
pp. 223
Meeting Set Systems
pp. 225
Dense Sets of Reals – An Application of the Baire Category Theorem
pp. 227
Partitions of Boxes
pp. 229
Distinct Representatives
pp. 230
Decomposing a Complete Graph: The Graham–Pollak Theorem – Take One
pp. 232
Matrices and Decompositions: The Graham–Pollak Theorem – Take Two
pp. 234
Patterns and Decompositions: The Graham–Pollak Theorem – Take Three
pp. 236
Six Concurrent Lines
pp. 237
ShortWords – First Cases
pp. 239
ShortWords – The General Case
pp. 241
The Number of Divisors
pp. 243
Common Neighbours
pp. 244
Squares in Sums
pp. 245
Extension of Bessel’s Inequality – Bombieri and Selberg
pp. 247
Equitable Colourings
pp. 249
Scattered Discs
pp. 251
East Model
pp. 254
Perfect Triangles
pp. 256
A Triangle Inequality
pp. 258
An Inequality for Two Triangles
pp. 260
Random Intersections
pp. 262
Disjoint Squares
pp. 264
Increasing Subsequences – Erdõs and Szekeres
pp. 266
A Permutation Game
pp. 267
Ants on a Rod
pp. 268
Two Cyclists and a Swallow
pp. 270
Almost Disjoint Subsets of Natural Numbers
pp. 272
Primitive Sequences
pp. 274
The Time of Infection on a Grid
pp. 276
Areas of Triangles: Routh’s Theorem
pp. 282
Lines and Vectors – Euler and Sylvester
pp. 284
Feuerbach’s Remarkable Circle
pp. 286
Euler’s Ratio–Product–Sum Theorem
pp. 288
Bachet’s Weight Problem
pp. 291
Perfect Partitions
pp. 294
Countably Many Players
pp. 296
One Hundred Players
pp. 298
River Crossings: Alcuin of York – Take One
pp. 301
River Crossings: Alcuin of York – Take Two
pp. 303
Fibonacci and a Medieval Mathematics Tournament
pp. 305
Triangles and Quadrilaterals – Regiomontanus
pp. 308
The Cross-Ratios of Points and Lines
pp. 312
Hexagons in Circles: Pascal’s Hexagon Theorem – Take One
pp. 315
Hexagons in Circles: Pascal’s Theorem – Take Two
pp. 318
A Sequence in Zp
pp. 319
Elements of Prime Order
pp. 320
Flat Triangulations
pp. 322
Triangular Billiard Tables
pp. 324
Chords of an Ellipse: The Butterfly Theorem
pp. 326
Recurrence Relations for the Partition Function
pp. 328
The Growth of the Partition Function
pp. 332
Dense Orbits
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