Mathematical Recreations and Essays

Preface to the First Edition......Page 5
Note to the Fourth Edition......Page 7
Table of Contents......Page 8
Mathematical Recreations.......Page 17
Some Arithmetical Questions.......Page 18
Elementary Questions on Numbers (Miscellaneous)......Page 20
Arithmetical Fallacies......Page 36
Bachet\'s Weights Problem......Page 43
Problems in Higher Arithmetic......Page 45
Fermat\'s Theorem on Binary Powers......Page 47
Fermat\'s Last Theorem......Page 48
Geometrical Fallacies......Page 51
Geometrical Paradoxes......Page 58
Colouring Maps......Page 60
Physical Geography......Page 62
Three-in-a-row.  Extension to p-in-a-row......Page 64
Tesselation.  Cross-Fours......Page 66
Colour-Cube Problem......Page 67
Dynamical Games of Position......Page 68
Shunting Problems......Page 69
Ferry-Boat Problems......Page 71
Geodesic Problems......Page 73
Problems with Counters placed in a row......Page 74
Problems on a Chess-board with Counters or Pawns......Page 76
Guarini\'s Problem......Page 79
Paradromic Rings......Page 80
Some Mechanical Questions.......Page 82
Paradoxes on Motion......Page 83
Force, Inertia, Centrifugal Force......Page 86
Work, Stability of Equilibrium, &c.......Page 88
Perpetual Motion......Page 91
Models......Page 94
Sailing quicker than the Wind......Page 95
Boat moved by a rope inside the boat......Page 97
Results dependent on Hauksbee\'s Law......Page 98
Cut on a tennis-ball.  Spin on a cricket-ball......Page 99
Flight of Birds......Page 101
Curiosa Physica......Page 102
The Fifteen Puzzle......Page 104
The Tower of Hanoï......Page 107
Chinese Rings......Page 109
The Eight Queens Problem......Page 113
Other Problems with Queens and Chess-pieces......Page 118
The Fifteen School-Girls Problem......Page 119
Monge on shuffling a pack of cards......Page 125
Arrangement by rows and columns......Page 127
Determination of one out of ½n(n+1) given couples......Page 129
Gergonne\'s Pile Problem......Page 131
The Mouse Trap.  Treize......Page 135
Magic Squares.......Page 137
Notes on the History of Magic Squares......Page 138
Construction of Odd Magic Squares......Page 139
Method of De la Loubère......Page 140
Method of Bachet......Page 141
Method of De la Hire......Page 142
Construction of Even Magic Squares......Page 144
First Method......Page 145
Method of De la Hire and Labosne......Page 148
Composite Magic Squares......Page 150
Bordered Magic Squares......Page 151
Pan-diagonal or Nasik Squares......Page 152
Magic Pencils......Page 153
Euler\'s Officers Problem......Page 156
Coin Squares......Page 157
Euler\'s Problem......Page 159
Euler\'s Theorems......Page 161
Examples......Page 164
Mazes......Page 165
Notes on the History of Mazes......Page 166
Geometrical Trees......Page 170
The Hamiltonian Game......Page 171
Knight\'s Path on a Chess-Board......Page 174
Method of Euler......Page 175
Method of Vandermonde......Page 179
Method of Roget......Page 180
Method of Moon......Page 183
Paths of other Chess-Pieces......Page 184
Miscellaneous Essays and Problems.......Page 186
Medieval Course of Studies: Acts......Page 187
Subject-Matter of Acts at different periods......Page 188
Oral Examinations always possible......Page 190
Additional work thrown on Moderators. Stipends raised......Page 191
Right of M.A.s to take part in it......Page 192
Scheme of Examination in 1763......Page 193
Foundations of Smith\'s Prizes, 1768......Page 194
Description of the Examination in 1772......Page 195
System of Brackets......Page 198
Problem Papers in 1785 and 1786......Page 199
Description of the Examination in 1791......Page 200
The Poll Part of the Examination......Page 201
Problem Papers from 1802 onwards......Page 202
Description of the Examination in 1802......Page 203
Scheme of Reading in 1806......Page 205
Introduction of modern analytical notation......Page 208
Scheme of Examination in 1827......Page 211
Scheme of Examination in 1839......Page 213
Creation of a Board of Mathematical Studies......Page 214
Scheme of Examination in 1873......Page 215
Scheme of Examination in 1882......Page 216
Origin of term Tripos......Page 217
Tripos Verses......Page 218
The Three Problems......Page 220
Legendary origin of the problem......Page 221
Lemma of Hippocrates......Page 222
Solutions of Archytas, Plato, Menaechmus, Apollonius, and Sporus......Page 223
Solutions of Vieta, Descartes, Gregory of St Vincent, and Newton......Page 225
Solutions quoted by Pappus (three)......Page 226
Solutions of Descartes, Newton, Clairaut, and Chasles......Page 227
Incommensurability of pi......Page 228
Definitions of pi......Page 229
Geometrical methods of approximation......Page 230
Results of Archimedes and other Greek writers......Page 231
Results of Indian and Eastern writers......Page 232
Results of European writers, 1200--1630......Page 233
Results of European writers, 1699--1873......Page 236
Approximations by the theory of probability......Page 238
Mersenne\'s Enunciation of the Theorem......Page 240
Cases awaiting verification......Page 241
History of Investigations......Page 242
Methods used in attacking the problem......Page 246
By trial of divisors of known forms......Page 247
By indeterminate equations......Page 249
By the use of a Canon Arithmeticus......Page 250
By the use of the binary scale......Page 251
Mechanical methods of Factorizing Numbers......Page 252
Astrology.......Page 253
Houses and their significations......Page 254
Planets and their significations......Page 256
Zodiacal signs and their significations......Page 258
Knowledge that rules were worthless......Page 259
Flamsteed\'s guess......Page 262
Cardan\'s horoscope of Edward VI......Page 263
A Cryptograph. Definition. Illustration......Page 267
Essential Features of Cryptographs and Ciphers......Page 268
Order of letters re-arranged......Page 269
Use of non-significant symbols. The Grille......Page 272
Use of broken symbols. The Scytale......Page 274
Ciphers of Four Types......Page 275
Ciphers of the First Type. Illustrations......Page 276
Ciphers of the Second Type. Illustrations......Page 279
Ciphers of the Third Type. Illustrations......Page 281
Ciphers of the Fourth Type. Illustrations......Page 283
Requisites in a good Cipher......Page 284
Charles I......Page 285
Pepys......Page 287
Marie Antoinette......Page 288
The Code Dictionary......Page 290
Poe\'s Writings......Page 291
Hyper-space.......Page 293
Space of two dimensions and of one dimension......Page 294
Existence in such a world......Page 295
Arguments in favour of the existence of such a world......Page 296
Euclid\'s axioms and postulates. The parallel postulate......Page 300
Elliptic, Parabolic and Hyperbolic Geometries compared......Page 301
Non-Euclidean Geometries of three or more dimensions......Page 303
Time and its Measurement.......Page 304
Units for measuring durations (days, weeks, months, years)......Page 305
The Civil Calendar (Julian, Gregorian, &c.)......Page 308
The Ecclesiastical Calendar (date of Easter)......Page 310
Styles, Sun-dials, Sun-rings......Page 313
Clocks and Watches......Page 317
Watches as Compasses......Page 319
Matter and Ether Theories.......Page 321
Popular Atomic Hypothesis......Page 322
Hypothesis of an Elastic Solid Ether. Labile Ether......Page 323
The Vortex Ring Hypothesis......Page 324
The Vortex Sponge Hypothesis......Page 325
The Ether-Squirts Hypothesis......Page 326
Speculations due to investigations on Radio-activity......Page 327
The Bubble Hypothesis......Page 329
Conjectures as to the cause of Gravity......Page 330
Conjectures to explain the finite number of species of Atoms......Page 334
Size of the molecules of bodies......Page 336
Index......Page 339
Notices of some works---chiefly historico-mathematical......Page 351
Project Gutenberg Licensing Information......Page 371