Mathematical Recreations and Essays

W. W. Rouse (Walter William Rouse) Ball

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• Publié par: urgya
• Publié le: 08 décembre 2010
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• Langue: English
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Project Gutenberg’s Mathematical Recreations and Essays, by W. W. Rouse
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Title: Mathematical Recreations and Essays
Author: W. W. Rouse Ball
Release Date: October 8, 2008 [EBook #26839]
Language: English
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START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS*** ***First Edition, Feb. 1892. Reprinted, May, 1892.
Second Edition, 1896. Reprinted, 1905.MATHEMATICAL
RECREATIONS AND ESSAYS
BY
W.W.ROUSEBALL
Fellow and Tutor of Trinity College, Cambridge.
FOURTH EDITION
London:
MACMILLAN AND CO., Limited
NEW YORK: THE MACMILLAN COMPANY

[All rights reserved.]Produced by Joshua Hutchinson, David Starner, David Wilson and
the Online Distributed Proofreading Team at http://www.pgdp.net
Transcriber’s notes
Most of the open questions discussed by the author were
settled during the twentieth century.
*The author’s footnotes are labelled using printer’s marks ;
footnotes showing where corrections to the text have been
1made are labelled numerically .
AMinor typographical corrections are documented in the LT XE
source.
This document is designed for two-sided printing. Consequently,
the many hyperlinked cross-references are not visually
distinguished. The document can be recompiled for more
Acomfortable on-screen viewing: see comments in source LT XE
code.PREFACE TO THE FIRST EDITION.
The following pages contain an account of certain mathematical
recreations, problems, and speculations of past and present times. I
hasten to add that the conclusions are of no practical use, and most
of the results are not new. If therefore the reader proceeds further he
is at least forewarned.
AtthesametimeIthinkImayassertthatmanyofthediversions—
particularly those in the latter half of the book—are interesting, not
a few are associated with the names of distinguished mathematicians,
while hitherto several of the memoirs quoted have not been easily ac-
cessible to English readers.
The book is divided into two parts, but in both parts I have in-
cluded questions which involve advanced mathematics.
The first part consists of seven chapters, in which are included var-
ious problems and amusements of the kind usually called mathematical
recreations. The questions discussed in the first of these chapters are
connected with arithmetic; those in the second with geometry; and
those in the third relate to mechanics. The fourth chapter contains
an account of some miscellaneous problems which involve both num-
ber and situation; the fifth chapter contains a concise account of magic
squares; and the sixth and seventh chapters deal with some unicursal
iiiiv PREFACE
problems. Several of the questions mentioned in the first three chap-
ters are of a somewhat trivial character, and had they been treated in
any standard English work to which I could have referred the reader, I
shouldhavepointedthemout. Intheabsenceofsuchawork, Ithought
it best to insert them and trust to the judicious reader to omit them
altogether or to skim them as he feels inclined.
The second part consists of five chapters, which are mostly histori-
cal. They deal respectively with three classical problems in geometry—
namely, the duplication of the cube, the trisection of an angle, and the
quadrature of the circle—astrology, the hypotheses as to the nature of
space and mass, and a means of measuring time.
Ihaveinserteddetailedreferences,asfarasIknow,astothesources
ofthevariousquestionsandsolutionsgiven; also, whereverIhavegiven
only the result of a theorem, I have tried to indicate authorities where
a proof may be found. In general, unless it is stated otherwise, I have
taken the references direct from the original works; but, in spite of
considerable time spent in verifying them, I dare not suppose that they
are free from all errors or misprints.
I shall be grateful for notices of additions or corrections which may
occur to any of my readers.
W.W. ROUSE BALL
Trinity College, Cambridge.
February, 1892.NOTE TO THE FOURTH EDITION.
InthiseditionIhaveinsertedintheearlierchaptersdescriptionsof
several additional Recreations involving elementary mathematics, and
I have added in the second part chapters on the History of the Mathe-
matical Tripos at Cambridge, Mersenne’s Numbers, and Cryptography
and Ciphers.
ItiswithsomehesitationthatIincludeinthebookthechapterson
Astrology and Ciphers, for these subjects are only remotely connected
with Mathematics, but to afford myself some latitude I have altered
the title of the second part to Miscellaneous Essays and Problems.
W.W.R.B.
Trinity College, Cambridge.
13 May, 1905.
vTABLE OF CONTENTS.
PART I.
Mathematical Recreations.
Chapter I. Some Arithmetical Questions.
PAGE
Elementary Questions on Numbers (Miscellaneous) . . . . . . 4
Arithmetical Fallacies . . . . . . . . . . . . . . . . . . . . . . 20
Bachet’s Weights Problem . . . . . . . . . . . . . . . . . . . . 27
Problems in Higher Arithmetic . . . . . . . . . . . . . . . . . 29
Fermat’s Theorem on Binary Powers . . . . . . . . . . . . 31
Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . 32
Chapter II. Some Geometrical Questions.
Geometrical Fallacies . . . . . . . . . . . . . . . . . . . . . . . 35 Paradoxes . . . . . . . . . . . . . . . . . . . . . . 42
Colouring Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Physical Geography . . . . . . . . . . . . . . . . . . . . . . . 46
Statical Games of Position . . . . . . . . . . . . . . . . . . . . 48
Three-in-a-row. Extension to p-in-a-row . . . . . . . . . 48
Tesselation. Cross-Fours . . . . . . . . . . . . . . . . . . 50
Colour-Cube Problem . . . . . . . . . . . . . . . . . . . . 51
viTABLE OF CONTENTS. vii
PAGE
Dynamical Games of Position . . . . . . . . . . . . . . . . . . 52
Shunting Problems . . . . . . . . . . . . . . . . . . . . . . 53
Ferry-Boat Problems . . . . . . . . . . . . . . . . . . . . . 55
Geodesic . . . . . . . . . . . . . . . . . . . . . . 57
Problems with Counters placed in a row . . . . . . . . . . 58 on a Chess-board with Counters or Pawns . . . . 60
Guarini’s Problem . . . . . . . . . . . . . . . . . . . . . . 63
Geometrical Puzzles (rods, strings, &c.) . . . . . . . . . . . . 64
Paradromic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter III. Some Mechanical Questions.
Paradoxes on Motion . . . . . . . . . . . . . . . . . . . . . . . 67
Force, Inertia, Centrifugal Force . . . . . . . . . . . . . . . . . 70
Work, Stability of Equilibrium, &c. . . . . . . . . . . . . . . . 72
Perpetual Motion . . . . . . . . . . . . . . . . . . . . . . . . . 75
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Sailing quicker than the Wind . . . . . . . . . . . . . . . . . . 79
Boat moved by a rope inside the boat . . . . . . . . . . . . . 81
Results dependent on Hauksbee’s Law . . . . . . . . . . . . . 82
Cut on a tennis-ball. Spin on a cricket-ball . . . . . . . 83
Flight of Birds . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Curiosa Physica . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter IV. Some Miscellaneous Questions.
The Fifteen Puzzle . . . . . . . . . . . . . . . . . . . . . . . . 88
The Tower of Hano¨ı . . . . . . . . . . . . . . . . . . . . . . . 91
Chinese Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
The Eight Queens Problem . . . . . . . . . . . . . . . . . . . 97
Other Problems with Queens and Chess-pieces . . . . . . . . . 102
The Fifteen School-Girls Problem . . . . . . . . . . . . . . . . 103viii TABLE OF CONTENTS.
PAGE
Problems connected with a pack of cards . . . . . . . . . . . . 109
Monge on shuffling a pack of cards . . . . . . . . . . . . . 109
Arrangement by rows and columns . . . . . . . . . . . . . 111
1Determination of one out of n(n+1) given couples. . . . 113
2
Gergonne’s Pile Problem . . . . . . . . . . . . . . . . . . . 115
The Mouse Trap. Treize . . . . . . . . . . . . . . . . . . 119
Chapter V. Magic Squares.
Notes on the History of Magic Squares . . . . . . . . . . . . . 122
Construction of Odd Magic Squares . . . . . . . . . . . . . . . 123
Method of De la Loub`ere . . . . . . . . . . . . . . . . . . . 124d of Bachet . . . . . . . . . . . . . . . . . . . . . . . 125
Method of De la Hire . . . . . . . . . . . . . . . . . . . . . 126
Construction of Even Magic Squares . . . . . . . . . . . . . . 128
First Method . . . . . . . . . . . . . . . . . . . . . . . . . 129
Method of De la Hire and Labosne . . . . . . . . . . . . . 132
Composite Magic Squares . . . . . . . . . . . . . . . . . . . . 134
Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . 135
Hyper-Magic Squares . . . . . . . . . . . . . . . . . . . . . . . 136
Pan-diagonal or Nasik Squares . . . . . . . . . . . . . . . . 136
Doubly Magic Squares . . . . . . . . . . . . . . . . . .