Amusements in Mathematics

371 Pages·2014·1.09 MB·English

by  Henry Ernest Dudeney

Amusements in Mathematics
Henry Ernest Dudeney
Project Gutenberg’s Amusements in Mathematics, by possibly many within,) I am including a chart of relative
Henry Ernest Dudeney values.
This eBook is for the use of anyone anywhere at no cost The most common units used were:
and with
thePenny,abbreviated: d. (fromtheRomanpenny,denar-
almost no restrictions whatsoever. You may copy it, give ius)
it away or
the Shilling, abbreviated: s.
re-use it under the terms of the Project Gutenberg License
the Pound, abbreviated: £
included
There was 12 Pennies to a Shilling and 20 Shillings to a
with this eBook or online at www.gutenberg.net
Pound, so there
Title: Amusements in Mathematics
was 240 Pennies in a Pound.
Author: Henry Ernest Dudeney
To further complicate things, there were many coins which
Release Date: September 17, 2005 [EBook #16713] were various
Language: English fractional values of Pennies, Shillings or Pounds.
*** START OFTHISPROJECT GUTENBERG EBOOK Farthing 1/4d.
AMUSEMENTS IN MATHEMATICS ***
Half-penny 1/2d.
Produced by Stephen Schulze, Jonathan Ingram and the
Online Penny 1d.
Distributed Proofreading Team at http://www.pgdp.net Three-penny 3d.
[Transcribersnote: Manyofthepuzzlesinthisbookassume Sixpence (or tanner) 6d.
a
Shilling (or bob) 1s.
familiarity with the currency of Great Britain in the early
Florin or two shilling piece 2s.
1900s. As
Half-crown (or half-dollar) 2s. 6d.
this is likely not common knowledge for those outside
Britain (and Double-florin 4s.
Crown (or dollar) 5s. For he, by geometrick scale,
Could take the size of pots of ale;
Half-Sovereign 10s.
Resolve, by sines and tangents, straight,
Sovereign (or Pound) £1 or 20s. If bread or butter wanted weight;
And wisely tell what hour o’ th’ day
This is by no means a comprehensive list, but it should be
The clock does strike by algebra.
adequate to
BUTLER’S Hudibras.
solve the puzzles in this book.
1917
Exponentsarerepresentedinthistextby^,e.g. ‘3squared’
is 3^2. PREFACE
Numbers with fractional components (other than 1/4, 1/2 In issuing this volume of my Mathematical Puzzles, of
and 3/4) have a + which some have
symbol separating the whole number component from the appeared in periodicals and others are given here for the
fraction. It makes first time, I
thefractionlookodd,butyeildscorrectsolutionsnomatter must acknowledge the encouragement that I have received
how it is from many
interpreted. E.G., 4 and eleven twenty-thirds is 4+11/23, unknown correspondents, at home and abroad, who have
not 411/23 or expressed a desire
4-11/23. to have the problems in a collected form, with some of the
solutions
]
given at greater length than is possible in magazines and
AMUSEMENTS IN MATHEMATICS
newspapers.
by
Though I have included a few old puzzles that have inter-
HENRY ERNEST DUDENEY ested the world
In Mathematicks he was greater for generations, where I felt that there was something new
Than Tycho Brahe or Erra Pater: to be said
about them, the problems are in the main original. It is except by haphazard attempts can be brought under a
true that some method of what has
of these have become widely known through the press, and been called “glorified trial”—a system of shortening our
it is possible labours by
that the reader may be glad to know their source. avoiding or eliminating what our reason tells us is useless.
It is, in
On the question of Mathematical Puzzles in general there
is, perhaps, fact,noteasytosaysometimeswherethe“empirical” begins
and where
little more to be said than I have written elsewhere. The
history of the
it ends.
subject entails nothing short of the actual story of the be-
Whenamansays,“Ihaveneversolvedapuzzleinmylife,”
ginnings and
it is
development of exact thinking in man. The historian must
difficult to know exactly what he means, for every intelli-
start from the
gent
time when man first succeeded in counting his ten fingers
individual is doing it every day. The unfortunate inmates
and in
of our lunatic
dividing an apple into two approximately equal parts. Ev-
asylums are sent there expressly because they cannot solve
ery puzzle that
is worthy of consideration can be referred to mathematics puzzles—because they have lost their powers of reason. If
and logic. there were no
Everyman,woman,andchildwhotriesto“reasonout” the puzzles to solve, there would be no questions to ask; and if
answer to the there were
simplest puzzle is working, though not of necessity con- no questions to be asked, what a world it would be! We
sciously, on should all be
mathematical lines. Even those puzzles that we have no equally omniscient, and conversation would be useless and
way of attacking idle.
It is possible that some few exceedingly sober-minded a little consideration, for now and again it will be found
mathematicians, that there is
who are impatient of any terminology in their favourite somemoreorlesssubtlepitfallortrapintowhichthereader
science but the may be
academic, and who object to the elusive x and y appearing apttofall. Itisgoodexercisetocultivatethehabitofbeing
under any very
other names, will have wished that various problems had wary over the exact wording of a puzzle. It teaches exacti-
been presented tude and
in a less popular dress and introduced with a less flippant caution. But some of the problems are very hard nuts in-
phraseology. deed, and not
I can only refer them to the first word of my title and unworthy of the attention of the advanced mathematician.
remind them that Readers will
weareprimarilyouttobeamused—not,itistrue,without doubtless select according to their individual tastes.
some hope of
Inmanycasesonlythemereanswersaregiven. Thisleaves
pickingupmorselsofknowledgebytheway. Ifthemanner the beginner
is light, I
something to do on his own behalf in working out the
can only say, in the words of Touchstone, that it is “an method of solution,
ill-favoured
and saves space that would be wasted from the point of
thing, sir, but my own; a poor humour of mine, sir.” view of the
As for the question of difficulty, some of the puzzles, espe- advanced student. On the other hand, in particular cases
cially in where it seemed
the Arithmetical and Algebraical category, are quite easy. likely to interest, I have given rather extensive solutions
Yet some of and treated
thoseexamplesthatlookthesimplestshouldnotbepassed problems in a general manner. It will often be found that
over without the notes on
one problem will serve to elucidate a good many others in Ihavetoexpressmythanksinparticulartotheproprietors
the book; so of The
thatthereader’sdifficultieswillsometimesbefoundcleared Strand Magazine, Cassell’s Magazine, The Queen, Tit-
up as he Bits, and
advances. Where it is possible to say a thing in a manner The Weekly Dispatch for their courtesy in allowing me to
that may be reprint some
“understanded of the people” generally, I prefer to use this of the puzzles that have appeared in their pages.
simple
THE AUTHORS’ CLUB March 25, 1917
phraseology, and so engage the attention and interest of a
CONTENTS
larger
PREFACE v
public. The mathematician will in such cases have no dif-
ARITHMETICAL AND ALGEBRAICAL PROBLEMS 1
ficulty in
Money Puzzles 1
Age and Kinship Puzzles 6
expressing the matter under consideration in terms of his
Clock Puzzles 9
familiar
Locomotion and Speed Puzzles 11
symbols. Digital Puzzles 13
Various Arithmetical and Algebraical Problems 17
I have taken the greatest care in reading the proofs, and
GEOMETRICAL PROBLEMS 27
trust that any
Dissection Puzzles 27
errors that may have crept in are very few. If any such Greek Cross Puzzles 28
should occur, I Various Dissection Puzzles 35
Patchwork Puzzles 46
can only plead, in the words of Horace, that “good Homer
Various Geometrical Puzzles 49
sometimes
POINTS AND LINES PROBLEMS 56
MOVING COUNTER PROBLEMS 58
nods,” or, as the bishop put it, “Not even the youngest
UNICURSAL AND ROUTE PROBLEMS 68
curate in my
COMBINATION AND GROUP PROBLEMS 76
diocese is infallible.” CHESSBOARD PROBLEMS 85
The Chessboard 85
Statical Chess Puzzles 88
The Guarded Chessboard 95
Dynamical Chess Puzzles 96
Various Chess Puzzles 105
MEASURING, WEIGHING, AND PACKING PUZZLES 109
CROSSING RIVER PROBLEMS 112
PROBLEMS CONCERNING GAMES 114
PUZZLE GAMES 117
MAGIC SQUARE PROBLEMS 119
Subtracting, Multiplying, and Dividing Magics 124
Magic Squares of Primes 125
MAZES AND HOW TO THREAD THEM 127
THE PARADOX PARTY 137
UNCLASSIFIED PROBLEMS 142
SOLUTIONS 148
INDEX 253
AMUSEMENTS IN MATHEMATICS.
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Contents
1 ARITHMETICAL AND ALGEBRAICAL PROBLEMS. 1
2 MONEY PUZZLES. 2
3 AGE AND KINSHIP PUZZLES. 13
4 CLOCK PUZZLES. 19
5 LOCOMOTION AND SPEED PUZZLES. 23
6 DIGITAL PUZZLES. 26
7 VARIOUS ARITHMETICAL AND ALGEBRAICAL PROBLEMS. 33
8 GEOMETRICAL PROBLEMS. 48
9 DISSECTION PUZZLES. 49
10 GREEK CROSS PUZZLES. 51
11 VARIOUS DISSECTION PUZZLES. 59
12 PATCHWORK PUZZLES. 70
13 VARIOUS GEOMETRICAL PUZZLES. 73
14 POINTS AND LINES PROBLEMS. 81
15 MOVING COUNTER PROBLEMS. 84
16 UNICURSAL AND ROUTE PROBLEMS. 95
17 COMBINATION AND GROUP PROBLEMS. 102
18 CHESSBOARD PROBLEMS. 113
19 THE CHESSBOARD. 115
20 STATICAL CHESS PUZZLES. 119
21 THE GUARDED CHESSBOARD. 128
22 NON-ATTACKING CHESSBOARD ARRANGEMENTS. 130
23 THE TWO PIECES PROBLEM. 132
24 DYNAMICAL CHESS PUZZLES. 133
25 VARIOUS CHESS PUZZLES. 143
26 MEASURING, WEIGHING, AND PACKING PUZZLES. 147
27 CROSSING RIVER PROBLEMS 151
28 PROBLEMS CONCERNING GAMES. 155
29 PUZZLE GAMES. 160
30 MAGIC SQUARE PROBLEMS. 164
31 SUBTRACTING, MULTIPLYING, AND DIVIDING MAGICS. 170
32 MAGIC SQUARES OF PRIMES. 173
33 MAZES AND HOW TO THREAD THEM. 176
34 THE PARADOX PARTY. 183
35 UNCLASSIFIED PROBLEMS. 192
36 SOLUTIONS. 200
37 INDEX. 335
38 THE END. 351
39 Section 1. General Terms of Use and Redistributing Project Gutenberg-tm electronic works 353
40 Section 2. Information about the Mission of Project Gutenberg-tm 358
41 Section 3. Information about the Project Gutenberg Literary Archive Foundation 359
42 Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation 360