Dictionary of Mathematical Games, Puzzles, and Amusements

279 Pages·1988·9.98 MB·English

by  Harry Edwin Eiss

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HARRY EDWIN EISS
GREENWOOD PRESS
New York • Westport, Connecticut • London
Library of Congress Cataloging-in-Publication Data
Eiss, Harry Edwin.
Dictionary of mathematical games, puzzles, and
amusements.
Bibliography: p.
Includes index.
I. Mathematical recreations-Dictionaries.
2. Puzzles-Dictionaries. I. Title.
QA95.E37 1988 793.7'4'0321 87-280
ISBN 0-313-24714-5 (lib. bdg. : alk. paper)
British Library Cataloguing in Publication Data is available.
Copyright © 1988 by Harry Edwin Eiss
All rights reserved. No portion of this book may be
reproduced, by any process or technique, without the
express written consent of the publisher.
Library of Congress Catalog Card Number: 87-280
ISBN: 0-313-24714-5
First published in 1988
Greenwood Press, Inc.
88 Post Road West, Westport, Connecticut 06881
Printed in the United States of America
8
The paper used in this book complies with the
Permanent Paper Standard issued by the National
Information Standards Organization (Z39 .48-1984).
10 9 8 7 6 5 4 3 2 I
Copyright Acknowledgment
Illustrations appearing on pages 175 and 178 are taken from Lewis Carroll, Through
the Looking-Glass: And What Alice Found There, 1896. Reprinted in The Complete
Works of Lewis Carroll, Random House, n.d.
To nly wife, Betty

CONTENTS
Preface
IX
Dictionary of Mathematical Games, Puzzles, and Amusements 1
Bibliography 273
Index 275

PREFACE
Zeno of Elea gazes out over the beautiful country of ancient Greece. He has
recently defended the philosophy of his teacher Pannenides against the views of
Pythagoras, Heraclitus, and Anaxagoras. Perhaps Pannenides' view of the world
as a solid, finite, unifonn entity, without time or motion, has not answered the
questions of all existence once and for all, Zeno thinks, but surely it makes more
sense than the other philosophies.
Motion must be only an illusion. An arrow at each instant in its path must
occupy a specific place in space. It is, therefore, always at rest. Puzzling,
though, Zeno speculates. And what if space is divided up into infinitely smaller
particles? Then a man could never walk across the street, because before he
could get all the way across he would have to get half way across, and before he
could get half way across he would have to get half of half way across, and on
and on into infinity.
Simple paradoxes? Easily answered? Well, it took some 2,400 years for some
one to arrive at an answer. In 1873, Georg Cantor came up with the Continuum
Hypothesis, which is based on the theory that, when dealing with the infinite, it
is necessary to replace some of the rules used for finite mathematics. One of the
most important is that an infinite collection can be put into a one-to-one corre
spondence with a part of itself. In other words, it is possible for there to be an
infinite number of integers, and it is also possible for there to be an infinite
number of even integers. Thus, there are infinities beyond infinities. The first
infinite set Cantor designated as aleph-null, the first infinite set beyond aleph
null as aleph-one.
This is not the place to get into a highly sophisticated discussion of Cantor's
x Preface
theories (refer to the entry on Paradoxes of the Infinite for a more detailed
overview). The point is that it took millennia for someone to come up with a
solution to Zeno's paradoxes (refer to the entry on Zeno's Paradoxes for further
discussion of them), and that Cantor's theories are no more universally accepted
by contemporary mathematicians than Zeno' s were by his contemporaries.
David Hilbert, the brilliant leader of what is called the Axiomatic Group of
twentieth-century mathematicians, praises Cantor: HNo one shall expel us from
the paradise which Cantor has created for us." However, equally brilliant
I
mathematicians condemn Cantor's theories. Henri Poincare has stated, HLater
generations will regard [Cantor's] Mengenlehre as a disease from which one has
recovered."2 Kurt Godel, who proved that Cantor's assumption that there is no
aleph between aleph-null and the power of the continuum could be assumed to be
true, and Paul J. Cohen, who proved the opposite, both believe that Cantor's
Continuum Hypothesis will ultimately be proven false.
The questions remain, and they rest at the very foundations of mathematics.
But what does all this have to do with mathematical games, puzzles, and amuse
ments? First of all, Zeno's paradoxes are among the most commonly included
subject matter in recreational mathematics, and Cantor's theories serve as the
basis for numerous problems included in recreational mathematics. Furthermore,
since the first third of the twentieth century, there has been a shift in recreational
mathematics from less sophisticated amusements in number curiosities, mazes,
simple geometric puzzles, arithmatic story problems, card tricks, magic squares,
trisection of the angle, squaring the circle, duplication of the cube, Pythagorean
triples, board games, and so on, to at times highly sophisticated explorations of
number theory, game theory, graph theory, topological problems, flexagons,
logical paradoxes, fallacies of logic, paradoxes of the infinite, and so on.
Mathematical games, puzzles, and amusements go back to the beginning of
recorded history, many of them, such as Zeno' s paradoxes, reappearing time and
time again throughout history. Little is known about the mathematical recrea
tions in existence during the so-called Dark Ages, but by the Middle Ages,
notable names, such as Fibonacci (Leonardo of Pisa; 1202), who is famous for
the Fibonacci series, and Rabbiben Ezra (1140), appear. Gerolamo Cardan
(1501-1576) and Nicholas Tartaglia (Niccolo Fontana; 1500-1557) took part in
a mathematical duel as the result of Cardan stealing Tartaglia's solution to a
cubic equation.
In the seventeenth century, books began appearing devoted solely to recrea
tional mathematics. Claude Gaspard Bachet de Meziriac (1581-1638) wrote
Problemes plaisans et delectables qui se font par les nombres (1612), which
went through five editions, the most recent in 1959, and served as a source for
similar collections. It included card tricks, watch-dial puzzles, weight problems,
and difficult crossings.
Van Etten (Jean Leurechon) wrote Recreations mathematiques in 1624, based
Preface xi
mainly on Bachet. It went through thirty editions before 1700. Claude Mydorge
wrote Examen du livre des recreations mathematiques (1630), and Denis
Henrion wrote Les Recreations mathematiques avec l' examen de ses problemes
en arithmetique, geometrie, mechanique, cosmographie. . . (1659)-both
based on van Etten. In Germany, Daniel Schwenter put together a collection of
recreational problems based on Leurechon' s book. These were published after
his death as Deliciae Physico-mathematicae oder Mathematische und Philosoph
ische Erquickstunden (1636) and were extremely popular. Subsequently, an
enlarged edition was published.
Mario Bettini published Apiaria Universae Philosophiae Mathematicae in
Quibus Paradoxa et Nova Pleraque Machinamenta Exhibentur, a two-volume
set, in 1641-1642, and followed this with a third volume, R ecreationum M athe
maticarum Apiaria Novissima Duodecim ... in 1660.
In England, William Leybourn published Pleasure with Profit: Consisting of
Recreations of Divers Kinds, viz. Numerical, Geometrical, Mechanical, Stat
ical, Astronomical, Horometrical, Cryptographical, Magnetical, Automatical,
Chymical, and Historical in 1694.
In the eighteenth century, England began publishing mathematics recreations
in earnest. Edward Hatton, Thomas Gent, Samuel Clark, William Hooper, and
Charles Hutton are some of the more important authors. The most important
work of the time, however, was that of Jacques Ozanam, Recreations mathe
matiques et physiques (1694), a four-volume set that went through numerous
editions, eventually being translated into English by Charles Hutton (1803,
1814) and again by Edward Riddle (1840, 1844).
The second half of the nineteenth century produced a number of important
writers in recreational mathematics, the most famous being Lewis Carroll (C. L.
Dodgson). But others, notably Edouard Lucas, author of Recreations mathemati
ques (1882-1894), a four-volume set, were also producing classic works in the
field. By the tum of the century, Sam Loyd and his son, Sam Loyd, Jr., were
publishing hugely successful volumes of puzzles and holding a friendly rivalry
with Henry Ernest Dudeney, a British collector of games and puzzles. These
creators and collectors were the forerunners of contemporary recreational mathe
w.
matics. Others, however, were also important. W. Rouse Ball published
Mathematical Recreations and Essays in 1892, using a more scholarly approach;
it became a classic and is still a standard reference. Maurice Kraitchik, editor of
Sphilu, published Mathematical Recreations (1942), which has since been re
vised and remains a standard reference. Fred Schuh published Wonderlijke Prob
lemen; Leerzaam Tijdverdrijf Door Puzzle en Spel ( 1943), another important text
on mathematical recreations, translated and republished in English by Dover
Publications in 1968, once again, a more esoteric approach.
These books, and others, reveal the shift to more sophisticated recreations. H.
Steinhaus' Mathematical Snapshots (1950), Joseph S. Madachy's Mathematics