The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications

391 Pages·1997·19.23 MB·English

by  Martin Gardner

THE LAST
RECREATIONS

© 1997 Springcr-Verlag New York, Inc.
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without the prior
written permission of the publisher.
Published in the United States by Copernicus, an imprint of
Springer-Verlag New York, Inc.
Copernicus
Springer-Verlag New York, Inc.
175 Fifth Avenue
New York, NY 10010
USA
Cartoon credits: p. 237-.THE FAR SIDE copyright 1991 FARWORKS INC.
Dist. by UNIVERSAL PRESS SYNDICATE. Reprinted with permission.
All rights reserved.
p. 284—B.C. copyright Field Enterprises, Inc. All rights reserved.
Library of Congress Cataloging-in^Publication Data
Gardner, Martin, 1914-
The last recreations: hydras, eggs, and other mathematical
mystifications/Martin Gardner,
p. cm.
Includes index.
1. Mathematical recreations. I. Title.
QA95.G255 1997
793.7'4-DC21 96-51641
0-387-94929-1 hardback
0-387-25827-2 paperback
Manufactured in the United States of America.
Printed on acid'free paper.
Designed by Irmgard Lochner.
9 8 7 6 5 4 3 2 1
SPIN 10561414
To Persi Diaconis for his remarkable
contrihutions to mathematics and conjuring,
for his unswerving opposition to psychic
nonsense, and for a friendship going back to our
Manhattan years.
Preface
One of my greatest pleasures and privileges was writing a column, over
a period of some 30 years, for Scientific American. It began with an
article on hexaflexagons in December 1956 and concluded with a col
umn on minimal Steiner trees in May 1986-
Writing diis column was a marvelous learning experience. I took
no courses in madi when I was an undergraduate at the University of
Chicago-my major was philosophy-but I have always loved mathemat
ics, and now and dien regret I did not pursue it as a career. It takes
only a gjance through earlier book collections of the columns to see
how they gradually became more sophisticated as I learned more about
madiematics. Not the least of my delights was getting to know many
truly eminent mathematicians who generously contributed material and
who have since become lifelong friends.
This is die fifteenth and final collection. As in previous books in
the series, I have done my best to correa blunders, to expand and
update each column with an addendum, to add new illustrations, and
to provide fuller lists of selected references.
Martin Gardner
Vll
Contents
Preface vii
1 TTie Wonders of a Planiverse I
2 Bulgarian Solitaire and Odier Seemingjy Endless Tasks 27
3 Fun widi Eggs, Part I 45
4 Fun widi Eggs, Part II 59
5 The Topology of Knots 67
6 M'Pire Maps 85
7 Directed Graphs and Cannibals 101
8 Dinner Guests, Schoolgiris, and Handcuffed Prisoners 121
9 The Monster and Odier Sporadic Groups 139
10 Taxicab Geometry 159
11 The Power of the Pigeonhole 177
12 Strong Laws of Small Primes 191
IX
The Last Recreations
13 Checker Recreations, Part I 207
14 Checker Recreations, Part II 221
15 Modulo Arithmetic and Hummer's Wicked Witch 233
16 Lavinia Seeks a Room and Other Problems 247
17 The Symmetry Creations of Scott Kim 267
18 Parabolas 285
19 Non-Eudidean Geometry 303
20 Voting Mathematics 317
21 A Toroidal Paradox and Other Problems 331
22 Minimal Steiner Trees 345
23 Trivalent Graphs, Snarks, and Boojums 361
Index 381
I
I'.' M mi U»
The
Wonders
of a
Planivcrsc
"Planiversal scientists are not a very
common breed.**
—Alexander Keewatin Dewdney
A
s far as anyone knows the only existing universe is the one
we live in, with its three dimensions of space and one of time.
It is not hard to imagine, as many science-fiction writers have,
that intelligent organisms could live in a four-dimensional space,
but two dimensions offer such limited degrees of freedom that it has
long been assumed intelligent two-space life forms could not exist.
Two notable attempts have nonetheless been made to describe such
organisms.
2 The Last Recreations
In 1884 Edwin Abbott Abbott, a London clergyman, published his
satirical novel Flatland. Unfortunately the book leaves the reader almost
entirely in the daik about Fladand's physical laws and the technology
developed by its inhabitants, but the situation was gready improved in
1907 when Chailes Howard Hinton published An Episode of Flatland.
Although written in a flat style and with cardboard characters, Hinton's
story provided the first glimpses of the possible science and technology
of the twoKlimensional worid. His eccentric book is, alas, long out of
print, but you can read about it in the chapter "Fladands** in my book
The Unexpected Hanging and Other Mathematical Diversions (Simon &
Schuster, 1969).
In "Fladands** I wrote: "It is amusing to speculate on two-dimen^
sional physics and the kinds of simple mechanical devices that would
be feasible in a flat worid.** This remark caugjit die attention of Alexander
I&ewatin Dewdney, a computer scientist at die University of Western
Ontario. Some of his early speculations on die subject were set down in
1978 in a university report and in 1979 in ''Exploring the Planiverse,**
an article in Journal of Recreational Mathematics (Vol. 12, No. 1, pages
16-20; September). Later in 1979 Dewdney also privately published
"Twcniimensional Science and Technology," a 97'page tour de force. It
is hard to believe, but Dewdney actually lays the groundwork for what
he calls a planiverse: a possible twoKiimensional worid. Complete with
its own laws of chemistry, physics, astronomy, and biology, the planiverse
is closely analogous to our own universe (which he calls the steriverse)
and is apparendy free of contradictions. I should add diat this remark
able achievement is an amusing hobby for a mathematician whose seri
ous contributions have appeared in some 30 papers in technical jour
nals.
Dewdney's planiverse resembles Hinton's in having an earth that
he calls (as Hinton did) Astria. Astria is a disklike planet that rotates in
planar space. The Astrians, walking upright on the rim of the planet,