Tracking the Automatic ANT: And Other Mathematical Explorations

238 Pages·1998·10.47 MB·English

by  David Gale (auth.)

TRACKING
THE
AUTOMATIC
ANT
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DAVID GALE
TRACKING
1/ THE
AUTOMATIC
ANT
~
MA
AND OTHER THEMAT ICAL
EXPLORATIONS
of
AC ollection Mdthemdticdl Ent 'ftdinm nt C lumn from
~nterrigeficer
Springer
David Gale
Department of Mathematics
University of California
921 Evans Hall
Berkeley, CA 94720
USA
Appendix I-A Curious Nim-Type Game, reprinted from the American Mathematical Monthly, Vol. 81, No.8,
October 1974, pp. 876-879.
Appendix 2-The Jeep Once More or Jeeper by the Dozen, reprinted from N.J. Fine's The Jeep Problem, Amer
ican Mathematical Monthly, Vol. 54,1947, pp. 24-31.
Library of Congress Cataloging-in-Publication Data
Gale, David.
Tracking the automatic ant and other mathematical explorations / David
Gale.
p.cm.
1. Mathematical recreations. I. Title.
QA97.G127 1998
793.7'4-dc21 97-33272
Printed on acid-free paper.
© 1998 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover lst edition 1998
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987654321
ISBN-13: 978-1-4612-7453-7 e-ISBN-13: 978-1-4612-2192-0
DOl: 10.1007/978-1-4612-2192-0
Preface
This book consists mainly of material from the column Mathemat
ical Entertainments in The Mathematical Intelligencer, a feature I
edited from 1991 through 1996. The column had been devoted
almost entirely to problems, but when I started as editor, I was
encouraged by the editor of the magazine, Sheldon Axler, to do
with it whatever I pleased. As a result the problems became fewer
and fewer and eventually disappeared altogether. Instead I was able
to write about anything that caught my fancy. As a result the topics
taken up vary all over the lot, but almost always are concerned with
very elementary things. Thus three of the chapters are devoted
almost entirely to triangles, two others to tiling by rectangles, three
more to mysterious properties of sequences given by simple recur
sions, three to games and paradoxes, and three to a particular
automaton.
As can be seen from this listing, there seems to be no unifying
thread to the collection, but I did follow some general princi-
ples in deciding which material to use.
(1) Roughly speaking, one should not have any spe
cialized knowledge in order to understand the
y
vi TRACKING THE AUTOMATIC ANT
chosen topics (though occasionally the subsequent analysis becomes a bit more
advanced).
(2) A high priority was placed on mathematical surprises and unexpected twists, some of
which have explanations, while others remain mysterious.
Both of these principles are illustrated by examples from everyday life that turn out to
have unexpected mathematical content. Among these are familiar children's games and
two sections on tying shoelaces. In the same spirit, in a section on geometric construc
tions, it turns out that one can throwaway the unwieldy compass and use instead a
straightedge on which one can make marks. This apparatus allows one to make not only
the Euclidean constructions, but also any construction involving equations of degree at
most four, thus making it possible to trisect angles and duplicate cubes (see Chapter 10).
In the first three appendices, I present the original sources for some of the material.
Computers playa role in many of the sections, but in various guises. Sometimes they
are used simply for performing computations, such as finding the power of 2 whose first
four digits are 1492 (Chapter 7). Elsewhere they are used for experimentation and, per
haps most interestingly, for proving theorems. For example, one obtains dozens of new
theorems of plane geometry by discovering numerically that certain geometrically
defined points are collinear (Chapter 6). When these discovered collinearities are then
formulated algebraically, the computer again steps in, in its symbolic rather than numer
ical mode, and provides the proofs.
From the point of view of this book, however, the most intriguing consequence of the
use of the computer is how it affects the way we think about mathematics itself. Tradi
tionally, mathematics has been the example of a deductive science. Thus, no matter how
much "experimental evidence" one has for, say, the truth of Fermat's famous theorem, it
isn't considered mathematics until it has been proved, that is, shown to be deducible from
previously known results. But things started to change in 1931, when Kurt G6del showed
that there is no nontrivial mathematical system that can prove or disprove all the state
ments that can be made in that system. Today, by enabling us to amass enormous
amounts of empirical data, computers have thrown a spotlight on the crisis. For example,
at latest count we now know several billion digits of "/1', and one-tenth of them are 7s to
within a very small epsilon. Yet no one has any idea how to show that this behavior goes
on "forever." Indeed, for all we know, there may be no 7s at all from some point on. Opti
mists would say that there has to be a proof out there somewhere, and when we become
sufficiently clever, we will find it. But what if there is no proof? In Chapter 3 and 5 we look
at other types of questions about sequences where certain results seem overwhelmingly
likely yet the prospects for finding proofs seem overwhelmingly remote. Thus, mathe
maticians find themselves in a curious situation. At one extreme, by amassing data com
puters can first supply us with conjectures and then go on, using programs we design, to
settle them. This is the case with the geometrical results mentioned above. At the other
end, the computer may be giving us glimpses of true mathematical facts for which there
is no proof. Of course, between the extremes there is enough room to keep mathemati
cians busy for the foreseeable and even the unforeseeable future.
So the advent of computers has made mathematics more like the physical sciences, in
that it now has a strong experimental as well as theoretical component. To find out what
is going on in the physical universe, physicists use particle accelerators, and astronomers
Preface vii
use high-powered telescopes. For mathematicians, the computer now plays an analogous
role. The important difference is that the computer does not give us information about
the physical universe but rather about the "abstract" universe. The things we study, finite
groups, analytic functions, and the like are no less "real" than atoms or galaxies. The dif
ference is that the objects are abstractions, things that wouldn't be there if we hadn't
invented them. Of course many of these abstractions were originally created in an
attempt to understand the physical universe, the prototypical example being the books of
Euclid. But there is also so-called "pure" mathematics, in which the objects of study are
the abstractions themselves, and herein lies a great paradox. As an illustration let us look
again at the digits of 'iT and see what is involved in describing the problem. First of all we
need the notion of a circle. Nature offers suggestive examples: the sun, the moon, the
head of a mushroom. But nature does not supply the center. We had to invent that, like
wise the circumference and the diameter, the concept oflength as a "number" (which we
also invented), and finally decimal expansions. All of these things are creations of the
human mind, and yet they seem to possess a life of their own, as they rise up and present
us with puzzles, some of which we may never be able to solve. The last appendix to the
book explores further the philosophical implications of this state of affairs.
Finally, a word of clarification. As noted, the columns were originally published as
"entertainments;' and I have referred above to mathematical phenomena as providing us
with puzzles. The everyday connotation of these words suggests that one is here con
cerned only with lightweight matters as opposed to "serious mathematics." This distinc
tion is more apparent than real. In fact, one might say that the whole scientific enterprise
is about solving the puzzles nature keeps throwing at us. There is, however, one sense in
which I have tried to keep the presentation on the light side. The process of trying to
understand mathematical results can be very hard work. By contrast, in this volume I
tried to choose topics from which readers could reap the rewards of discovery with at
most a moderate expenditure of energy, in other words, to maximize the ratio of pleasure
to effort. Although the book does not attempt to read like a novel, my hope is that in dip
ping into its pages, the reader will find things that are entertaining, yes, but also illumi
nating.
University of California David Gale
Berkeley
Acknowledgments
I would like to acknowlege contributions from people who played
important roles in helping to put together the material for this
book.
I am grateful for the encouragement and support of Sheldon
Axler and Chandler Davis, the two editors of The Mathematical
Intelligencer who gave me free hand to write about whatever caught
my fancy during my time as editor of the Mathematical Entertain
ments column. Some of the chapters were written in part or entirely
by "guest" columnists, including Donald Newman, Jim Propp,
Scott Sutherland, Serge Troubetzkoy, Sherman Stein, John Halton,
Solomon Golomb, Michal Misiurewicz, and Armando Machado.
In addition, material for many of the chapters was obtained
from discussions, email messages and telephone conversations with
many people, among them Michael Somos, Imre Barany, Clark
Kimberling, Elwyn Berlekamp, Joe Keller, Clifford Gardner and
especially the late Raphael Robinson who made key contributions
to almost all of the early chapters.
All of these people have made the writing of the columns and
the resulting book a sort of cooperative venture. Ih ope the readers
will now share some of the pleasure that we had in putting this
book together.
viii
Contents
Preface v
Acknowledgments
Vlll
CHAPTER 1
Simple Sequences with Puzzling Properties 1
CHAPTER 2
Probability Paradoxes 7
CHAPTER 3
Historic Conjectures: More Sequence Mysteries 13
CHAPTER 4
Privacy-Preserving Protocols 19
CHAPTERS
Surprising Shuffles 25
CHAPTER 6
Hundreds of New Theorems in a
Two-Thousand-Year-Old Subject:
Where Will It End? 31
ix
x TRACKING THE AUTOMATIC ANT
CHAPTER 7
Pop Math and Protocols 37
CHAPTER 8
Six Variations on the Variational Method 45
CHAPTER 9
Tiling a Torus: Cutting a Cake 55
CHAPTER lO
The Automatic Ant: Compassless Constructions 63
CHAPTER 11
Games: Real, Complex, Imaginary 73
CHAPTER 12
Coin Weighing: Square Squaring 83
CHAPTER 13
The Return of the Ant and the Jeep 91
CHAPTER 14
Go 101
CHAPTER 15
More Paradoxes. Knowledge Games 113
CHAPTER 16
Triangles and Computers 119
CHAPTER 17
Packing Tripods 131
CHAPTER 18
Further Travels with My Ant 137
CHAPTER 19
The Shoelace Problem 151
CHAPTER 20
Triangles and Proofs 163