The Unexpected Hanging and Other Mathematical Diversions

262 Pages·1991·7.04 MB·English

by  Martin Gardner

A N D O T H E R
MATHEMATICAL
n I V E R S I O N S
OLLECTION OF
GI ES FRON
A ERIC"
The
Unexpected
Hanging
A Partial List of Books by Martin Gardner:
The Annotated "Casey at the Bat" (ed.)
The Annotated Innocence of Father Brown (ed.)
Hexaflexagons and Other Mathematical Diversions
Logic Machines and Diagrams
The Magic Numbers of Dr. Matrix
Martin Gardner's Whys and Wherefores
New Ambidextrous Universe
The No-sided Professor
Order and Surprise
Puzzles from Other Worlds
The Sacred Beetle and Other Essays in Science (ed.)
The Second Scientific American Book of Mathematical
Puzzles and Diversions
Wheels, Life, and Other Mathematical Entertainments
The Whys of a Philosopher Scrivener
MARTIN GARDNER
The
Unexpected
Hanging
And Other
Mathematical Diversions
With a new Afterword
and expanded Bibliography
The University of Chicago Press
Chicago and London
Material previously published in Scientific American is
copyright 0 1961, 1962, 1963 by Scientific American, Inc.
All rights reserved.
The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
O 1969, 1991 by Martin Gardner
All rights reserved. Originally published 1969.
University of Chicago Press edition 1991
Printed in the United States of America
Library of Congress Cataloging-in-Publication Data
Gardner, Martin, 1914-
The unexpected hanging and other mathematical diversions :
with a new afterword and expanded bibliography 1 Martin
Gardner. -
University of Chicago Press ed.
p. cm.
Includes bibliographical references.
ISBN 0-226-28256-2 (pbk.)
1. Mathematical recreations. I. Title.
QA95.G33 1991
793.7'4-dc20 91-17723
CIP
@ The paper used in this publication meets the minimum
requirements of the American National Standard for Information
Sciences-Permanence of Paper for Printed Library Materials,
ANSI 239.48-1984.
For My Niece
Dorothy Elise Weaver
CONTENTS
NOTE TO THE 1991 EDITION 9
1 The Paradox of the Unexpected Hanging 11
2 Knots and Borromean Rings 24
3 The Transcendental Number e 34
4 Geometric Dissections 43
5 Scarne on Gambling 52
6 The Church of the Fourth Dimension 65
7 Eight Problems 76
8 A Matchbox Game-Learning Machine 90
9 Spirals 103
10 Rotations and Reflections 114
11 Peg Solitaire 122
12 Flatlands 136
13 Chicago Magic Convention 147
14 Tests of Divisibility 160
15 Nine Problems 170
16 The Eight Queens and Other Chessboard Diversions 186
17 A Loop of String 198
18 Curves of Constant Width 212
19 Rep-Tiles : Replicating Figures on the Plane 222
20 Thirty-Seven Catch Questions 234
AFTERWORD
BIBLIOGRAPHY
NOTE TO THE 1991 EDITION
The Unexpected Hanging was the fifth book collection of the Mathe-
matical Games column that I wrote for Scientific American, starting in
1957. The first two collections-Hexajexagons and Other Mathemati-
cal Diuersions, and The Second Scientific American Book of Mathe-
matical Puzzles and Diuersions-are currently available as University
of Chicago Press paperbacks.
I have updated the chapters as best I can with an afterword, as well
as added more than a hundred new references to the bibliography.
Martin Gardner
CHAPTER ONE
The Paradox
of the
Unexpected Hanging
"A NEW AND POWERFUL PARADOX has Come to light." This is
the opening sentence of a mind-twisting article by Michael
Scriven that appeared in the July 1951 issue of the British
philosophical journal Mind. Scriven, who bears the title of
"professor of the logic of science" at the University of In-
diana, is a man whose opinions on such matters are not to be
taken lightly. That the paradox is indeed powerful has been
amply confirmed by the fact that more than twenty articles
about it have appeared in learned journals. The authors, many
of whom are distinguished philosophers, disagree sharply in
their attempts to resolve the paradox. Since no consensus has
been reached, the paradox is still very much a controversial
topic.
No one knows who first thought of it. According to the
Harvard University logician W. V. Quine, who wrote one of
the articles (and who discussed paradoxes in Scientific Amer-
ican for April 1962), the paradox was first circulated by word
of mouth in the early 1940's. It often took the form of a puzzle
about a man condemned to be hanged.
11
12 The Unexpected Hanging
The man was sentenced on Saturday. "The hanging will
take place at noon," said the judge to the prisoner, "on one
of the seven days of next week. But you will not know which
day it is until you are so informed on the morning of the day
of the hanging."
The judge was known to be a man who always kept his
word. The prisoner, accompanied by his lawyer, went back to
his cell. As soon as the two men were alone the lawyer broke
into a grin. "Don't you see?' he exclaimed. "The judge's sen-
tence cannot possibly be carried out."
"I don't see," said the prisoner.
"Let me explain. They obviously can't hang you next Satur-
day. Saturday is the last day of the week. On Friday afternoon
you would still be alive and you would know with absolute
certainty that the hanging would be on Saturday. You would
know this before you were told so on Saturday morning. That
would violate the judge's deoree."
"True," said the prisoner.
"Saturday, then is positively ruled out," continued the
lawyer. "This leaves Friday as the last day they can hang
you. But they can't hang you on Friday because by Thursday
afternoon only two days would remain : Friday and Saturday.
Since Saturday is not a possible day, the hanging would have
to be on Friday. Your knowledge of that fact would violate
Figure 1
The prisoner eliminates all
possible days