MATHEMATICAL BRAIN BENDERS 2nd Miscellany of Puzzles STEPHEN BARR

DOVER PUBLICATIONS, INC. NEW YORK

Copyright © 1969, 1982 by Stephen Barr. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC2H 7EG. This Dover edition, first published in 1982, is an unabridged and corrected republication of the work originally published in 1969 by The Macmillan Company of New York under the title 2nd Miscellany of Pu:z;:z;lesMathematical and Otherwise. Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014

Library of Congress Cataloging in Publication Data Barr, Stephen. Mathematical brain benders. Reprint. Originally published: 2nd miscellany of puzzles. New York : Macmillan, 1969. 1. Puzzles. 2. Mathematical recreations. I. Title. GV1493.B28 1982 793.73 81-19527 ISBN 0-486-24260-9 AACR2

Add another hue unto the rainbow ... Shakespeare

We subtract faith and fallacy from fact ... Samuel Hoffenstein Be fruitful and multiply ... Genesis Round numbers are always false ... Samuel Johnson

Vicious circle ... George Du Maurier

Straight down the crooked lane and all around the square ... Thomas Hood Every cubic inch of space is a miracle ... Walt Whitman

The god delights in an odd number ... Virgil Euclid alone has looked on beauty bare ... Edna

st. Vincent Millay

Measure your mind's height by the shade it casts ... Browning Gaul is all divided in parts three ... Caesar Is 5 ...

e. e. cummings

CONTENTS PUZZLES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Rollers Owls' Eggs Conical Helix The Owl Island Flag Pseudo-Moebius Strip The Butler and the Crumbs The Three Clocks Slit Strips The Pot on the Crosspiece For Scrabble Players Area of Roof Paper-Folding The Two Pyramids The Man Who Gave Up Smoking Tetrahedron Angles Hypocycloids Squares on a Circle Three Coins Two Coins The Coin Collector's Nightmare The Hi-q, Set Cryptarithmetic

19 20 21 22 23 24 25 26 28 29 31 31 32 33 37 38 39 40 40 41 42 43 9

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 SO 51 10

Origametry The Hauberk More Origametry Unique Parts of Letters For

//'

/0/

F L

V FIGURE 2

193

til ~

u.J

3: til Z

« vertical, and a and b are parallel, all the triangles are similar:

:. alb = ylx :.ax=by From before, x

=

y't

Substitute for x, aiw

.

= by

Lb

.. a=W Since band Ware constant when L changes, a is changed proportionally; e.g., if we double L, then

~{;J- = 2a, etc.

It will be noticed that c

need not equal b. There are many other arrangements of the scale which embody the above principle.

194

57 HYDRAULIC INFERENCE A squat cylinder; approximately 5 by 8. (The old style of coffee cans would be very near, both in proportion and size.) Most, if not all, thin kitchen pots are round. It was stated that the water, after stopping, continued to rise at exactly half its previous rate, which strongly implies that its previous rate was constant; therefore its sides were vertical. Since the first rate was twice that of the second, the pot's area of cross section was half that of the cubical casserole, and the pot's height is of course half. Taking the width of the casserole as 2, the radius, T, of the pot is got from 7TT2 = 2 (half the area of the casserole); therefore T = ~, almost .8, as shown in the figure. (The actual capacity of the casserole is not needed for the above.) Side View

Plan

r

~

2

)

I

~

~

r 1

v2/rr

2

)

1 195

58

Vl ~ LU

3: Vl z

I< *Note to model maker: since the surfaces are warped, and cannot be opened out to a true flat plane, the cardboard must be thin, and soaked in water so that it will stretch.

196

Continued

58

CJ')

~ LU

~

CJ')

Z