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MATHEMATICAL BRAIN BENDERS
Copyright ([) 1969, 1982 by Stephen Barr. All rights reserved under Pan American and In...
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MATHEMATICAL BRAIN BENDERS
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 Puzzles-
Mathematical and Otherwise.
Add another hue
unto
the rainbow.
.
.
Shakespeare We subtract faith and
fallacy
from fact.
.
.
Samuel HoHenstein
CONTENTS
23
44
Origametry
24 The Hauberk
45
25 More
46
26
46
Origametry Unique Parts of Letters
27 For Fans
47
28 The Truck Gardens
48
29
50
30
51
31
Origametry The Cockeyed Kite More Origametry
52
32 The Bookmark
33 Snow 34
on
52
the Roof
Drafting Puzzle The Psychedelic
53 54
Cube
55
36 Construction Problem
56
35
37 A Walk in 38 More
a
Field
41 42
43
58
Diagonals
39 Short Proof (Cut 40
57
Cube)
60
Word-Changing Polyhedral Model Dihedral Angles Cheese Wedges
61 62
63
44 The Poisoned Glass
45 To Cover 46 Pure
a
64
Circle
65
Origami Solution
66
47 The Flat Pan
48 The Siamese Moebius 49 The Nine
59
67
Strip
68
Digits
69
50 Minimum Area
70
51
71
Cocyclic
Points
52 The Tilted Carton
53
Topology
Puzzle
72
74
54 The Vanadium Steel Clothesline
75
55
77
Visualizing
56 The Balance 57
78
Hydraulic Inference Striped Whatsis
79
58 The
79
59 The Terri Turnover
80
60 The
81
Heavy Chest
61 The Pile and the Patriot
83
62 The Pillar of Chios
88
63
89
literary Quiz
64 Thunder 65 Two
on
the
Triangles
Right
90 91
A SUBMISCELLANY OF SHORT
PUZZLES,
MORE OR LESS FROM EVERYDAY LIFE 1
Transparent Object
21
95
Doffing
or
Donning
100
2 Icicles
95
22 Gibraltar
100
3 Rodin's Thinker
95
23 Bible
100
4 Blue Drink
95
24 Box
100
5 Verbalized
25
Quote Fitting
Folding Times Tilting Dish
Number
95
26
6 Chain Reaction
95
27 Corn in
7 Count the
96
28 Same
8
Objects Barley, Marbles,
96
etc.
9 North America
and Europe 10 Great Circle
96
Bag
Height
100 100 101 101
29 Torn Card
101
30 Bear and
101
Penguin
31
Remove Water
32
Changed
97
Equation
101
101
11
Next to New York 97
33 Northernmost
101
12
Floating Coffee
34 Letter Code
102
13
Two-Digit Numbers
18
Measuring Tree How Many Kings?
36
98
37 Tesselation
98
38
Leaky Balloon Midpage in
99
39
Manuscript
99
State
99
on
15 Box at
17
35 97
14 Rain
16
97
the Roof
Edge
Oblong
Quiz in Verse Opposite States
40 Panama Canal 41
Dousing Cigarette
102 102 102 102
103 103
103
19 Water from Pan
99
42 Easternmost
103
20 Bookshelf
99
43
103
Alphabet Spoken
44
Rhyme
Puzzle
45 Southernmost 46 Wheels 47 Wet
Spoon
MATHEMATICAL BRAIN BENDERS
PUZZLES
Figure
1 shows the side view of two
between two
parallel rails;
ping, the wheels
can
equal wheels no slip-
if there is
roll either way and still
their vertical relative positions. The would be true if they were spheres and the
maintain same
horizontal lines
were planes. If, however, the top sphere were much largthan the lower (Fig. 2) and they were er pushed to one side, which would go ahead of ,
which?
2
3
4
5
6
7
8
Continued
9
10
30
11
13
14
14
Continued
14
15
16
17
18
20
21
22
23
24
25
The ratio between the parts of a line, divided so that the lesser is to the greater as the greater is to the
whole,
the Golden Section,
is called
the Divina proporzione,
or
in
some
geometry
books, the Extreme and Mean proportion. It is
represented by the Greek letter
(phi), and figure):
we
derive it from this equation (see
a
b
:
=
a
+ b
:
a
a, ...
=
a
/5 +
1
2
In
equal?
==
,
the
(1.618034
+ b .
.
!?., If a
b
=
1,
a
=
.)
following equation, what does
x
28
Continued
29
30
31
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
Continued
53
54
76
55
56
57
59
60
60
61
84
Continued
61
Continued
61
62
63
64
65
92
A SUBMISCELLANY OF
SHORT PUZZLES
These
puzzles are slightly inconsequential and do not quite strictly follow the rules of style. They must be read attentively, because the wording is sometimes tricky or overliteral. Some depend on observation, some on previous knowledge, some on rather offbeat reasoning. The reader is warned to watch for the unexpected.
7 count
is two
limit,
the
COUNT THE OBJECTS. Without moving your hands, things shown below (Fig. 1); a dot inside a square
things; the letter 1 1/2 min.
A is
one.
Give the method used. l'ime
10
you
were
GREAT CIRCLE. If all the
standing
at
the
corner
seas were
frozen over, and
of Fifth Avenue and
Forty-
second Street in New York City (Fig. 3), which of the four routes would take you nearest, and soonest, to Rome, Italy,
provided est" here
you walked in
a
straight path
means soonest to
or
great circle? ("Soon-
the nearest point to
Rome.)
RAIN ON THE ROOF. When rain falls
straight down (Fig. 4), there is less rain per unit area than if the roof were level (Fig. 5). This being the case it would seem that rain falling vertically on level ground would give it more of a wetting than the same rain would if it fell at an angle, because of wind. Why is this not so? 14
on
a
roof which is tilted at 45°
16
LEAKY BALLOON. A child's rubber balloon
in the attic a
resting
on
hemicylindricallid,
a
as
box and in the
an
was
left
old-fashioned trunk with
figure. The balloon
was so
thin
that its outline may be considered a circle, but it leaked and sank as it shrank. What is the locus of its center, C, as it low-
ered until it fell? The
edge, E,
may be considered
as a
point.
DOFFING OR DONNING.
21 to
Why is it so much easier fairly tight sweater off, if you wear glasses, without being displaced, than it is to put the sweater on without being displaced?
take
them them
a
22
GIBRALTAR.
Figure
Gibraltar. In between it and
7 shows us
is
an
a
view of the Rock of
old weather
vane
with
the pointer and letters missing, but here indicated as A, B, C, D, with B toward us. Which arm points most nearly toward Morocco?
27 can one
CORN IN BAG. How many shucked
fit into
a
very thin kind that
ears
of fresh
polyethylene bag, about 8 by some bread comes wrapped in),
thin
10 in.
corn
(the
in 10 sec.?
34
LETTER CODE. If
we are
told that certain letters in
the words "Standard Oath" represent something to do with number series, what should the next letters be? 35
MEASURING TREE. There is
a
pine
a
tree that looks
about 100 ft. tall, about 150 ft. away from the house, and slightly downhill. We can get to it by walking around a pond. Our only means of measuring is a 6-ft. tape measure. What is the
simplest 36
way to find its
height within
10
percent?
HOW MANY KINGS? Guess within 30 percent how crowned in England since the Norman
kings have been Conquest. many
37
TESSELATION. In the floor of
a
Gothic church is the
tesselated figure shown here, consisting of three equal an equilateral triangle, measuring 2 ft. to the side. As as
possible, give, and
prove, the
area
of the dark part.
arcs
in
simply
39
that
are
OPPOSITE STATES. What two very familiar
alike in
states in that
40
a
certain
important respect,
respect when described
PANAMA CANAL. If you
the Panama Canal and sail many miles
west
along
the
by
are
at
are
same
in
term?
the Atlantic end of
it to the Pacific
will you have traveled?
things, opposite
end, how
"No; I like take,
it
48
by
an exact
comes out
PINCH-PENNY. A
airmail. He had
took the letters 49
measurement, but with the number I
just right." How
to
man
many did he take?
sending a lot of letters weighing, but he always the post office. Why? was
an
accurate scale for
be
weighed
at
TWO LIQUIDS. What two
liquids,
which
are
to
be
found in most average homes, when poured into the same clean empty container, neither mix, flavor, nor contaminate each
other, and lowed) ? 50
can
be
quickly
and
easily separated (no heating al..
LONGEST SENTENCE. What is the
in two words in
English?
longest
sentence
ANSWERS TO PUZZLES
Neither. Since the top surface of the lower ball is in nonskidding contact with the bottom surface of the
top ball, we can imagine a sheet of paper, s, between them, moving to the left. When s moves x inches, the center, u, of the bottom ball moves 1/2 X inches in the
same
direction. The
same
is
of the center, b, of the top ball; thus b remains vertically above u, regardless of the ditrue
rection of motion.
V)
z
