Mathematical Brain Benders- 2nd Miscellany of Puzzles

,

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 <:

2

3

Continued

Z <:

3

4

z

<

5

6

7

z

<:

THE

THREE

CLOCKS

Continued

Continued

en

Z <

8

9

V) UJ

V)

z «

Continued

9 Answer 2

10

V')

0::: w

V')

z

<

11

z <:

12

------

I I

I I

I I I ex

_________1

z

<

13

14

Continued

V)

z

<:

14

Continued

14

V) LU

V)

Z

<:

Continued

Continued

Z

<:

14

Continued

z

<

15

Continued

V') w

V)

z

<:

17

V')

0::: w

V')

z

<

3

oz.

(troy).

When the small coin reaches the point where it tips, its diameter AD must be a chord of the

circular hole. A

point

on

the coin's

edge that

of the hole, C, will be on CB, with B the center of the coin, and CB J... AD,

touches the because AC

center

=

DC, radii of the hole. Since they

also radii of the coin, AB = CB; call this length 1. Then AC = v'2: and since the coins are

have the

portional portional

to to

Hence the small

thickness, their weights

same

one

are

pro-

their areas, which in turn are prothe squares of their respective radii.

big

coin

(2

=

weighs 2).

twice

as

much

as

the

V)

z <:

20

Continued

Continued

V')

Z <:

2a

Continued

2a

z

<:

21

22

z

<

23

Continued

V)

z

<:

24

Continued

V) w

V)

z

<

26

V)

z

<:

27

28

V)

z

<

29

30

z <:

31

32

en w

en

Z <

33

V)

z <:

34

35

V)

Z

<

35

Continued

Continued

35

V') LU

V')

Z

<

36

z <:

37

Continued

(f)

37 Answer 2

LU

S <J)

z «

z

<:

38

39

z

<:

40

41

z

<

42

43

V) LU

V)

Z

<:

Continued

43

z

<:

44

45

Continued

Z

<:

45

Continued

45

Continued

V)

z

<:

45

Continued

45

z

<:

46

Continued

46

z

<

47

48

V)

z

<:

49

50

Z <:

50

Continued

51

z <:

52

53

V) LU

V)

Z

<:

Continued

54

z

<:

55

56

V)

Z <:

56

Continued

57

V')

z

<:

58

Continued

z

<:

59

4

Continued

Z

<

59

Continued

59

z

<

60

61

V)

0::: u.J

V)

z

<:

Continued

61

Continued

Z

<:

61

Continued

61

z

<:

62

63

V) LoU

V')

z

<:

Continued

64

z «

65

Continued

65

ANSWERS TO

SUBMISCELLANY

1

ground glass. The clean cloth must be dry. The example in a kitchen might be a Pyrex baking dish become worn or scratched on the bottom, so as to be Wet

commonest

that had

merely translucent. When

wet it

is clear.

8

According to size, not weight. Counting from the fragments, barley, buckshot, marbles.

bottom up: cork

They overlap. Iceland (according to the National Geographic Society et ai.) is in Europe, and its west coast is 200 mi. west of 17° west meridian, which runs through Shannon Island, the eastern outpost of Greenland (part of North America). 9

10

Europe,

Up Fifth but

east

Avenue. It would take you on

Forty-second

Street

through northern

just misses South

Africa, and doesn't hit land until Asia. To be test it on a

Ice

large globe

with

a

seen, one must

great-circle ring (or

wait for the

Age). 11

Shanghai. (Tokyo

is

biggest, London

next, New York

third.) 12

stays

on

Almost

none.

It doesn't

always work, but usually

the bottom until bumped

or

heated.

it all

16

of this

A

hyperbola. If the

curve

in the

answer

reader remembers the definition to Puzzle

63, he will realize that

the center, C, of the balloon will move so that its distance from the edge, E, will always equal the balloon's radius, and at any

given position its distance from the center of the hemicylinder always equal the radius of the balloon plus the radius of the cylindrical lid. This fulfills the definition.

will

Peter. The missing word-which (Matt. 16: 23).

23 is Satan

Matchbooks.

24

ends the

alternating,

same

25

so

They

are

was not

arranged

that the missing

one

with

asked for-

their

narrow

leaves two pointing

way.

Neither. It is

totally impossible

A

the middle of the

to fold it ten

times.

(Try it.) 26 not

point, P,

at

that the distance from rim

bears

a

when

we

greater proportion press at P than if

to

to

side, because

most if

width. Figure 5 shows base, B, which acts as fulcrum,

all oval dishes have rims of

constant

the distance from B to center, C,

we

press at E.

31

Yes; by stirring the

32

6

==

SIX becomes 9

water in ==

IX.

the small bowl very fast.

38

Work.

47

Two

teaspoonfuls. Call the

amount

that sticks to the

inside of the spoon, x: we have to assume that x also sticks to the under side, otherwise the problem cannot be solved, and anyway it has

spoonful 1, 1 +

x

nearly the same area. Therefore, calling the dry spoonful is put into the cup with the spoon,

if 1 wet

is delivered: with 2 it would be 1

with 3 it would be it would be 4

1

-

x

+ 1

-

x

-

x

+ 1 + x,

and 1 + x, or

3

-

x:

or

2:

with 4

2x. The expression is, if n number of spoonings, the delivered amount is n( 1 x) + x, which is asymptotic to n( 1 x). In other words he wanted, and got, two. ==

-

-

-

48

Airmail is

figured by fractions of ounces. He gave his unstamped to be weighed by the clerk, who then sold him the right stamps to be put on afterward. Their weight could frequently have made just enough difference to require more

letters

postage.

Mercury and water. The former can be got from a by breaking. The water stays on top and can be and poured off, any traces removed with blotting paper or just Oils a nearly unremovable film on both water and leave waiting. 49

thermometer

mercury.

ENVOI (with apologies

to

John Lyly)