Great Book Of Math Puzzles

lb,4

1 k1 I I

GREAT BOOK OF

Philip

Heafford

ff Sterling Publishing Co., Inc.

ew York

al those who love to solve problem

Library of Congress Cataloging-in-Publication Data Available 10

8

3

Published in 1993 by Sterling Publishing Company, Inc. 38 Park Avenue South, ew York, N.Y. 10016 Originally published in Great Britain under the title Mathematics or 1959, 1987 by Philip Heafford Distributed in Canada by Sterling Publishing Canadian Manda Group, P.O. Box 920, Station 5P Toronto, Ontario, Canada Manufactured in the United States of America ll rights reserved Sterling ISBN 0-8069-8814-2

CONTENTS 1. Quickies 2. Th

.............................

Printer's Nightmare

3. Simple? Perhaps .......................

ou at Home in

4. Are

Rome?...............8

5. Easy Teasers ......

...................

Triangle 'lest ....................

Th

.10

7. Teasers ................................

.11

8. Some Old

.12

Some

Spot the Mistakes

10 11 12

13

What's

.................. .......................

Line? .....................

Mathematical Mixture

.................

Lighter Limericks .................... Math Medley .........................

.1 .1

.15 .1

.17

14 "C Gets the

It ..................

.1

15 Letters for

..................

.19

16 Some Short Stories ......................

.2

17

Brevity in Mathematics .................

.2

18

as Charlie Coping? ..................

.23

19

Can

ou Arrange These? .................

Puzzle These Ou Browsing in Books

.......................

.2

......................

.2

Strange Figures Formed Fun with Problems

.24

Figures.......

......................

.2 .2

Tackle These Twisters ....................

.3

Some Statistical Studies ...................

.3

ew Fast Ones .....................

.32

27 Calculus Cocktails ........................

33

28 Track the Term .........................

29 Arches 30

.............................. More Circles .............

Answers ................................ Index ..................................

35

36 37 95

Quickies these numbers ring bell? For instance, the number 365 would mean only one thing to an that is the number of days in a year. Ask someone to test you with this quiz. Six seconds for each question. Ho many can you get right in the time limit two minutes for all the questions?

1. 1,760

.4771

2. 2,000

12. .4971

3. 4,840

1.6

4. 640

14. 1.414

5. 1.732

15. 1,728

6. 2.54

16. 3-4-5

7. 3.1416

17. 6,080

8. 36

18.

9. .3010

19. 90

10. 1492

20 88

Answers

page 38

he Printer's Nightmare Before th days of th typewriter, the printer's lot was always happy one. Imagine difficult it must have been fo th unfortunate printer trying to se up th type for il arithmetic book when the hand-written manuscript putting printer overcame this difficulty legible. could "stars" fo th figures decipher. ee if yo could finding what th figures really are. have helped im 1. Addition: *22*

113

1**1

14

3489

*410 2. Subtraction: 4**2 *35*

6*35 *82* 4*

12 3. Multiplication:

*7

***7

**

****6 **203 *37**

**91

*kh**

Equations:

.:. x

*x x =

Answers on page 39.

4x ** or*

Simple? Perhaps! solve these problems?

1.

five girls pack five boxes of flowers in five minutes, ho many girls ar required to pack fifty boxes in fifty minutes?

2.

bo ha long cardboard strip inch wide an 48 inches long. It is marked at 1-inch intervals so that ca cu off series of square inches. If th bo takes on second fo each cut, how long will it take to cu th 48 square inches?

3.

move safe, tw cylindrical steel bars 7 inches in diameter ar used as rollers. How far will th safe have moved forward when th rollers have made on revolution?

town in India ha population of 20,000 people. pe cent of them are one-legged, an half the others go barefoot. How many sandals are worn in th town?

5. Without introducing signs, arrange six "nines" in ay that they add to 100. such 6. What is there peculiar about th 50 100? 4937.

left-hand side of

tail as long as its head plus quarter th length of its body. It body as three-quarters of its total length. It head was 4 inches long. What wa the length of th fish? fish

Answers on pages 39-42.

Are

at Home in Rome?

will have to know or most of th answers to this quiz the Roman figures. As they zero to give their numbers "place value," it must have been very awkward when it came to multiplication! as used by th Romans to help with cal1. What ai culations?

on famous monument: MDCCLXXVI. What year does this represent?

The following is

3. Write 1789 in Roman figures.

4. What is th largest number yo ca write using these Roman numerals once each, I,C,X,V,L?

5. What is th smallest number yo ca write using th same Roman numerals once each, I,C,X,V,L? 6. Without changing to our Hindu-Arabic notation, CCLXV. find th value of CXVI XIII 7. What Roman numbers of tw integers between on twenty become larger when th left-hand in teger is omitted?

8.

"groma" used by th veyor, cook, or sailor? Answers

Roman merchant, sur-

pages 42-44.

Easy Teasers 1. During vacation it rained on thirteen days, when it rained in the morning the afternoon was fine, an every rainy afternoon was preceded by fine morning. There were eleven fine mornings an twelve fine afternoons. ow long was th vacation? o'clock will th two 2. At what time between an hands of clock be in straight line?

3. If

1,728, what is the cube perfect cube 1,442,897?

1,331 an

root of th

4.

cents. bottle of cider costs cider cost 15 cents more than the bottle. Ho much should you receive on returning th bottle?

5.

lengths of the sides of right-angled triangle measure an exact number of feet. If the hypotenuse is foot longer than th base and the perpendicular is feet long, how long are th sides?

6.

spruce tree when planted was feet high and it grew by an equal number of feet each year. At the en of th seventh year, it was one-ninth taller than at the en of the sixth year. Ho tall was the tree at th en of the twelfth year?

7. Without doing th actual division state whether 13,972,536 is exactly divisible by 8. 8.

cement mixture costs $33 ton. It is composed of Grade cement at $36 ton an Grade cement at $24 a ton. Ho were these two cements mixed? Answers on pages 44-46.

6.

Test

triangle is figure bounded three straight lines and having three angles. Such a definition may be corth idea that triangle is decidedly rect, but it gives uninteresting figure. There re many different kinds of triangles and each one has it interesting peculiarities. From th information given, can state the names of these triangles?

1.

am readily suggested when yo look at

trillium.

2.

appear when man stands on level ground with is legs straight hi feet slightly apart.

3.

have special name derived from meaning "uneven."

4.

am formed by joining th feet of th perpendiculars from th vertices of to th opposite sides.

Greek word

sides equals 5. The su of th squares on tw of third side. th square constructed on

6. There ar at least two of us responding angles ar equal proportional. 7. The sides an th diagonals used to construct e. 8.

find that ur corur sides ar

quadrilateral ar

th sum of my

straight lines sides ar angles is greater than 1800.

9. I have gained th title ponss asinorum" for a certain proposition in Euclid. 10.

am connected with the stars Answers

10

th zenith.

pages 46-48.

Teasers 1. There ar three books, each on inch thick. They III. stand side by side in order-Volumes I, II A bookworm starts outside the front cover of Volan eats its way through to th outside of th back cover of Volume III. th worm travels fa does it travel? in straight line,

an built cubical house with ordinary windows found whenever in al th upright walls. looked of window that he was looking south. Where did he build hi house? 3.

merchant ha two large barrels. smaller barre holds 33 liters but is only five-sixths full of wine. empties this wine into th other barrel an finds that th wine fills only four-ninths of it. How much wine would th larger barrel hold when full?

4. What three curves ar produced making sections of right circular cone in directions other than parallel to th base?

5.

en play card game and the stake is on penny game. At the one has on three games th other ha three pennies. many games id they play?

number consists of three digits, 9, 5, another. If these digits are reversed an then subtracted from th original number, an answer will be ob tained consisting of th same digits arranged in different order still. What is that other digit? Answers

pages 48-49. 11

8.

Some

Some

1. Find quantity such that th sum of it seventh of it shall equal nineteen.

one-

2.

Chinese party many guests were present if every two used dish fo rice between them, every three dish for broth, every four dish fo meat, there were 65 dishes altogether.

3.

quarter of is life as retired colonel lived young man, one-third as boy, one-fifth as with responsibilities, and thirteen years on pension. ol as when died? in club outnumber th thin sixteen. Seven times the number of thin ceeds nine times th number fat and thin thirty-two. Find th number in th club.

The fa

ex-

en

5.

explorer grew beard during is travels. th of his journeys, found that double th length of hi whiskers added to its square plus twenty exactly equalled th number of days been away. If measured th length of is beard in centimeters, if he had been away 140 days, how long as his beard at the of his travels?

6.

cathedral tower 20 feet high is 25 feet from th to of each church tower 150 feet high. th tower is pigeon. The tw pigeons fly off at th same speed directly to some same time grain on th level straight road between th th same towers. The pigeons reach th grain th fa is th grain from th foot instant. cathedral tower? Answers

12

pages 49-51.

9.

Spot the Mistakes

Merely because statement appears in print it is not necessarily accurate! often one hears th remark, "I'll show it to in black white," as if that is sufficient to decide whether something true. mathematician must always accurate. Are th following statements true false?

1. The pentagram Pythagoras is formed by drawing al th diagonals of regular pentagon. Archimedes as th originator of th well-known puzzle of Achilles and the tortoise. 3. 1:05 p.m. is sometimes written as 1305 hours. The curve in which

uniform cable hangs when suspended from tw fixed points is parabola.

5.

pantograph is mechanical device fo drawing figures similar to given figures.

6.

histogram is kilograms, this standard unit is kept th International Bureau of Weights Measures at SRvres, near Paris. cantilever beam is beam supported extending horizontally. only

one end

parameter is independent variable in terms of which th co-ordinates of variable point ay be expressed. Answers on pages 51-53.

What's

Line?

purposes identification certain lines have been given special names, e.g. tangent, an arc, and a radius. have to name th line referred to in each these questions.

1. join th vertex of opposite side.

triangle to th mid-point of th

2. was said to be th points.

shortest distance between tw

3. subtend

right angle

th circumference

circle.

4. am the line so drawn in circle that th angle be tween an certain tangent is equal to th angle in th alternate segment.

5. "touch" 6.

hyperbola at an infinite distance.

circle in two points.

7. join al th points of th same latitude

th earth.

8. am th locus of point from which the tangents drawn to tw given circles ar equal.

9.

th essential straight line which, together with th special point or focus, enables points on an ellipse or parabola to be determined.

10. pass through the feet of th perpendiculars drawn to th three sides of triangle from an point on circumcircle of triangle.

Answers

pages 53-55.

1.

Mathematical Mixture

of questions. Some are easy and some This is a mixed connection between them whatsoever. re hard. There is Get busy as th proverbial ee many yo count

can answer correctly. Perfect marks will qualify fo the award th Pythagorean star which draw fo yourself. know

1. th

number of barleycorns in

inch?

2. th instrument used by ir Francis Drake to find th altitude of th su an hence th time? 3. the instrument used in th sixteenth century to tell th time at night by observing th constellation

Ursa Major?

th

name of th

mathematician

Vs(s-

(s-

first proved

b) (-c)

5. th name given to th figure like five-pointed star often used in th Middle Ages to frighten away witches? 6. what "meter" is used to measure th area contained plane curve? 7. th name of th solid formed by cutting cone by two parallel planes?

pyramid

8. to what us Simpson's rule is put?

9. th common name fo

regular hexahedron?

will take to strike "twelve" if long takes five seconds to strike "six"?

Answers

pages 55-57.

15

Lighter Limericks 1.

dear ol Grandpa named Lunn Is twice as ol as is son. Twenty-five years ago Their ag ratio Strange enough as three to one. When does Grandpa celebrate his centenary?

Said

certain young lady called Gwen tally of smitten young men, "One less and three more Divided by four Together give ne more than ten." she? many bo friends

There as young fellow named Clive, is bees numbered te to th power five. The daughters to each son Were as nineteen to one, A truly remarkable hive! many sons (drones) were in th hive? team's opening batter named Nero Squared is number of hits, th hero! After subtracting hi score, took off te an tw more, th final result as "zero." many hits id Nero make?

5.

Some freshmen from Trinity Hall Played hockey with wonderful ball; They found that two times it weight, Plus weight squared, minus eight, Gave "nothing" in ounces at all. What wa th of th ball? Answers

16

pages 57-59.

Math Medley 1. What is the name of th small metal frame with fine black glass or plastic front on which is line? used to facilitate th reading of slide rule.

th followratio 2. What is constructed in th in numbers? 24: 27: 30: 32: 36: 40 45 48 3. The minute hand of clock is distance does th ti of th

minutes?

inches long. What hand move in 22

4. What curve ha been called the "Helen of Geometers"? 5.

ca yo plant te tulips in ten straight rows with three tulips in each row?

6. The diameter of long-playing record is 12 inches. unused center as diameter of inches and there is smooth outer edge inch wide around to th inch, th recording. If there ar 91 th needle move during th actual fa playing of th recording? 7.

men, Mr. Henry Mr. Phillips, ar ap pointed to similar positions. elects to receive beginning salary of $3,000 pe year with increases of $600 each year, r. th second, Phillips, beginning salary of $1,500 pe increase of $300 every six months. half-year an person is better paid? Answers

pages 59-61.

17

Gets th

Worst

It

Below will find some problems that were common in arithmetic textbooks fifty years ago. often A, B, appeared, and th unfortunate and seemed to th loser, th person th worst everything! If ever single person deserves lasting credit from authors it is surely C. There re rivals fo that honor! Turn the clock back fifty years and solve th following:

1.

field is owned by three people; as three fifths of it, an has twice as much as What fraction of th field belongs to C?

2. In

mile race beats by 40 yards. By mile race?

3.

ca do piece of work in te days; ca do it in twelve days; can do it in an twenty days. long will take to do th work alone?

by yards, and he beats in much could beat

4. During game of billiards A can give 10 points in 50, ca give 10 points in 50. many points in 50 ca to make an even give game?

5.

form partnership. furnishes $1,875, furnishes $1,500, $1,250 capital. The partnership makes profit of $1,850 in th first year. What should take as his share of th profit? B,

6. Pipes ca fill tank in tw hours three hours respectively. Pipe ca empty it in five hours. If all be turned on when th tank is empty, how long will it take to fill?

Answers 18

pages 61-63.

Letters for Numerals Some simple sums were prepared using the numerals to Then all the numerals were changed to letters. You have to used for the change. You discover the code which this if you look carefully for every possible clue. There is no need to guess. Work these clues methodically, trying each possibility one after the other. There is only one solution to each sum. Th code has been changed for each sum. Don't peep at the answers until you have finished an checked your calculation, because the knowledge of one single change will make it to easy an spoil your fun.

1. Addition xxxx Y y y y z z

3. Division

IL)PHIL

YXXXZ

2. Multiplication NX

L

4. Division AY)NEL

L

X NX

Answers on pages 63-65.

Some Short Stories 1. When it Who was it? This is th story of well-known born years ago. ha influenced fo many generations th thoughts an th minds of en women in many different lands. We ca tell yo that the first last digits of to th year during which wa born ad th second digit, an that the third digit is that three times larger than th second digit, th fourth digit equals tw times th third digit. is yo calculate th year of hi birth? this gentleman? 2. Who caught the bus? Juliette and her sister Lucile lived together in that beautiful town of Montreux by La Leman in th Swiss Alps. In th springtime on of their favorite to th lovely fields of narcissi walks was to go growing on th mountain slopes nearby. long straight on occasion they came to stretch of road, certain point on it, they left th road right angles across walked field to large clump of narcissi. Juliette stopped to pick some of th flowers 40 meters away from th road, while Lucile also collected some flowers another meter farther on Suddenly they looked going along th road to Montreux. up to see decided to ride home, th When they as 70 meters away from th point where they left th road to walk across th field. half th speed th They ra traveled to missed th th point where they left the road bus! There is least one point on that stretch of road where th bus could have been caught.

bus?

calculate where they should have if both of th sisters could have caught th

3. How

this done? Arab when he died left to is three sons seventeen camels, giving to th eldest four ninths, to th second one third, to th youngest one sixth of them. three young en sat in front their house contemplating ow they could fulfill their father's wish without killing an of th animals. They did not find solution to this problem. Suddenly dervish came riding along camel. They asked im to si down on it them fo moment told im of their troubles. The dervish pondered fo moment, smiled cunningly, an said, "I know how yo ca carry ut your father's wish without having to kill even one of th animals." yo guess what suggestion th dervish made?

4. Can

sheet paper have side only? page on which this is printed ha tw sides edge all th ay around. yo tear it of th book yo ca easily trace th edge with pencil. Nevertheless it would be pity to spoil th book by doing this! If yo want to go from on side of th paper to th other, yo must go through the paper or over on of th edges. you design piece of paper that ha only ne side an also only one edge? yo ca do this, then yo ca paint th whole surface with brush (if the brush held enough paint) without removing it from th surface or going over an edge.

Answers on pages 65-67.

Brevity in Mathematics The mathematician frequently uses abbreviations in

work. th word "logarithms" he uses the shortened term "logs," and fo "simple harmonic motion" he uses th initial these words and writes "S.H.M." What abbrevialetters tion does use for . . . ?

1. "which was to be proved or demonstrated'? 2. th

cosine

th angle B?

expression which depends fo it value value yo give to x?

3.

4. th

5. th

th

with respect to x?

integration of

smallest number which is exactly divisible tw or more numbers?

6. th hyperbolic sine of x? 7. th square root of -1

8. th greatest number which will divide exactly into tw or more numbers?

9. th derivative of

th

eccentricity of conics? Answers

22

with respect to x?

pages 67-69.

Charlie Coping? Some rather surprising correct results are often found in Charlie'swork, which frequently is good only in parts. Here are some examples from Charlie's homework. You have to correct these quickly as possible. Are they right wrong? 144

122=

2. 132=

/i

3.

..

212=

..

312=

5VA

27-=

2-v/

lines joining the mid-points the sides of parallelogram form parallelogram. Therefore the lines joining the mid-points the sides of any convex quadrilateral also form parallelogram.

5. Th

in (a

b)-sin (a

in (a Solve

b)-sin (a

and

y-1

.-.

+ sin b)

(sin

b)

x1 x

si

b) _2

x--=-1=2 -V

x=

+1

x=5,andy=6 How mn.v t r i I n a n .ro. there in this figure? There are twelve lines. Each triangle three sides

12 :.

10

triangles

Answers on pages 69-70.

(sina

si

sinb)

Yo Arrange These? 1.

bo is to chosen president and a girl vice-president of th senior class of school. In many ways is this possible if the class ha twelve boys te girls?

photographed in row. 2. Six boys ar to many different arrangements ca be made order in which they ar to sit?

th

3. The same six boys ar to si around table fo lunch. many different arrangements ca made of th order in which they ar to sit?

If th first three letters of telephone number indicate th name of th exchange, ho many such arrangements of three is it possible to devise from th twenty-six letters of th alphabet? 5.

6. 7.

many different forecasts must be made of four football games in order to ensure that ne forecast is correct? ow many different ways ca tw dice, on blue, come up when thrown?

of th crews in th Harvard-Yale race as problem fo its captain. Three of th crew are tw of them ar bowstroke-side oarsmen only side oarsmen only. Ignoring weights personal preferences, in ho many ways ca th captain arrange hi eight to form th crew? The co is selected does change. Answers

24

re

pages 71-73.

Puzzle These 1.

water lily doubles itself in size each day. From the time it first leaf appeared to the time when the the pond completely covered took surface the pond to be forty days. How long did it take half covered?

quart bottle had ll it dimensions doubled. What the new bottle? is the volume

3. From Philadelphia to Atlantic City is 60 miles. Two trains leave at 10:00 A.M., one train from Philadelmiles phia at hour and the other from Atlanti City at 50 miles an hour. When they meet, re to Atlantic City? they nearer to Philadelphia Spot the wrong number in these series

(a (b (c

5. What (a) (b (c

1, 2, 4, 8, 15, 1, 7, 27, 64, 125, 10 15, 21 25

numbers:

is the missing number in these series: 81 2 7 , - , , , 1, 4, 9 , - , 25 2, 6, 12 -, 30

greatest and which the least of lo 4), (log 3), and (log lo 4) lo (6 lo )?

6. Which is th (2

7. Write down the Roman numerals from "one" to "six" clock face. as seen

Answers on pages 73-74.

Browsing in Books It is always interesting to look at ld books. have th following been fortunate enough to have seen some in ld arithmetic books. What could the author have meant

..

567342452

36418

98525476

5. by

3.

83 4' 2" 5"'

feet

inches

inches

18

19 feet

8.

25

Answers 26

inches

74-76.

Strange Figures Formed Figures 3

21

56

10

10

35

35

56

21

1

28

1. Write down th seventh line of figures in th metical triangle.

arith-

2. What ar th missing numbers in th last line arithmetical triangle?

th

3. Where in th arithmetical triangle do th coeffi(x a) cients of th terms of (x appear?

4. Use the (x

5.

2)4.

is th triangle?

to work out the coefficients of

mathematician

associated with this

Find su of the numbers in each column, each row, each diagonal of th square printed above. What name is given to square built in this way? 27

7. Complete as th

number square built in th printed above, given:

same

ay

12

16

-8--

four 8. Construct number square of four rows columns such that th su of each column, row, given that th to diagonal is th same, the left-hand column 4, 14, ro is 1, is 1, 12, 8, Answers

pages 76-78.

Fu

with Problems

1. Th first five terms the series 10 up to 150. What five terms without fractions, add up to 153?

30

50 another series,

2. Find three vulgar fractions the same value using ll the digits to once only. Here is one solution of the problem: 3. =

only three weights, bu with 3. A boy selling fruit pounds them weigh any whole number from pound to 13 pounds inclusive. What weights

mathematical process, such al and only the as +, -, x, ., A/ etc., digits 9, 9, to make (a 1, (b 4, (c) 6?

Can you,

adopting

the surface of the earth ou 5. From where travel 10 miles due south, then 10 miles due west, and finally 00 miles due north to arrive again at your starting point?

miles an hour takes three train traveling at seconds to enter tunnel and further thirty seconds to pass completely through it What is the length the (a train, (b tunnel?

Answers

pages 78-79.

Tackle These Twisters quantities terms re faced with succession quantity, re formed according which, after th first term to common law. This sounds very complicated, but confidence is all grain of common sense plus tw grains that is necessary to have some fun with th following series. Here

1.

reciprocals ar in arithmetical progression, an am of some interest in the theory of hope sound. What is my name?

2.

ratios of successive terms of this series ar conleaves of head of nected with plant growth. lettuce th layers of an onion grow like this. What is my name?

3. What is th series?

su

of th 7x3

5x2+

3x

first twenty terms of this

4. What is the eighth term an also the sum of th first eight terms of this series? 9-11-13 7-9-11 11-13-15 5-7-9

5. Is th logarithmic series, loge (1

X)

-3_X

logarithms to th

useful fo working

6. What is th name of this series?

5!

3!

7!

7. What is th name of this series? 1+

+-

2!

Answers

3!

+

pages 79-81.

base e?

Some Statistical Studies

atJ

4l

VZ uJ

VAR/ALE QUANTITY What is the name

...

1. this special column graph? 2. th

shape formed by joining th tops of th columns?

3. th frequency curve shaped like 4. th arithmetical average of th quantity?

5. th

mid-points

th

cocked hat?

values of

most frequently observed value of quantity?

variable variable

6. that which most satisfactorily indicates th spread of th observed values of variable quantity? 7. th

sample chosen such that every sample equal chance of being picked? Answers

as an

pages 81-83.

31

Fast Ones fa ca yo go into forest? drives along main highway which regular service of buses is in operation. He notices that every three minutes he meets and that every six minutes overtakes him. often does leave th terminal station of th route? 3. There are twelve dollars in dozen. many dozen? dimes ar airplane flies around the equator at height of 200 feet. If the radius of th earth is 4,000 miles ow much farther than th circumference of th earth will th airplane have to travel? 5. In small town of 50,000 inhabitants, it ha been cent of th males an 28 pe counted that 42 cent of th females married people from their ow town. Assuming these numbers have remained fairly constant over th years, many males ar there in th town? 6. You are standing at the center of circle of radius 9 feet. You begin to hop in straight line to th is 4% Your first each second 21/4 feet, an yo continue to time half th length of your previous hop. many hops will yo make before you ge of th circle? Three students have tw boxes of candy which they want to share equally among themselves. Neither the number of pieces in th first bo th numin th second is divisible by three. et ne of the students noticed that there were seven more then pieces in th second bo than in th first said, 'W ca share this candy equally between us." correct?

1.

Answers 32

pages 83-85.

Calculus Cockils There re some problems which anyone with elementary knowledge calculus can solve with utmost ease. After tackle interesting and practical short time a beginner problems. The feeling accomplishment, and even fun,

which th subject brings -when rightly used is also increased it methods. Below are four problems which th beauty th id be solved this admirable instrument.

1.

miles from th nearest hiker on th moors is on straight road. miles from point, along th road is an inn. hiker ca walk at miles hour over th grassy moors miles er hour along th good road. what distance from must ai to strike th road in order to et to th in as quickly as possible?

2.

Dare the space-ship pilot wears a space in paraboloid of revolution. The dith shape of ameter of th circular base is inches and the is 12 inches. What volume of height of th heavy water will it hold?

3. Equal squares are cut out each of th corners of rectangular sheet of tinfoil whose dimensions ar 32 inches 20 inches. Find the maximum lined by volume of wooden bo which ca suitably bending the tinfoil to cover th base an th sides of th box. 4. A pleasure steamer 150 feet long has changed its direction through 30 degrees while moving through distance equal to twice its ow length. What is th radius of th circle in which it moved? Answers

pages 85-88.

Track th

Term

in There re large number mathematical terms that cluded in expressions in common use. How frequently

hear "equal rights," "shooting a line," "hot rod," "100-percent effort," "integral part," "vicious circle. will able to find many more if listen carefully. In the following, the whole, of find th mathematical term that is part, an everyday expression suggested by

1. The area over which anything exerts influence. 2. Having equal scores when playing golf.

3. The hour at which an operation is timed to begin. 4. The Great

5.

between Atlantic

excess in size, amount, etc.

Pacific.

having

relation

6. The modern measuring unit of intelligence. 7. A famous military building. 8.

ho sets forth in words, expounds, or interprets.

9. The rejection of 10.

judge recapitulates th evidence case. Answers

34

person proposed fo some office.

pages 88-89.

th

of

Arches application of geometry to architectural drawings is vious, an mathematician can an interesting companion students when on sight-seeing tour. One thrilled group he showed them how particulararch in an old church could pair of compasses an drawn readily with the aid the followruler. Ca you spot the arch that is suggested ing statements?

Th

1.

sounds like exclamation, but is an arch double curves that rise to point.

tw

It is common in Early English churches, and is also the name surgical instrument.

3. Very good food is suggested! The Mohammedan race inhabiting Northwest Africa never used this arch. 4. This arch is not connected with heraldry, bu is used solid steps. to support flight

5. Obviously very much connected with triangle. Th

certain kind

commonest brick arch in house construction.

rounded arch

more than

semicircle.

Most likely to be found in spacious buildings constructed in England between 1485 and 1546. am semicircular in shape and often have ornamentation.

Answers

pages 90-92.

chevron

Circles, Circles Circles

More

Given th following clues, can you name the circle which is implied?

1. It seems to "a manager," but th tw tangents from any point it to an ellipse ar at right angles. 2. It seems as if this circle could be helpful to an ellipse. 3. Is

4.

circle very much tied with the feet of th altitudes th mid-points of th sides of a triangle. What size shoe id Clementine wear? circles which

"right" across each other.

5. The circle which touches all th 6. King Alfred id

sides of

polygon.

really name this circle!

7. The circle which seems to be suffering from "spring fever."

8.

triangle is greedy enough to have more than of these circles.

9.

circle which passes through the vertices angle. Answers on pages 92-94.

36

tri-

ANSWERS

1. Quickies 1. Yards in

mile.

Pounds in

ton.

3. Square yards in

4. Acres in

acre.

square mile.

5. The square root of 3. 6. Centimeters in

inch.

7. w-the ratio of th circumference eter. 8. Days in

a circle to its diam-

leap year.

9. The logarithm

to th base 10.

10. The year in which Christopher Columbus found land (in th Bahamas) by sailing west from Spain. The logarithm of

logarithm of

12.

1.6 kilometer

to th

base 10.

to th base 10.

mile,

0.6214 mile

kilometer.

14. The square root of 2.

Cubic inches in 16. Ratio of th 17. Feet in 18. 62

cubic foot. right-angled triangle.

sides

nautical mile.

pounds is th weight of

19. Degrees in

feet

cubic foot of water.

right angle. second

th same as 60 miles

hour.

Printer's Nightmare 1.

11

2228 1261

14 52

3489

1410

6235 5828

4472 4351

07

12

5467 89

53 14 23

43736 49203 43736

2491

5x

.-.

4909366

4x

4x

.-.

21

-3

Simple? Perhaps! 1. FIVE GIRLS Five girls pack five boxes in five minutes, Five girls pack ne bo in on minute (working on th same box!), Five girls pack fifty boxes in fifty minutes.

2. 47 SECONDS The time taken will 47 seconds, because th produces th last tw squares.

47th cu

3. 44 INCHES Steel bars ar often used as rollers in this way. forward twice th length of th circumference of ne of th

steel bars. This distance is therefore

2-22-7

inches,

which is 44 inches. With three or an number of rollers forward de th safe it will still inches. The best ay to see this is to consider this problem in tw parts: (a) th

motion forward caused by ne revolution of th rollers if they were rolling of the ground,

(b) th

motion forward of th centers th rollers cause they themselves roll forward on the ground.

both cases th motion amounts to 22 inches, so that th th safe mounted total movement th rolling rollers is 44 inches. 20,000 id yo et this right? It really does matter what percentage of the population one-legged! ll th one-legged people will require only one shoe in an case. th remainder, half will wear no th other half will carry tw shoes on their two feet. This works at on shoe person fo th "others." shall therefore need fo th whole population on th average shoe er person. 4.

5. 99 99 99

This is on

40

th

ol

trick problems. If recognized signs

were introduced, you could arrange six "nines" to give 100 as follows: [(

X 9)+

91

99

ou could also arrange four "nines" to give 10 as follows: 99

6.

TH

TO

NUMBERS

APPEAR

This is interesting, it is by no means th only example taken once to of composing 100 from all th numbers only. Examine these solutions: + (b

78 (c) 89

+ 2+ 3

4+ 5

+ 7 + (8

9)

21 6i%

43

(d (e

90 836 i4 96 1 23-A (f) 9740 26 (g) 9796 2-5--

yo

design still more solutions?

128 INCHES

represent th head, total length of th fish. th problem Looking facts: B3=L It is also true that

th body, ar

th tail,

given th

th

following three

1B

inches

+

41

In this equation keep L

.W

substitute for everything else.

4inches

inches inches inches

-L + 'L

UL

(H

IB

inches

-w

16inches 128 inches 8

Thus

see that th fish

as 128 inches long.

at Home in Rome?

Are

ABACUS 1. The merchants traders ancient days in Egypt an Mesopotamia used to set pebbles in grooves of sand to "units" accounts. There would calculate groove, groove for "tens," on fo "hundreds." Such th word derived from Greek as a simple abacus, Roman times, calculating frame word meaning "tablet." this as also wires as made in which pebbles slid called an abacus. size of th abacus determined th size th numbers which could be dealt with. The Roman multiplication merals made simple addition, subtraction, very complicated. Calculations were done by slaves using an abacus. It is interesting to note that th Roman word for here we have th derivation of "pebble" as "calculus" word "calculate."

2. 1776 Letters were used by th Romans to represent various numthey seem to follow th pattern set by th Greeks. bers, 1, 5, 10, 50, I,V,X,

1000

The Roman numerals on

1000,

500, 5,I=

monument therefore read as: 100,

100,

50,

10,

These added together make 1776, an knows this date!

every American

3. MDCCLXXXIX 1000,

10, 500, 100, 100, 50, and IX 9. Ad these al together and th result is 1789.

CLXVI This number is 166. 4.

5. CXLIV This number is 144. XL is te before fifty, which is forty. Similarly, IV is on before five, which is four. 6.

CC

II VI

CD

answer of th addition is four C's, which is four hundred. It was th custom no to write four similar numerals consecutively. Hence, instead of writing four "hundreds" (CCCC), th Romans wrote one hundred less than five less before the D meant hundred (CD). Placing the than D, an placing it after the D, in DC, meant more than D. that is four hundred an is six hundred.

7.

Roman numeral for four is IV. Therefore, when the left-hand integer is removed there remains the integer V. is the Roman numeral for five. Hence the four changes to five. Similarly, the Roman numeral for nine is IX, and when the left-hand integer I is removed there remains th Roman numeral X, which is ten.

43

8. SURVEYOR The "groma" as important surveying instrument used th Roman surveyors or agrimensores. As fa as mathematics was concerned, th Romans were practical more. them mathematics was a tool that helped them to construct an to measure. agrimensor as land- or field-measurer, in this work used th groma. It as frequently carved on the tombstones Roman surveyors.

Easy Teasers 1. 18 DAYS There ar three possible types (a) Rain in th morning (b) Fine in th morning (c) Fine in th morning

day: fine in th afternoon rain in th afternoon fine in th afternoon

number such days in each category be an c. number of days which rain falls + 13 number days having mornings b+ 11 number days having fine afternoons + 12 From these equations, we derive 7, 5. 6, an .-. number vacation is days + 18.

5% MINUTES PAST o'clock th minute hand is 35 divisions behind th hour hand. be opposite on another th minute hand must gain divisions on th hour hand. But the minute gains 55 divisions in 60 true minutes. th minute gains divisions in 55/1 true minutes. ll problems concerning the positions of the hands on clock faces will be solved readily if you draw sketch rememer that th accurate position is "something" an "something"-elevenths. 2.

44

3.

11

1,442,897 is seven-figure number an thus th cube root must lie between 110 an 120. As th last figure is 7, th cube root must end in 3.

FIVE CENTS Let your algebra help youl

4.

25 15 5.

5. 9, 40 and 41 FEET The procedure is as follows: square the length of th perpendicular, subtract 1, divide by 2. The result is th length of th base. then that is th length of th hypotenuse. This applies when the perpendicular is an od numbe an combinations numbers ar sometimes Pythagorean series. Other combinations ar 3-4-5, 5-12-13, so on indefinitely. 7-24-25, 11-60-61, 13-84-85, method is derived from th theorem of Pythagoras concernin right-angled triangle. (B +1

2B+1 -1

6.

FEET

th tree grow feet each year. th sixth year, th height At the en growth

At the en

the twelfth year, the height th tree

th tree (3 (3

feet

-91

(3

12x) feet 15 feet.

7.

ou know th tests divisibility? Numbers will divide exactly if they en with an even digit if th su of th digits is divisible if th last tw digits ar divisible by if th last digit is or " if divisible by both if th last three digits are divisible " What about the tests of divisibility fo 7, 9, 12?

8. GRADE A: GRADE be mixed with parts of Grade cement, then the equation is: 36

24B

parts

Grade

33 (A 9B

he Triangle Test 1. EQUILATERAL th name suggests, all th sides ar equal. Equilateral triangles have only equal sides but have equal angles to ca thus be called equiangular triangles. The trilof symmetry suggesting an equilateral lium ha three triangle. Many plants have geometrical shapes in their roots, flowers. What shape is found in the crossstems, leaves, section of th stem of th sedge family? 2.

ISOSCELES

word means "equal legs." Any triangle which ha tw equal is called an triangle. of it three look around us we ca often find geometrical figures in na ture. th construction of houses, ships, aircraft, th isosceles triangle is often encountered.

3.

SCALENE This is triangle which is "uneven" because it has ll it sides unequal. Th word "scalene" nothing to with drawing to scale, bu is derived from the Greek word "skalenos," which means "uneven" "unequal."

4. PEDAL This triangle is sometimes called the orthocentric triangle. If AD BE, and CF re the perpendiculars dropped from the vertices the triangle ABC to the opposite sides, then the is the pedal triangle. Th three perpendiculars triangle common point called the orthocenter. pass through

5.

RIGHT-ANGLED he unique property the right-angled triangle ABC is that if angle A is the right angle then Th geometrical proof this property is associated with the Greek mathematician Pythagoras, lived in the sixth century B.C. Triangles whose sides re in the ratios 3-4-5 5-12-13 re right-angled.

6.

SIMILAR

similar if they have their angles Two triangles re said to another and have their sides, taken in order, equal to about the corresponding equal angles proportional. The similar triangles re proportional to the squares of areas their corresponding sides. 7.

HARMONIC ABCD is the quadrilateral and is the harmonic triangle. It is formed joining the in tersection points the sides and the intersection noint ra ___ th diagonals.

8.

SPHERICAL

spherical triangle consists of portion of sphere bounded conby three arcs of great circles. Obviously th sides do sist of straight lines. su of th three angles of such 47

triangle lies between 1800 work with spherical triangles

5400. Much of th early as done by Menelaus about

A.D. 1 00

ISOSCELES th base The particular proposition proved that th angles of an isosceles triangle ar equal. The title "pons asinorum" means th "bridge of asses." It is said that in the Middle Ages th "donkey" could pass over this bridge to conhi study of Euclidean geometry, th name ay be to th fact that th figure in Euclid resembles simple truss bridge. 9.

10.

ASTRONOMICAL

This is spherical triangle on a celestial sphere which fo its vertices th nearest celestial pole, the zenith, th star under consideration.

Teasers 1. ON INCH Surely th bookworm ha only to go from as in th figure! This distance is th to thickness of Volume II-one inch.

2.

TH

NORTH POLE

cannot imagine that there is cubical igloo th North if there were if it windows they would Pole! al have southern outlook.

30 LITERS th volume of th will be

second barrel. Your equation

36 :.

4.

336

x=-

ELLIPSE, PARABOLA, HYPERBOLA

These three curves are often

referred to as "conic sections." Th sections re shown in the figure-the hyperbola is paralle to the axis, the parabola is parallel to the slant height, and the ellipse is oblique. Th base is, of course, circle.

ELLIPAR

PARAB~

5. NINE thus gains three pennies. wins three games ha to in back these three pennies which takes another wins three more to in three games, th total su of three pennies. 6.

If th missing digit, then th solution is found from this equation: 900

50

(100x

Thus th number is 954,

8. Some 1.

50

95

9)

45

100x

90 +

495.

Some

16% found in the ancient Rhind papyrus This problem is to Ahmes papyrus written more than thousand years before Christ. Th original wording appears strange: "heap, it seventh, it whole, it makes nineteen."

The solution is found thus: X+

X=

19

13 16

.-

2.

This problem is traditionally attributed to Sun Tsu, who solving the lived in the first century A.D. His method problem did not involve the usual unknown quantity "x." the solution but only the answer-perhaps He did not give he guessed. The modern solution is the number guests. Let Equating the number of dishes from the information

given,

+ -3+

65 12 12

65 13x= 65 3x

.-

60

YEARS example of the type of problem which This is popular in the fourth century. It is easily solved letting is years. represented 3.

13

Then 20x

15x

.-.

13

60x-(60

60

and THIN MEN 56 FAT ME To solve this problem, ll need to do is to put the story down in mathematical symbols thus:

F-16=

7F-32

T

...... .......

.1 2

......

3

Multiply line

9F-144

away from line 3:

Take line

2F-112=

Hence

and

56

10 CENTIMETERS-not too long! Translate this problem into mathematical symbols and then there is easy quadratic equation to be solved. Let the length centimeters. beard e 5.

L2

2L 12)(L

(L

6. A

Hence

12 10

14

10 centimeters

90 FEET

tneorem ot Pythagoras to both of th right-angled triangles: Applying tM

Y2

2002 +

15 (250-x) .- 40,000 + 22,500 62,500

Y2

.. 500x .-.

150'

SO(x

45,000 90

Spot the Mistakes 1.

TRUE

pentagram Pythagoras is the five-pointed star formed drawing al the diagonals regular pentagon deleting the sides. Pentagrams, heptagrams, and monograms

Th

very early date to have magical and were considered at used mystical properties. Th pentagram in particular symbol health and happiness. The Pythagoreans as (disciples Pythagoras) used the pentagram as their badge of recognition. 51

2.

FALSE the paradox of Achilles and the Zeno as the originator tortoise. Zeno Elea lived in the fifth century B.C., and been associated with many paradoxes time, space, and motion. Zeno argued that no matter bow fast Achilles ran he could never catch the tortoise, fo the latter would move short distance on as Achilles covered the distance between him and where the tortoise was!

3. TRUE Th sensible means A.M. and P.M. is to

avoiding any confusion between the twenty-four-hour clock. The the continent railway timetables Europe leave no possible doubt about the time of the train night when station. Midnight is expressed as 0000 is due to leave hours and every time during the is expressed in four digits. For instance, 9:30 A.M. 0930 hours, and 12 noon 1200 hours, and 4:15 P.M. 1615 hours.

FALSE Th curve assumed cable when hanging chain, rope, freely between tw supports is very much like parabola, catenary, but is really quite different. Th curve is called which is derived from the Latin word "catena," chain. Galileo thought this curve parabola, and it not until late in the seventeenth century that the Bernoullis and the catenary. Leibnitz discovered the peculiar properties 4.

5. TRUE Th pantograph any combination

used to copy any figure composed of straight lines and curved lines. It be to the same size adjusted to cause the copy to be enlarged or reduced. Th instrument consists essentially of freely-jointed parallelogram of hinged rods. Th lengths the sides of the parallelogram varied to produce the different sizes the copy. Th original and the copy are in tw dimensions and li in the same plane.

FALSE ay connected with th unit of weight, A histogram is in th gram. It is true that the metric standards ar kept at th Measures at Sevres International Bureau of Weights near Paris. The histogram is connected with the method of graphical representation of data, is described elsewhere in this book. 6.

7. TRUE In architecture the cantilever is projecting bracket which is used to support balcony. There are very many examples of its us in buildings of several generations. It is also of cantilevers stretch great us in th building of bridges. from piers on opposite sides an these ar joined together by girder to complete th span. Quebec Bridge over th t. Lawrence River in Canada as th longest cantilever span in th world-1,800 feet. 8. TRUE ay in which th term "parameter" is used in This is th mathematics when dealing with the equations of curves an surfaces. It came into se about hundred years ago. The se of parametric equations frequently simplifies calculations in algebraic geometry calculus. Parameter means "side measure." For instance, th co-ordinates of point on parabola expressed in terms of one parameter "t" ca be written as at

2at.

What's

Line?

1. MEDIAN This is th definition which is usually given of median. The three medians of triangle ar concurrent, an th point which they intersect is ay along each third of th median measured toward th vertex from th mid-point of th opposite side. 53

2.

STRAIGHT LINE

This as th ol definition of straight line. It is doubtful whether navigator would agree with this entirely, as fa as mathematics is concerned this is th meaning usually better definition would attached to it in simple geometry. be "a straight line is on which keeps th same direction throughout its length." DIAMETER

circle is straight line drawn through th diameter center terminated at both ends by th circumference. Any diameter cuts circle into tw equal parts, each being called a semicircle. The angle in a semicircle is right angle.

CHORD important theorem in geometry states that if tangent be drawn to circle at the point of contact chord drawn, then th angles which th chord makes with this tangent ar equal to th angles in th alternate segments of th circle. 4.

5. ASYMPTOTE This peculiar word is derived from th Greek language and appeared in mathematics in 1656. It as used fo th name of th line to which curve continually approaches does meet within finite distance. Frequently an asymptote is called tangent at infinity. 6.

SECANT

tw distinct points, it is If straight line cuts any curve called secant. The beginner in geometry must always tangent, fo th latter, differentiate between secant no matter ho far it is produced either way, as only one point in common with curve. PARALLEL OF LATITUDE This is small circle drawn through places of th same at right angles to th latitude. It is parallel to the equator 7.

the line joining the North and South Poles. earth's axis Latitudes are expressed in degrees and minutes either side north the equator. south

8. RADICAL AXIS

point which moves so that the This is the locus or path tangents drawn from it to tw fixed circles are equal. Actually it is straight line perpendicular to the line joining the centers the tw circles.

9.

DIRECTRIX Th term came into

in 1702. Th distance from conic (ellipse point parabola) to the directrix bears constant ratio to the distance the same point from the focus determining the standard equation the conic. for conic the directrix-focus property is generally used.

10

SIMSON

PEDAL LINE

Robert Simson, professor of mathematics at Glasgow University in the eighteenth century, been honored by having this particular line triangle named after him. He made many contributions to mathematics, and most the Euclid re based his work. English editions

Mathematical Mixture 1. THREE

Early linear measurements were defined in terms body course, vary from man to man! The earliest sizes which, English la defining length made during the reign of read, "Three barley corns, round and Edward II in make inch." dry, this to more man's thumb. standard than th width ASTROLABE

Th

name is derived from tw

Greek words meaning "star

taking," but the instrument used to take the altitude the fifteenth century, quite complithe sun moon. cated astrolabes were being used astronomers, bu about 1480 made fo mariners. Th simpler instrument mariner had to know the sun's declination from tables and he could then calculate is latitude from his own observathe altitude tion the midday sun using the astrolabe. This instrument did not give accurate results and in th Hadley's sextant. eighteenth century it superseded Th maximum angle which could Hadley's measured sextant 90°, hence it as sometimes called quadrant. NOCTURNAL nocturnal is an instrument which used fo finding the night time observing the relative positions of th North Star and the pointers the ig Dipper. In the British Humfray Cole in seen nocturnal made Museum not difficult to use, and when th 1560. This instrument adjusted so that the tw pointers movable the it the time could ig Dipper appeared to li read of from time disk graduated in hours and minutes. Th nocturnal the chronometer at sea, bu used replaced until quite late in the eighteenth century. 3.

4.

HERO OF ALEXANDRIA Hero's work we do not know, but he Th exact period know that belonged to the First Alexandrian School. We he an able mathematician. About B.c., he pu gineering and surveying more scientific basis. He is credited with the discovery this formula for the area triangle where the sides half the the triangle.

5. PENTACLE

pentacle looks like tw interlaced triangles. Th penused as tacle and the pentagram the same figure. It symbol the Greeks, and various societies have mystery used the symbol. In the Middle Ages, many people thought the pentacle had the power to keep away evil spirits.

Th

6.

PLANIMETER This instrument is used fo

mechanically measuring the

area irregular plane figure. Th hatchet planimeter is probably the simplest type and the wheel and disk, or Amsler type, is the most common. 7.

FRUSTUM Frustum means "a piece off." It refers to that portion regular solid left after cutting of the upper part plane parallel to the base, bu it used describe

tw

the portion intercepted between

planes.

APPROXIMATING AN AREA Th area is divided into any even number of equal breadth.

8.

9.

CUBE

solid figure which regular hexahedron is cube, fo it hexahedron is

10.

parallel strips

six faces, so that the si equal faces.

ELEVEN SECONDS

"strike" occurs instantaneously and then there is a "rest" before the next strike. Hence, to strike si times requires five rests and this takes five seconds. There eleven rests when the clock strikes "twelve."

Lighter Limericks 1.

and also

Lunn's or

2S

or or and

10

(L-25)

son's

(S-25) 3S-75 25 3S-50 50

Thus drink Grandpa's health just now-he is one hundred years old-a centenarian.

42 or 21 This problem admits 2.

two solutions, according to the interpret the wording the limerick.

(a)

B-1

+

10

or

42

or

(b

+ 10

(B

(B-1)

44 21

or or

What lucky young lady! More than enough friends her to have different on each of the month!

5,000 Clive had in his hive queen who laid the eggs headed the colony, as many as 95,000 workers (daughters the queen-undeveloped females), and only 5,000 drones (sons the queen-fully developed males). 3.

Workers + drones 10 But in every 20

100,000

was

Number of drones

drone

100,000

5,000 4.

Let the number hits scored the information given in the form

(H-4)(H .-.

Nero. Then, writing equation, obtain:

3)

Clearly the answer needed here is cance the -3?

or

-3

but what is the signifi-

5.

one.

OUNCES

solution of this problem is similar to that of th last

Let the weight the ball e ounces. Th equation obtained is another quadratic thus: 2x +

(x-2)(x

..

4)

or

-4

Math Medley 1. CURSOR part of mathematical instrument, cursor is defined which slides backwards an forwards. Newton suggested cursor or runner should be used on slide rule bu his idea was no taken up for hundred years. It is obvious when one is trying to read the graduations on two different scales that some convenient straight line is necessary, an th cursor is no an essential part of any slide rule.

2. MUSICAL SCALE From ou childhood we are accustomed to "do, re, mi fa, sol, la, ti do," which is known the diatonic scale. Other scales are to be found in Turkish an Persian music! he frequencies of th eight notes of diatonic scale are in th following ratio: 4 24

27 30

2, which is th 36: 40 45

same as

giving th

notes

16.13 INCHES

In 60 minutes the tip of the minute hand of the clock makes one complete revolution tracing ou the circumference of circle whose radius is inches. 59

In

minutes the tip moves through

7r inches

or

2)X

inches

In 22 minutes the tip moves through -2

60

inches

or

16.13 inches

CYCLOID the class of curves known This is the simplest member not known before the fifteenth century and roulettes. It many not seriously studied until the seventeenth century. brilliant mathematicians like Descartes, Pascal, Leibnitz, the the Bernoullis, and others have investigated the properties sometimes named the "Helen of Geomcycloid that it eters." It is matter opinion whether the curve, like Troy, is surpassing beauty. Helen

5. DESARGUES' ANSWER

OA, OB, and OC re three concurrent straight lines with and Z any points each respectively. Join CA and Similarly, join and and produce them to meet in to meet in S, and

and ZY to meet in

T. G6rard Desargues,

seventeenth-cen-

tury French engineer, T, and proved that

always lie straight line. Hence, if plant your ten tu lips in the positions the ten points A,

Y,

they will

rows 60

three.

C,

in ten

5I

6. ABOUT THREE INCHES Th

needle moves from the outermost groove to the inner-

most groove in an arc whose radius is th pickup arm.

length of th

7.

PHILLIPS Mr. Henry will earn during the first year $3,000. r. Phillips will earn during the first year $1,500 $300 $3,300. Mr. Henry will earn during the second year $3,000 $3,600. Mr. Phillips will earn during th second year $1,800 $2,100 $4,500. $300 Hence it is clear that Mr. Phillips will earn

$1,500 $600 $300

$1,200 more than Mr. Henry the time they have both worked tw years. the years go Mr. Phillips will earn in creasingly more than r. Henry.

Gets the Worst

It

The two relevant equations are: and

From these we derive that 3C =-

or

15

20 runs 1,760 yards while

runs 1,740 yards

1,720 yards. runs 1,740 yards while

runs 1,720 yards

runs

runs 1,760 yards while

beats

3.

yards in

by

runs

'7

1,760 1,740

or 1,739g7 yards mile

DAYS

(A + B) do of the work in one day do of the work in day (A -2j' of the work in day (B these all together and we have: -j of the or (A C) the work in ne

or

yards yad

work

in one day

-lo th work in on ay (A th work in on -1-) alone does alone does 6-I of work in can do all the work in sixty days

18

scores 50 while scores 50 while scores

scores scores 40

while

scores or

scores 50 while scores give 18 points to

50

32 and while

scores 32

5. $500

The profit should shared in the following proportions between A, B, and C: 1,875: 1,500: 1,250 or 10

or

15

12

10

37

37

37

's share of the profit should

37

1,850 or

$500

HOURS In fills fills hour and with all pipes working (' hour tank is filled in

1"

with al th pipes working in

(30)

empties 3-

of th

of the tank. or ("-) of the

tank is filled

31 hours. (191

Letters

Numerals

1. X = 9 , Y = 1 , Z = 8 which Examine the units column. 10. Examine the tens column. means that + Z+ the units column) (from lO 9. Examine the ten-thousands which means that column and the equation obtained is (from the thousands column) lOY Substitute 9, and the equation then becomes: 10 lOY, or = 1. From this it follows that 8, and 9. 9999

1111

8888 19998

2. X = 5 , N = 2 , P = l , S = O , R = 6 , a n d Z = 3 Examine the second ro multiplication times .'. must 1, #& 6. gives another 0, 5, because the product is not XXX. #& because the product #& because there is is not PNX. product NX NX

which will give

375. multiplication

.'-

N N.

5, because 25 x 5 12 or Examine the first row of 2 or #5 because the product is not

-.

2. The remainder of th letters follow fairly easily from this stage. four

-.

unknown

125

25

25

3125

3. P = 3 , H = 1 , 1 = 2 , L = 5 , T = 6 , a n d S = 0 .

From the first division it is clear that = 1. From the third division L times L gives another in the units place. *must 5, 1, #A because the product is not 0, 6. LL. #Z because the product is not IL, and in any case because whatever value is given to = 1. between and 9, the product times I6 will not 1I6. -. 5. HIL reads 115. Examine the third division again. times I5