.-'
~1ATHEMATICAL
TEACHER:
PUZZLES FOR THE SECONDARY
MATHE~1ATICS
A COLLECTION, CLASSIFICATION,
AND EVALUATION
A Thesis Presented to the Graduate Faculty
of
California State University, Hayward
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Education
By
Randall R. Baumback
\
November,\ 1980
Copyright
@
1980
By Randall R. Baumback
ii
ABSTRACT
The purpose of this study is to provide secondary
mathematics teachers with a collection of classified
mathematical puzzles which when used in conjunction with
the appropriate level and concept in the mathematics
curricula, will serve to stimulate, illustrate, and
enrich the desired objectives of study.
As a basis for this study, three assumptions were
made:
(1) attitudes are fundamental to, and contribute
to what students learn,
(2) mathematical puzzles intrigue
and motivate students, and (3) the use of puzzles as an
integral part of the mathematics instruction is an
effective way to teach concepts and skills.
Teachers recognize that students' mathematical
achievement and interest has suffered, and that
recreational materials provide a welcomed relief from
the rigorous mathematics curricula.
However, a brief
review of the literature will show that it does not
provide the secondary mathematics teacher with a practical,
effective classroom resource.
From the available literature on mathematics
recreations, the author made a final selection, edition,
and classification of the 117 puzzles in Chapter 3
iii
according to four criteria:
standability,
(1) clarity and under-
(2) time factor practicality,
(3) motiva-
tional value, and (4) illustration of a mathematical
concept.
An evaluation of the puzzles was made by five
secondary mathematics teachers.
Each teacher rated each
puzzle with respect to the four criteria on a four-point
scale.
An analysis of the data followed revealing that
nearly all of the puzzles were rated very highly.
Hence,
the puzzle collection was judged to be an excellent
resource for secondary mathematics teachers by those
teachers who served on the evaluation panel.
iv
Y..)._THEl1ATICAL ?GZZI.ES FOR THE SECONDARY :1J:._THE:1ATICS
TEACHER:
A COLLECTION, CLASSIFICATION
AND EVALUATION
3y
?,andall ?. 3a.umbac;·:
APPROVED:
DATE:
Lk'.2. 7:
>
1'180
~e~:f::(/(~ 9 ///0
/
v
ACKNOWLEDGMENTS
I am indebted to Dr. John Hancock for his
immeasurable guidance in developing this work, and which
is in large part a reflection of his professional concern
for the student's and teacher's needs.
Also, I wish to thank Dr. Albert Lepore for his
crucial assistance, and Dr. Gerald Brown for his
invaluable references.
A special thanks to Cathy Vulpi,
Vicki Reese, Bill Oliver, Rich Cordes, and Janice Carter
for their participation, and a word of appreciation is
due the James Logan High School Mathematics Department for
their cooperation in this study.
I
am most grateful to my Mother and Father whose
love, and encouragement has enabled me the fulfillment of
undertakings such as this one.
vi
TABLE OF CONTENTS
Page
ABSTRACT .
.
.
.
iii
ACKNOWLEDGMENTS
vi
LIST OF TABLES .
xv
Chapter
1.
INTRODUCTION
1
PURPOSE OF THE STUDY .
1
BACKGROUND OF THE STUDY
1
Inaccessibility of Appropriate Puzzle
Literature . . . .
. . . ..
ASSUMPTIONS
3
REVIEW OF THE LITERATURE .
2.
2
.
4
Use of Mathematical Puzzles to
Illustrate a Concept . . . .
4
Use of Mathematical Puzzles to
Stimulate Student Interest
11
METHOD OF STUDY
15
PROCEDURE FOR DEVELOPING PUZZLE COLLECTION
DISCUSSION OF CRITERIA .
. . .'.
15
16
Clarity and Understandability
16
Time Factor Practicality
16
Motivational Value.
17
.
.
Illustration of a Mathematical Concept.
vii
17
TABLE OF CONTENTS (Continued)
Chapter
Page
DESCRIPTION OF POPULATION
INSTRUMENTATION .
. .
.
...
.
. . . .
18
PROCEDURE FOR COLLECTING DATA
3.
MATHEMATICAL PUZZLES
.
CONCEPTS COVERED
. ..
. .
ARITHMETIC PUZZLES
.
17
.
18
.
20
.
20
. .
24
1.
A Snail in the Well . . . .
24
2.
A Tennis Tournament.
25
3.
Ten Million
4.
How Many Squares?
25
5.
The Bookworm
26
6.
Squares.
7.
A Cake of Soap
8.
The Puzzled Driver
9.
Thirty Flasks.
. ..
Bacter~a
25
...
26
...
27
...
27
.
. ..
28
10.
How Many Sandals?
....
28
11.
From 1 to 1,000,000,000
12.
Grandfather's Faded Bill
13.
A Pile of Paper . .
29
14.
The Deleted Checkerboard
30
15.
Twenty-One Pigs
30
16.
Me Pay You? . .
30
17.
Ten Vacant Rooms
31
viii
28
....
29
TABLE OF CONTENTS (Continued)
Chapter
Page
18.
Three Men in a Hotel . .
32
19.
Guess an Age . .
33
20.
A Trick with Three Dice
34
21.
A Right Answer . . .
34
22.
A Phonograph Record
35
Solutions to Arithmetic Puzzles .
ALGEBRA PUZZLES .
.
.
.
.
35
42
23.
De Morgan's Birthday.
42
24.
Chain Letter . .
42
25.
If Half of Five is Three
43
26.
A Bottle and a Cork
43
27.
Brothers and Sisters.
28.
How Much does the Brick Weigh? .
43
29.
The Annual Pioneer Pancake Eating
Contest
. . . . . . .
43
30.
Algebra Homework . .
44
31.
How Many Students Did Mr. Einstein
Have?
. . . . .
. ..
44
.
.
.
. . . .
43
32.
A Basket of Eggs
33.
Jack and Jill
45
34.
Phil Anthrope
45
35.
More or Less?
. . . .
45
36.
Cinch Around the Earth
46
ix
44
TABLE OF CONTENTS (Continued)
Chapter
Page
37.
Counting Sheep
46
38.
Find the Flaw
46
39.
Is 4
40.
What is the Error? .
47
4l.
The Lucas Problem
47
42.
A Mountain Climber's Journey.
48
43.
A Card Trick . .
48
44.
Crossing Tiles
49
45.
The Refreshed Runner .
49
=
8?
47
.
Solutions to Algebra Puzzles
GEOMETRY PUZZLES
.
.
50
.
61
46.
Finding a Proof
47.
Irish Friends
48.
The Game of "Stogey"
49.
Toe-Tac-Tic
50.
Medians of a Triangle
51.
A Paradox
52.
What is the Length?
64
53.
To Find the Center of a Circle.
64
54.
The Geometry Club's Badge
65
55.
Remaining Metal
66
56.
Yang, Ying and Yung
66
61
.....
61
62
.
62
.
x
62
63
TABLE OF CONTENTS (Continued)
Chapter
Page
57.
A Grazing Goat
67
58.
Separating the Sheep .
67
59.
Koch's Triangles . .
68
60.
Overlapping Figures
68
61.
A Square Peg in a Round Hole .
69
62.
The Carpenter's Rope . .
69
63.
The Spider and the Fly
69
64.
The Hunter's Dilemma.
70
65.
The Fly and Honey
70
66.
A Familiar Object
71
67.
A Bottle's Volume
71
68.
A Curious Sphere .
69.
A Geometrical "Vanish"
72
70.
What's My Angle? .
73
71.
Planting Trees .
73
.
Solutions to Geometry Puzzles .
INTERMEDIATE ALGEBRA PUZZLES
72
73
88
72.
Three Daughters
88
73.
How Many Pages?
88
74.
The Evasive Engineer .
88
75.
Watches
89
76.
Two Candles
89
xi
TABLE OF CONTENTS (Continued)
Chapter
Page
77.
Two Horses and a Fly . .
89
78.
Draining a Water Tank
90
79.
The Physicist and the Escalator
90
80.
Average Speed
90
8l.
The Confused Teller
91
82.
How Long was the Vacation? .
91
83.
The Restaurant Bill
91
84.
Scales . .
92
85.
Bank Shot
92
86.
An Infinitude of Twos
93
87.
The Golden Ratio . .
93
88.
A Swarm of Bees
89.
An Algebra Error . .
94
90.
An Elephant and a Mosquito
95
9l.
Is
92.
Explain the Paradox
96
93.
Another Paradox
96
. .
1 = -l?
.
93
96
Solutions to Intermediate Algebra Puzzles
ADVANCED MATHEMATICS PUZZLES
97
107
94. Two Trees with the Same Number of
Leaves
. . .
107
95.
108
Slicing a Cube.
xii
TABLE OF CONTENTS (Continued)
Chapter
Page
96.
Tower of Hanoi . . .
108
97. A Deceitful Proof by Mathematical
Induction . .
. . .
109
98.
Glasses
110
99.
Faulty Scale.
110
100.
Time to Trisect any Angle
110
101.
The Triangle is Equilateral
110
102.
Four Bugs
110
103.
A House Number
III
104.
A Calculus Paradox.
111
105.
How Many Handshakes? .
112
106.
Triangles
112
107.
Thirty-one Flavors.
112
108.
How Many Routes? . .
113
109.
The Prize Contest
113
110.
A Fast Deal
114
111.
Four Letters
114
112.
The Pentagon Building
115
113.
The Same Birthday
115
114.
Three Darts
115
115.
Professor of Ancient History
115
116.
Are You Certain? . .
116
xiii
TABLE OF CONTENTS (Continued)
Chapter
Page
117.
A Chance for Survival
.
Solutions to Advanced Mathematics Puzzles
PUZZLE SOURCES .
.
. .
RESULTS
5.
ANALYSIS, SUMMARY, AND RECOMMENDATIONS
.
130
ANALYSIS OF EVALUATIONS
. .
.
. .
. .
. .
.
. .
139
139
.
140
RECOMMENDATIONS
BIBLIOGRAPHY
116
128
4.
SUMMARY
116
.
141
.
142
APPENDICES
I .
II.
DATA SHEET .
.
. .
.
.
. .
.
. .
.
RATING FORM AND PUZZLE EVALUATIONS
xiv
.
. .
148
152
LIST OF TABLES
Table
1.
2.
3.
Page
Distribution of Ratings of Criteria for
Puzzles . . . . . . .
154
Puzzle Ratings:
Respondents .
132
Summarized Ratings by
Summary of Ratings of Puzzles . .
xv
138
Chapter 1
INTRODUCTION
PURPOSE OF THE STUDY
The purpose of this study is to provide secondary
mathematics teachers with a collection of classified
mathematical puzzles which, when used in conjunction with
the appropriate level and concept in the mathematics
curricula, will serve to stimulate, illustrate, and
enrich the desired objectives of study.
BACKGROUND OF THE STUDY
There are several strategies available for
teaching most secondary mathematical
concepts~
however,
rarely is an amusing mathematical puzzle used to stimulate
the interest or attention required to demonstrate the
desired topic.
Although most experienced mathematics
teachers know a few such puzzles, well organized collections
of mathematical puzzles are scarce at this level.
At the
same time, teachers recognize that student achievement
and interest has suffered froITl the most abstract of all
the secondary subjects, and that recreational materials
provide a welcomed relief from the rigor of the
1
2
mathematics curricula.
This study is designed to partially
free that gap--to promote both learning and enthusiasm for
students and teachers alike of secondary mathematics.
Inaccessibility of Appropriate
Puz £IeLT"terature---The pleasure and stimulation found in grappling
with mathematical puzzles is not new, attested by the many
books in the field.
However, a brief review of the
literature will show that it does not provide the secondary
mathematics teacher with a practical, effective classroom
resource.
In general, the available books suffer from one
or more serious pedagogical defects.
Many books offer
a collection of puzzles with no deliberate classification
or continuity.
Others classify according to difficulty
level but not concept.
In nearly all cases the teacher
must spend a disproportionate amount of time searching
through unsuitable material in order to collect a few
appropriate puzzles.
Frequently, such books pose puzzles
with no or partial mathematical analysis accompanying the
answers.
Unfortunately, the occasional book that escapes
these defects is narrow in scope, and cannot be used
throughout the secondary mathematics curricula.
Given
the importance of such available materials, a need clearly
exists for a more detailed resource of puzzles.
The
3
collection of mathematical puzzles herein should meet this
purpose.
ASSUMPTIONS
In conducting the study, the author has proceeded
on the following assumptions:
1.
Attitudes are fundamental to, and contribute
to what students learn.
There is little debate among
educators that a positive attitude is highly correlated
with meaningful learning.
However obvious this assumption
appears, it is necessarily stated here since one of the
purposes of this study is to stimulate student interest.
2.
students.
Mathematical puzzles intrigue, and motivate
Although some teachers say that students do not
respond to challenging puzzles, it is the author's belief
that this is due to the circumstance in which the puzzles
are presented.
Fear of failure is commonplace.
And the
pressure of solving a puzzle in the presence of a group
may be a fear-producing situation.
However, when allowed
to attempt appropriate puzzles under their own conditions,
students are often consumed by, and appreciate them.
3.
The use of puzzles as an integral part of the
mathematics instruction is an effective way to teach
concepts and skills.
It is assumed that puzzles provide
4
pleasant, and usually successful experiences which result
in efficient learning.
REVIEW OF 'I'HE LITERATURE
Use of Mathematical Puzzles
to Illustrate a Concept
The introduction of recreational materials such as
puzzles and games into the mathematics classroom is
frequently discussed in the professional journals, and is
often proposed as a solution to combat declining mathematics
achievement and student apathy.
As early as 1923 Simmons quotes Longfellow as
saying:
There is something devine in the science of
numbers. Like God, it holds the sea in the
hallow of its hand.
It measures the earth;
it weighs the stars; it illumines the universe;
it is law, it is order, it is beauty, and yet
we imagine--that is, most of us--that its
highest end and culminating point is bookkeeping
by double entry.
It is our own way of teaching
that makes it prosaic. l
This same attitude, fifty years later, is still
reflected in the literature.
Bradfield, in 1970, as he
proposes to enliven the mathematics classroom by providing
lLao G. Simmons, "The Place of the History of
Mathematics in Teaching Algebra and Geometry," The
Mathematics Teacher, XVI (January, 1923), 94-l0~
5
enrichment problems, suggests that "a teacher is limited
only by his own ingenuity and creativity.,,2
The use of mathematical recreation in the classroom to develop mathematical skills has been suggested by
many writers.
Johnson and Rising claim that mathematical
puzzles introduced at the appropriate time in the
curricula not only play an important role in building
positive attitudes but also aid as a device in the
cognitive process.
3
Taking this claim a step further,
Allen, Jackson, Ross, and Hhite explored the extent to
which the effectiveness of instructional recreation has in
facilitating the learning of specific mathematical ideas.
They found that it is a highly effective tool for
enhancing both motivation and achievement in the learning
of mathematics.
Furthermore, they assert that mathematical
recreations "should be used more, and they deserve to be
studied more.,,4
2Donald L. Bradfield, "Sparking Interest in the
Hathematics Classroom," The Arithmetic Teacher, XVII
(Harch, 1970), 239-242.
3Donavan A. Johnson and Gerald Rising, Guidelines
for Teaching Mathematics, 2nd ed. (Belmont: Wadsworth
Publishing Co., Inc., 1972), p. 265.
4Layman E. Allen, Gloria JacKson, Joan Ross and
Stuart White, "what Counts is How the Game is Scored One
Way to Increase Achievement in Learning Hathematics,"
Simulation and Games, IV (December, 1978), 371-389.
6
'I'he connection between puzzle strategies and
mathematical concepts is illustrated by Steen, who
suggests that "mathematical games offer more than
fun~'-'
they can provide insight into mathematical theory."
He
claims that both the mathematician and the puzzlist rely
on hypothetical reasoning to develop sound strategy.
As
evidence, he cites that Conway of Cambridge University
uses puzzles and games for understanding the concept of
numbers.
S
Polya, in his book, How to Solve It, feels that
challenging problems that brings into play inventive
faculties, when solved, such experiences "at a susceptible
age may create a taste for mental work and leave their
imprint on mind and character for a lifetime."
He believes
that drilling students with routine operations "hampers
their intellectual development," and that curious problems
at the appropriate intellectual level may give the
student "a taste for, and some means of, independent
thinking. ,,6
evidence.
Rutherford supports Polya's suggestions with
He reports on three experiments carried out
S
Lynn Arthur Steen, "What's in a Game?," Science
News, CXII (March, 1978), 204-206.
6
G. Polya, How to Solve It (Princeton:
University Press, 1973), p. v.
Princeton
7
independently.
The purpose of each investigationuwas to
answer the question of whether spending weekly class time
on mathematical puzzles has an advantageous effect on
achievement in mathematics.
In all three studies, achieve-
ment and attitude were improved through the use of puzzles.
Results of a year-long study in a Detroit innercity school indicate that mathematics classes using
instructional recreation tournaments have more peer
tutoring, and that students perceive the class as
significantly more satisfying and less difficult than
students in control classes.
8
Not surprisingly, in the
same series of studies there were also significantly
greater gains in mathematics achievement, as measured by
the Stanford Achievement Test.
9
7porter B. Rutherford, "The Effects of Recreations
in the Teaching of Mathematics," School Review, XLVI
(June, 1938), 423-427.
8D. L. DeVries and K. J. Edwards, "Student Terms
and Instructional Games: Their Effects on Cross-Race and
Cross-Sex Interaction" (research report on effects in the
classroom in John Hopkins University Center for Social
Organization, Baltimore), report no. 137, 1973, cited
by Layman E. Allen and others, "What Counts is How the
Game is Scored One Way to Increase Achievement in Learning
Mathematics," Simulation and Games, IV (December, 1978),
371-389.
9 K . J. Edwards, D. L. DeVries and J. P. Snyder,
"Garnes and Teams: A Winning Combination," Simulation and
Games, III (September, 1972), 247-269.
7
8
In the judgment of authors Hollingsworth and
Dean, puzzles are a clever technique for building
mathematical skills.
They found drill in the guise of
challenging puzzles to be an important by--product of the
puzzle solving process.
It is their experience that
"students at many different grade levels find the puzzles
fun, challenging, and rewarding_"lO
The value of
recreational mathematics has long been recognized by
teachers as well.
Brandes affirms that the variety and
inherent interest in recreational mathematics items "can
be utilized to improve pupil attitudes and to make
learning more effective."ll
The significance of puzzles
in the mathematics curricula goes without question according
to Professors Chein and Averback of Temple University.
They support this claim by offering a four-credit
undergraduate course called "Math Recreations".
Students
taking this class study unusual mathematical puzzles,
games, and problems.
They feel that "making math enjoyable
will help students overcome the fear of math that often
lOCaroline Hollingsworth and Eleanor Dean,
"Factoring Puzzles," The Hathematics Teacher, LXVII (Hay,
1975), 428-429.
llLouis Grant Brandes, "Using Recreational Mathematics in the Classroom," The Mathematics Teacher, XLVI
(April, 1953), 326-329.
9
remains with them throughout their lives.
1i
Another
rationale for the course comes from their belief that
the discipline required to solve the puzzles will
"facilitate problem solving in all fields."
Furthermore,
they both agree that "the methods and philosophy of
teaching used in this course definitely could be adapted
to teaching mathematics at the K-12 level.,,12
Bellman, Cooke, and Lockett, authors of
Algorithms, Graphs, and Computers, discuss the value of
mathematical recreation in their preface, "the principal
medium we have chosen to achieve our goals is the
mathematical puzzle.,,13
Interest in puzzles in the teaching of mathematics
has increased in recent years.
This
~nterest
is
evidenced by the growing number of presentations on the
subject at meetings of professional organizations of
mathematics teachers.
The reason for this birth of
enthusiasm, according to Smith and Eackrnan, is that
mathematical puzzles are viewed to serve particular
purposes.
II
They cite four instructional functions inherent
12"puzzling Problems," Nation's Schools and Colleges,
(Hay, 1 97 5), 4 9 .
13Richard Bellman, Kenneth L. Cooke and Jo Ann
Lockett, Algorithms, Graphs, and Computers (New York:
Academic Press, 1970), pp. vi-vii.
10
in puzzles.
These are:
the development of concepts and
perceptual abilities, provision for drill and reinforcement,
opportunities for logical thinking, and problem solving.
14
A parallel view of the usefulness of recreational
materials is delineated by Hoffman.
She feels strongly
that mathematical recreations are an excellent component
in the individualization of instruction.
Although she
emphasizes that mathematical recreations "can contribute
to mathematical learning," she cautions that teachers
should select materials carefully.
Their use should be
well planned, and keyed to the concept being taught.
15
Essentially with the intent to teach the basic
principles of arithmetic, Hurwitz, Goodard, and Epstein
wrote their book, Number Games to Improve Your Child's
Arithmetic.
They assert that their thoroughly tested set
of puzzles and games serve a greater purpose than
immediate entertainment for laying the foundations of
mathematics but they "help to develop a lifelong love of
the subject."16
l4Seaton E. Smith, Jr. and Carl A. Backman, eds.,
Games and Puzzles for Elemen~ary and Middle School Mathematics (The National Council of Teachers of Mathematics,
1975), pp. 1- 2.
15 RUth 1. Hoffman, "Mathematics: Learning Through
Games," Instructor, LXXXIV (October, 1974), 69-70.
16Abraham B. Hurwitz, Arthur Goddard and David T.
Epstein, Number Games to Improve Your Child's Arithmetic
(New York:
Funk and Wagnals, 1975), pp. 2-6.
11
Use of Mathematical Puzzles to
Stimulate Student Interest
Of course, learning mathematics is not all play,
and no student should bypass the hard work of the subject.
But much good mathematics can be learned from enjoyable
recreations.
Such outstanding mathematicians as Gauss,
Liebnitz, and Euler found many new ideas and even fields
of mathematics through such pastimes.
Two examples of
such topics growing out of recreation are probability
theory and game theory, both of which are used extensively
today.
The recognition by educators that motivated
students learn best and the pedagogical value of
recreational mathematics have been established.
Gardner
addresses the problem of boredom in the mathematics
classroom by claiming that "there is no better way to
relieve the tedium than by injecting recreational topics
into a course, topics strongly tinged with elements of play,
humor, beauty, and surprise."
More specifically Gardner
asks, "What is mathematics, after all, except the solving
of puzzleS?,,17
Using puzzles to entertain many mathematical
17Martin Gardner, Mathematical Puzzled and
Diversions (New York: Simon and Schuster, Inc., 1959),
pp. ix-xi.
12
subj ects,
~'lhi t.e
makes this same point.
He uniquely argues
that "amusement is one of the fields of applied mathematics. fl18
Teac~!ng
In The
of Secondary
Mat~ematics,
by
Butler, Wren, and Banks, the authors write as follows:
It may be taken as axiomatic that students
will work most diligently and most effectively
at tasks in which they are genuinely interested.
To create and maintain interest becomes, therefore, one of the most important tasks of the
teacher of secondary school mathematics.
It
is also one of the most difficult problems the
teacher encounters.
They go on to point out that students do not actually hate
mathematics.
Rather, they hate the drudgery, boredom, and
frustration.
The use of mathematical recreations is
offered as a specific suggestion for motivating the
student.
19
Sobel expresses similar sentiments.
Realizing
the correlation between student interest and achievement,
he claims that "mathematical garnes, puzzles, and tricks
can almost always be counted on to generate excitement
in a class.,,20
l8williarn F. White, A Scrap-Book of Elementa~
Mathematics (Chicago: The Open Court Publishing Co., 1908),
p. 7.
19Charles H. Butler, F. Lynwood Wren and J. Houston
Banks, The Teaching of Secondary Mathematics (New York:
McGraw-Hill Book Co., 1970), p. 118.
20Max A. Sobel, "Junior High School Mathematics:
Motivation vs. Monotony," The Mathematics Teacher, LXVIII
(October, 1975), 479-485.
13
Professors Metzner and Sharp are also concerned
with functions puzzles and games serve in the classroom.
They present six recreations relevant to the learning and
retention of basic mathematical concepts.
They think that
puzzles and games influence students and "add a certain
'pizzaz' to the mathematics program.,,2l
Additional support
for the idea of using mathematical puzzles, games and
riddles is offered by Dohler, who illustrates her ideas
with sample activities.
According to Dohler, such
activities need not take up much class time but are an
excellent way to begin or end a class period.
22
Similarly,
Nies reminds teachers not to overlook the role recreational
devices play in the improvement in students' mathematical
skills and attitudes.
She goes as far as to suggest
their use not be limited to classroom time but also
between periods and other times of the school day.23
2lseymour Metzner and Richard M. Sharp,
:Cardematics I--Using Playing Cards as Reinforcers and
Hotivators in Basic Operations," The Arithmetic Teacher,
XXI (May, 1974), 419-421.
22
Dora Dohler, "The Role of Garnes, Puzzles, and
Riddles in Elementary Mathematics," The Arithmetic Teache:r:.,
X (November, 1963), 450-452.
23 RUth H. Nies, "Classroom Experiences with
Recreational Arithmeti:::::," The Arithmetic Teacher, III
(Ap r i 1, 1956), 90- 9 3 .
14
Kerr provides a refreshing discussion about the
involvement of students with a mathematical recreation
that resulted in some excellent analytic thinking.
He
asserts that mathematical puzzles and games in school
are important because, in using them, "children have fun
associated with a topic that is not always considered to
be enjoyable."
Although it is not possible or desirable
to organize the mathematics curricula solely around
puzzles, Kerr believes that they "complement the regular
mathematics instruction and thereby justify more classroom
t
'
,,24
lme.
24Donald R. Kerr, Jr., "!-1athematics Games in the
Classroom," The Arithmetic Teacher, XXI (Harch, 1974),
172-175.
Chapter 2
METHOD OF STUDY
PROCEDURE FOR DEVELOPING PUZZLE COLLECTION
From available literature on mathematical
recreations, the author mathematically analyzed approximately 3,000 puzzles, and made a selection according to
the criteria discussed in the next section.
From this
examination, 123 puzzles were judged to meet the criteria.
These were then edited and recorded on 5 x 8-inch file
cards for further analysis and evaluation.
A classification of these puzzles with respect to
content area and grade level was undertaken by the author
and a team of nine high school mathematics teachers.
The
nine teachers were divided into three groups of three
each.
Each group examined approximately one-third of the
puzzles, and recorded their analyses on data sheets.
sample data sheet, Appendix I, page 150.)
(See
From this
information puzzles were assigned to one of five content
areas:
Arithmetic, Algebra, Geometry, Intermediate
Algebra, and Advanced Mathematics.
Within each content
area the puzzles were clustered according to concept,
and ordered based on a combination of two factors:
15
(i) from
16
least difficult to most difficult, and (ii) by concept
hierarchyly in the mathematics curriculum.
A final selection of 117 puzzles was made by the
author and five other teachers.
These puzzles are
presented in Chapter 3.
DISCUSSION OF CRITERIA
In evaluating whether or not a given puzzle was to
be included in this collection, four selected criteria
were applied.
These criteria evolved from previous
experience, and were further developed to be the prime
instrument used in selecting the most appropriate puzzles
of those available.
Clarity and Understandability
Essential in successful problem solving is clearly
defining the problem before attempting a solution.
Thus,
the first criterion established whether or not the puzzle
was clear and understandable at the level which they would
be used.
Time Factor Practicality
Since this study was prepared for use in the
secondary school classroom, the many curricular and
administrative time consuming demands burdening today's
teacher had to be considered.
Also, because of the problem
17
of sustaining student attention, acceptable puzzles had to
be restricted in length.
As an initial screening process
all puzzles and solutions had to be able to be written on
a single 5 x 8-inch file card in order to be considered
for evaluation.
Motivational Value
Puzzles containing intrigue, amusement, humor,
and challenge were sought since tllese are the eloments
which seem to divert and motivate students toward learning.
Mathematics teaching often lacks these features, suggesting
one reason interest in the subject has frequently proven
difficult to generate.
Illustration of a Mathematical Concept
The previous criteria have dealt with nonmathematical characteristics, however the puzzles must
instruct as well as entertain.
Hence, prime consideration
was given to the puzzle's ability to illustrate a
mathematical concept.
Therefore, each puzzle had to meet
unconditionally this criterion in order to be included
in the collection.
DESCRIPTION OF POPULATION
Five high school mathematics teachers, known by
the author, were asked to evaluate the puzzles.
All five
18
respondents had taught mathematics in the public schools
at both the junior high and senior high school levels.
Four of the five teachers hold bachelor degrees in
mathematics.
The other teacher received his undergraduate
degree in aeronautical engineering.
At the time of this
study, all five respondents were teaching in the San
Francisco Bay Area.
INSTRUMENTATION
A rating form was designed to register the
respondents' evaluation for each puzzle.
rating form, Appendix II, page 153.)
(See sample
The teachers were
instructed to use their professional judgment to rate each
puzzle based on how well it met the following criteria:
l.
possession of clarity and understandability;
2.
possession of a time factor practicality;
3.
possession of motivational value; and
4.
illustration of a mathematical concept.
The degree to which the puzzle met the criteria
was to be based on a four-point scale:
excellent, good,
fair, or poor.
PROCEDURE FOR COLLECTING DATA
Each respondent was individually contacted in
person.
At the time of the conference, the purpose of the
19
study was summarized.
The respondent was given a rating
form, and the criteria for evaluation of the puzzles were
read aloud by the author.
A brief discussion of the
criteria, as stated in this chapter, was then presented
for further clarification.
The amount of time required to
read and rate t,he puzzles by each respondent was
approximately twelve hours.
Chapter 3
MATHE~mTICAL
PUZZLES
From the available literature on mathematical
recreations, the author made a final selection (based on
the four criteria discussed in Chapter 2), of 117
mathematical puzzles, and classified them according to
five content areas:
Arithmetic, Algebra, Geometry,
Intermediate Algebra, and Advanced Mathematics.
Within
each content area the puzzles were clustered according to
concept, and ordered based on a combination of two factors:
(i) from least difficult to most difficult, and (ii) by
concept hierarchyly in the mathematics curriculum.
This
chapter contains the collection of classified mathematical
puzzles.
CONCEPTS COVERED
Due to the cumulative nature of the mathematics
discipline, many of the puzzles contained in this chapter
illustrate more than one concept.
As a result, some of
the puzzles classified under one content area may also be
appropriate under another content area depending on the
depth and emphasis of the desired concept.
20
21
The following list is presented here to act as a
guideline for selecting a puzzle to introduce or review
a particular mathematical concept or skill.
Puzzle No.
Concept
Arithmetic
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Adding integers
One to one correspondence
Deductive reasoning--doubling a number
Squaring a number
Addition of fractions with like denominators
Addition of fractions with unlike denominators
Multiplication of fractions
Deductive reasoning--distance = rate x time
Deductive reasoning--classifying information
Average of numbers--deductive reasoning
Discovering patterns
Divisibility by 8 and 9
Powers of 2
Deductive reasoning--one to one correspondence
Sets of numbers--subsets of numbers
Deductive reasoning--whole number subtraction
and multiplication
One to one correspondence
Deductive reasoning--adding integers
Binary numeration
Deductive reasoning--addition of whole numbers
Whole number addition with regrouping-deductive reasoning
Diameter of a circle
Algebra
23.
24.
25.
26.
27.
28.
29.
30.
31.
Square root
Exponents
Solving a proportion
Naming a variable--solving
Naming a variable--solving
Naming a variable--solving
fractional equation
Naming a variable--solving
Naming a variable--solving
equation
Naming a variable--solving
equation
a linear equation
a linear equation
a linear
a linear equation
a linear decimal
a linear fractional
22
Puzzle No.
Algebra
32.
33.
34.
35.
36.
37.
38.
39.
40.
4l.
42.
43.
44.
45.
Naming a variable--solving a linear fractional
equation
Naming a variable--distance = rate x time
Naming a variable--solving a linear decimal
equation
Naming a variable--solving a linear decimal
equation
Naming a variable--circumference and radius
of a circle--substitution property
Naming a variable--solving simultaneous linear
equations
Field properties--division by zero
Principal and negative values of square roots
Operations with units of measure
Graphing--one to one correspondence
Graphing--ordered pairs
Naming variables--simplifying a polynomial
expression
Greatest common factor
Radius and circumference of a circle--formulas
Geometry
46.
47.
48.
49.
50.
5l.
52.
53.
54.
55.
56.
57.
58.
59.
60.
6l.
62.
63.
64.
65.
66.
Proof--equilateral triangles
Indirect proof
Reflection through a point
Reflection through a point
Triangle inequality theorem--auxiliary lines
Area of a parallelogram--conservation of area
Diagonals of a rectangle
Right angle inscribed in a semicircle
30-60-90 triangle and 45-45-90 triangle
Area of a circle
Area of a circle
Radius and area of a circle
Pythagorean theorem
Perimeter of a polygon--conjecturing a formula
Area of overlapping figures
Pythagorean theorem
3-4-5 Right triangle
Pythagorean theorem (3 dimensional)
Diagonal of a cube
Pythagorean theorem (3 dimensional)
Visualization of solids
23
Puzzle No.
Geometry
67.
68.
69.
70.
71.
Volume of a solid with circular or
recta.ngular base
Surface a.rea and volume of a sphere
Deductive reasoning--conservation of length
Solving angle measures--auxiliary lines
Collinear points and concurrent lines
(Desargue's theorem)
Intermediate
Algebra
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
Deductive reasoning--factors--combinations
Estimation--place value--solving a linear
equation
Naming a variable--solving a linear equation
Naming a variable--solving a linear fractional
equation
Relating two variables--solving a proportion
Distance = rate x time
Nanling a variable--solving a linear fractional
equation
Distance = rate x time--solving simultaneous
linear equations
Distance = rate x time
Naming a variable--solving linear equations
for integral values
Naming variables--solving simultaneously
several linear equations
Naming variables--solving a quadratic equation
by factoring
Naming variables--solving simultaneously
several linear equations
Finding an equation of a line given 2 points-solving intersection point of 2 lines
Naming a variable--solving a quadratic
equation by factoring
Naming a variable--solving a quadratic
equation by formula
Naming a variable--solving a quadratic
fractional equation by factoring
Property of equal fractions having equal
nonzero numerators
Principal and negative values of square roots
Squaring the imaginary number i
Squaring the imaginary number i
loglOa = x
24
Puzzle No.
Advanced
Mathematics
94.
95.
96.
97.
98.
99.
100.
10l.
102.
103.
104.
105.
106.
107.
108.
109.
110.
Ill.
112.
113.
114.
115.
116.
117.
Generalizing from a pattern
Conjecture of a formula--induction
Conjecture of a formula--induction
Principle of mathematical induction
(-x)y
Geometric mean--solving equations by
substitution
Sum of infinite geometric series
Proof by algebraic factoring
Logarithmic spiral
Discovering patterns-- Gcri thmet.ic progress:!. on-triangular numbers
Maxima and minima
Counting (combinations)
Counting (combinations)
Counting (combinations)
Counting (combinations)
Counting (combinations)--power set
Permutations--binomial probability
Deductive reasoning--binomial probability
Deductive reasoning--probability
Binomial probability
Binomial probability--combinations
Probability of dependent events
Probability of independent events
Probability of compound events
ARITH~£TIC
1.
PUZZLES
A Snail in the Well (4)*
A snail climbing out a well advances three feet
each day but slips back two feet at night.
How many days
does it take the snail to climb out, if it starts 30 feet
below the top of the well?
*Each puzzle is referenced by number.
SOURCES at end of Chapter.
See PUZZLE
25
2.
A Tennis Tournament (21)
If 78 players enter a tennis tournament for a
singles championship, how many matches have to be played
to determine the winner?
3.
Ten Million Bacteria (9)
Each bacterium in a certain culture divides into
two bacteria once a minute.
If there are 20 million
bacteria present at the end of one hour, when were there
exactly 10 million bacteria present?
4.
How Many Squares? (4)
How many squares, of any size, are made by the
lines of a standard chessboard?
26
5.
The Bookworm
,---- (21)
The two volumes of Gibbon's "Decline and Fall of
the Roman Empire" stand side by side, in order, on a
bookshelf.
A bookworm commences at page 1 of volume I
and bores his way in a straight line to the last page of
volume II.
If each cover is 1/8 of an inch thick, and
each book without the covers is 2 inches thick, how far
does the bookworm travel?
6.
Squares (10)
•
•
•
•
• • • •
• • ••
• • • •
••••
Start at any point and connect each point with a
straight line, completing as many whole squares as possible,
never taking your pencil off the paper.
You may not
retrace or cross any previously formed line.
Every square completed
=
1 point;
3 sides of a square formed
=
3/4 point;
2 sides of a square formed
=
1/2 point;
1 side of a square formed
= 1/4 point.
How many points can you get?
27
7.
A Cake of Soap (17)
If you place 1 cake of soap on a pan of a scale,
and 3/4 cake of soap and a 3/4-pound weight on the other,
the pans balance.
8.
How much does a cake of soap weigh?
The Puzzled Driver (17)
The odometer of the family car shows 15,951
miles.
The driver noticed that this number is palindromic:
it reads the same backward as forward.
driver said to himself.
that happens again."
"Curious," the
"It will be a long time before
But 2 hours later, the odometer
showed a new palindromic number.
traveling in those 2 hours?
How fast was the car
28
9.
Thirty Flasks (9)
Thirty flasks, ten full, ten half-empty, and
ten entirely empty are to be delivered among three sons
so that flasks and contents should be shared equally.
How may this be done?
10.
How Many Sandals?
(13)
A city in India has a population of 20,000 people.
A small percentage of them are onG legged, and half the
others go barefoot.
How many sandals are worn in the
city?
11.
From I to 1,000,000,000 (17)
When the celebrated German mathematician Karl
Friedrick Gauss was nine he was asked to add all the
integers from 1 to 100.
He quickly added 1 to 100, 2 to
99, and so on for 50 pairs of numbers each adding to 101;
Answer:
50 x 101
=
5,050.
Now find the sum of all the
digits in the integers from 1 through 1,000,000,000.
That's all the digits in all the numbers, not all the
numbers themselves.
29
12.
Grandfather's Faded Bill (20)
Among grandfather's papers a bill was found:
turkeys at $ 67.9.
72
The first and last digits of the
number that obviously represented the total price of those
fowls are replaced here by blanks, for they have faded and
are now illegible.
What are the two faded digits, and what
was the price of one turkey?
(Assume each turkey cost the
same.)
13.
A Pile of Paper (4)
I decided to make a pile of paper.
The paper I
used was one-hundredth (.01) of an inch thick.
put down a single sheet.
down, making 2 sheets.
sheets.
I first
Then I doubled what I have put
A second doubling gives me 4
A third doubling makes 8 sheets, or eight-
hundredths of an inch.
If I double the number of sheets
thirty times in all, how high is the pile?
100 inches, 1,000 inches, or more?
10 inches,
30
14.
The Deleted Checkerboard (14)
Two squares are removed from opposite corners of
a checkerboard leaving 62 squares.
Can the checkerboard
be filled with 31 dominoes, each domino covering two
adjacent squares?
15.
Twenty-One Pigs (10)
How can you put 21 pigs in 4 pigpens and still
have an odd number of pigs in each pen?
16.
Me Pay You?
(10)
A man was recently discharged from his job
because his employer decided the man wasn't worth the
$50,000 a year he was getting.
and you want me to pay you?
way:
You don't work at all,
The employer reasoned this
31
365
365 days in a year
-122
(1) Sleeps 8 hrs/day
243 days left
(2) Rests 8 hrs/day (meals, reads,
recreation, etc.)
-122
121 days left
(3) 52 Sundays, 52 Saturdays
-104
17 days left
(4)
2 weeks annual vacation
-10
7 days left
(5) 1/2 hr lunch (50 wks amounts to
5 days, 2 1/2 hrs/wk)
-5
2 days left
(6) Christmas and New Years off
2
0 days left
What is wrong with his reasoning?
17.
Ten Vacant Rooms (10)
A clerk had only 10 vacant rooms left in the
hotel.
There were 11 men who went into the hotel at the
same time, each wanting a separate room.
The clerk,
settling the argument, said "I'll tell you what I'll do.
I will put two men in room 201 with the understanding that
I will come back and get one of them a few minutes later."
The men agreed to this.
The clerk continued:
put the rest of you men as follows:"
"I will
32
The third man in room 202;
The fourth man in room 203;
The fifth man in room 204;
The sixth man in room 205;
The seventh man in room 206;
The eighth man in room 207;
The ninth man in room 208;
The tenth man in room 209;
The the clerk went back to get the extra man he had left
in room 201, and put him in room 210.
Everybody is happy.
What is the fallacy of this plan?
18.
Three Men in a Hotel (10)
Three men entered a hotel and asked for a room.
The hotel clerk said there was only one room available.
The room cost $30, and each man paid $10.
After the men
had left for their room, the clerk decided that he had
overcharged them.
He called the bellboy and instructed
him to "take this five dollars back to the three men.
Tell them I overcharged them and that they divide it among
themselves."
The bellboy thought that dividing $5 among
three men would be fairly difficult and being dishonest,
kept $2 for himself.
Then he returned $1 to each man, so
the cost to each man was $9.
the bellboy kept
=
$29.
Now 3 x 9
=
$27 and the $2
What happened to the other $l?
33
19.
Guess an Age (9)
1
3
5
7
2
3
6
7
4
5
6
7
9
11
13
10
11
14
15
12
13
14
15
17
19
21
15
23
18
19
22
23
20
21
22
23
25
27
29
31
26
27
30
31
28
29
30
31
Card 1
Card 3
Card 2
8
9
10
11
16
17
18
19
12
13
14
15
20
21
22
23
24
25
26
27
24
25
26
27
28
29
30
31
28
29
30
31
Card 5
Card 4
Using 5 cards numbered exactly as they are here,
ask someone (must be 31 years old or younger) to identify
each card in which their age appears.
Then, by adding the
first number of each card that their age appeared will
equal their age.
For example, if the person identifies
cards #4 and #5, then the person's age
How does this trick work?
=
8 + 16
=
24.
34
20.
A Trick with Three Dice (11)
A
magician turns his back and asks someone to
roll three standard dice and add the top faces.
The
spectator then picks up any die and adds its bottom
number to that total.
The same die is rolled again and
its top number is added to the previous total.
The
magician turns around for the first time to glance at the
dice.
Although he has no way of knowing which cube was
picked for the extra roll, he is able to state the final
total correctly.
21.
A Right
How does he do it?
~nswer
(23)
Things are not always what they seem, and here
is a puzzle to prove it:
S EVE N
+
N I N E
RIG H T
If we regard it as a coded addition in which each different letter represents a different digit, then at first
glance the answer appears to be right but in fact is not.
What is the reason that this simple addition cannot be
decoded?
35
22.
A phonograph Record (4)
The outer track of a phonograph record is 9
inches in diameter, and the unused portion in the center
is 3 inches in diameter.
If the record has 20 grooves to
the inch, how far does the needle travel while the
record is playing?
Solutions to Arithmetic
Puzzles
1.
Day Number
123 4
Height Gained (ft.)
3 4 5 6 . . . n+2
Therefore, the equation
n + 2
.
=
30
n
gives the
number of the day on which the snail reaches the top of the
well, so
n
=
28
days.
2.
Since 77 of the entrants must be eliminated,
then 77 matches are required.
3.
At the end of 59 minutes since 10 million is
half of the present 20 million, and will double to 20
million one minute later, i.e., at the end of one hour.
36
4.
If the small squares have I-inch sides, there are
8
2
=
64 I-inch squares; 7
down to 1 8-inch square.
2
8 = 64
2
7 = 49
62
=
2
= 49
2-inch squares; etc.,
The total number is 204.
36
52 = 25
2
4 = 16
2
3 = 9
2
2 = 4
2
1 = 1
-204
5.
When two volumes are in order (left to right) on
a bookshelf, the first page of volume I and the last page
of volume II are separated only by two covers.
correct answer is
1/8 + 1/8
=
1/4
inch.
The
37
6.
Illustration of 11 points, largest solution known
to author.
There are several designs for any given score.
START
5
1/
,
" Lt
J
13
11.
7
a
15
21 "
17
I'"
\0
2S
Cf
END
18
20
2.'2.
2'1
'....t7
/
11
~~
let
7.
Since
weighs
4 x
i
·1
4
43
cake weighs
pounds, an entire cake
= 3 pounds.
8.
The first digit of 15,951 could not change in 2
hours.
nlli~ber.
Therefore, 1 is the first and last digit of the new
The second and fourth digits changed to 6.
the middle digit is
traveled
110,
210,
0,
310,
1,
.
2,
.
.
.,
. .,
If
then the car
miles in 2 hours.
Clearly the first alternative is the correct one, and the
car traveled
110 = <::;5
-yoJ
miles per hour.
38
9•
Evenly divide the 10 full flasks between 2 sons
and evenly divide the 10 empty flasks between the same 2
sons.
Thus, these 2 sons each have 5 full flasks and 5
empty flasks, yielding 10 flasks each.
Hence, leaving the
remaining son with the 10 half-empty flasks--equivalent
to 5 full and 5 empty flasks.
10.
It doesn't matter what percentage of the population is one-legged.
All the one-legged people will require
only one shoe in any case.
Of the remainder, half will
wear no shoes and the other half will wear two sandals.
This works out at one shoe per person for the "remainders."
Thus, the whole population on the average will wear one
sandal.
Therefore, there are 20,000 sandals worn.
11.
The numbers can be grouped by pairs:
999,999,999
and
0;
999,999,998
and
1;
999,999,997
and
2;
and so on.
There are half a billion pairs, and the sum of the
digits in each pair is 81.
The digits in the unpaired
number 1,000,000,000 add to 1.
(500,000,000 x 81) + 1
Then:
=
40,500,000,01.
39
12.
We know that the total price of the turkeys,
_67.9 , is divisible by 72.
by both 8 and 9.
7.9
Therefore, it is divisible
If it is divisible by 8, the number
must be divisible by 8, and so 7.9
The last faded digit is 2.
If
must be 7.92:
67.92 is divisible by 9,
the sum of its digits must be divisible by 9, and so the
first faded digit must be 3.
Thus, the price of one
=
turkey was (in grandfather's time) $367.92772
$5.11.
13.
3
The pile is 2 °(.01) > 10 million inches
> 100 miles;
and will, of course, topple over long before completed.
14.
• •••
• •••
• ....
•.:.
••
..
·•••••
.•.•. •.•
'
Removing the opposite corners leaves 32 squares
of one color and 30 squares of the other color.
each domino covers one square of each color.
it can't be done.
However,
Therefore,
40
15.
A big pen around the three smaller ones meets the
conditions of the problem!
There are several solutions
in addition to the one shown such as
(5, 7, 9),
(3, 5, 13), etc.
16.
Many days are being counted more than once.
For
example, 8 hrs. sleepjday--one also sleeps on weekends
which were counted separately again.
17.
The second man and the eleventh man are counted
as the same man.
18.
The problem is wrongly stated.
The men didn't
actually pay $27 for the room but $25.
If the problem
is correctly stated, the amount is $30 since
$25 + $3 + $2
=
$30.
Another way to state the problem is that the room
costs $30,
less a $3 rebate, leaving $27.
Now, here is
41
the fallacy of the original problem.
The $2 the bellboy
took is not added to $27 to get $29, but subtracted from
$27 to get $25 plus the $5 hotel clerk rebate equals $30.
19.
The magic in this trick is a result of the binary
system of numeration.
card:
Look at the first number in each
1, 2, 4, 8, 16.
They are all powers of 2.
On the
iirst card are all those numbers whose last digit in the
binary system is 1; the second
contains all those numbers
whose second digit from the end is 1; the third contains
all those numbers whose third digit from the end is 1;
and so on.
Thus, it is easy to announce the age by
merely adding the top left numbers on the cards where it
appears.
Example:
16 8 4 2 1
24
=
1 1 0 0 0 (base 2)
20.
The magician names the final total by simply adding
7 to the sum of the three top faces of the dice.
This
total is the sum of the three top faces plus the previous
top and bottom of one die.
Since opposite sides of a
die total 7, the working is obvious.
42
21.
In three different columns
E
and
N
are
added together, each time with a different result.
This
is impossible since only two different totals can arise,
depending on whether 0 or 1 is carried over from the
column on the right.
22.
Just over 3 inches.
ALGEBRA PUZZLES
23.
De Morgan's Birthday (9)
Augustus DeMorgan, who lives in the nineteenth
century proposed the following:
the year
24.
x2 . "
"I was
x
years old in
When was he born?
Chain Letter (27)
You receive a chain letter that instructs you to
send your math teacher to the person whose name appears
at the top of a list of six names.
You then scratch that
name off the top and add your name to the bottom of the
list.
Next, you send the new letter with your name at the
43
bottom of the list to six friends.
If everyone that gets
a letter follows the instructions, how many mathematics
teachers will you receive when your name gets to the
top of the list.
25.
If Half of Five is Three (10)
If half of five is three, what is a third of ten?
26.
...
--~
A Bottle and
a Cork (10)
...
~-----
'_._~.
A bottle and a cork cost $1.10.
$1.00 more than the cork.
27.
The bottle cost
How much does each cost?
Brothers and Sisters (17)
A boy has as many sisters as brothers but each
sister has only half as many sisters as brothers.
How
many brothers and sisters are there in the family?
28.
How Much
doe~?
the_.Brick Weigh?
(10)
A brick weighs 6 pounds plus half its weight.
How
much does the brick weigh?
29.
The Annual Pioneer Pancake Eating Contest (26)
A team of five represented the Electric Eleven in
the annual Pioneer Pancake Eating Contest.
Janice was
first up for the team followed by Vicki, Cathy, Rich and
Bill in that order.
Each member ate six more pancakes
than their previous team member.
Together they combined
44
for 100 pancakes--easily winning the contest.
How many
pancakes did each member eat?
30.
Alge~!a
Homework (9)
In order to encourage his daughter in the study of
algebra, a father agrees to pay her 8 cents for every
problem correctly solved and to fine her 5 cents for each
incorrect solution.
At the end of 26 problems neither
owes anything to the other.
How many problems did the
girl solve correctly?
31.
How Many Students Did Mr. Einstein Have? (10)
When Einstein was asked how many students he had,
he replied,
"~Of them study mathematics, } of them study
geometry, ~ of them study chemistry, and 20 of them don't
study at all."
32.
How many students did Einstein have?
A Basket of Eggs (2)
A farmer had a basket of eggs to sell.
To his
first customer he sold half his eggs plus half an egg.
He sold half of what he had left plus half an egg to a
second customer.
To a third customer he sold half of
what he had left plus half an egg.
The farmer went home
happy since he sold all his eggs, yet he had not broken a
single egg.
How many eggs were in the basket when the
farmer started his sales?
45
33.
Jack and Jill (16)
Jack and Jill left home at three o'clock to fetch
a pail of water.
They walked along a level road, up a
hill, back down the hill, and home without stopping.
arrived home at five o'clock.
They
Their speed was four miles
an hour on the level, three miles an hour uphill, and
six miles an hour downhill.
34.
How far did they walk?
Phil Anthrope (16)
Phil Anthrope, the eccentric millionaire, recently
revisited his home town.
As a memento of his visit he
offered ten dollars to each of the boys in the town and
six dollars to each of the girls.
All the girls
accepted his offer, but for some reason 40 per cent of
the boys declined.
Assuming there was a total 2,240 boys
and girls living in the town, how much did Mr. Anthrope
give away?
3 ;:)
C'
.
More or Less?
(4)
Pitcher "A" contains a litre of water, and pitcher
"B" contains a litre of milk.
A decilitre of water is
removed from pitcher "A" and added to pitcher "B", which
is then well stirred.
A decilitre of the mixture in
pitcher "B" is now added to pitcher "A".
Is there more
or less milk in pitcher "A" than water in pitcher "B"?
46
You'll have to use your imagination on this one!
Tightly cinch a tape around a ping pong ball.
tape, and add to it 10 more feet of tape.
cut the
If the tape is
equally spaced around the ball, how far is the tape from
the ball?
Now, cinch a tape around the earth (assuming it
to be a perfect sphere), and cut it and add 10 more feet
of tape.
earth?
If equally spaced, how far is the tape from the
Could you slide a piece of paper between the earth
and the tape?
Could you crawl between them?
Could you
drive a truck between them?
37.
Counting Sheep (21)
The same farmer who counted sheep by counting the
legs and dividing by 4 also kept track of his cows and
chickens by counting both the legs and the heads.
If he
counted 78 legs and 35 heads, how many cows and chickens
did he own?
38.
Find the Flaw (27)
Let a = b
2
a = ab
?
a 2 _b 2 = ab-b(a-b) (a+b) = (a-b)b
a+b = b
b+b = b
2b = b
2 = 1
47
39.
Is 4
8?
=:
(19)
= 64-96 (since both sides = -32)
16-48
=
=
=
=
16-48+36
(4-6)2
4-6
4
64-96+36 (adding 36 to both sides)
(8_6)2 (factoring)
8-6 (taking square root)
8 (adding 6)
What is wrong here?
40.
What is the Error? (2)
We all know that .
1 dollar
=
.
.
100 cents
and that .
i
dollar
=
25 cents
taking the square root
J~
dollar
so .
.
=
125 cents
.
~ dollar = 5 cents
What is the error?
41.
The Lucas Problem (17)
This problem was invented by Edward Lucas, a
French nineteenth-century mathematician.
"Every day at noon," Lucas said, "a ship leaves
Le Havre for New York and another ship leaves New York for
Le Havre.
The trip lasts 7 days and 7 nights.
How many
48
New York-Le Havre ships will the ship leaving Le Havre
today meet during its journey to New York?
Can you
answer graphically?"
42.
A Mountain Climber's Journey (26)
A mountain climber leaves the base of the mountain
at 6 am and walking at a constant rate reaches his
destination at the peak at 6 pm.
The following morning
at 6 am he begins his journey down the trail from which h8
came.
Since he is able to maintain a faster pace hiking
downhill, he frequently stops to take in the spectacular
views, and yet he reaches the base of the mountain at
6 pm.
Later that night, while relating his experience to
a group of friends, he recalls seeing on his return hike
the same spectacular view at the exact same location and
time as he saw on his uphill hike the previous day.
this possible or is he mistaken?
43.
Is
Can you answer graphically?
A Card Trick (15)
From a deck of regular playing cards, place 9
cards face down.
From these cards select 1 card (look at
it) and place it on top of the stack of the remaining 8
cards.
9 cards.
Place the remaining deck on tops of the stack of
Deal cards one at a time from the top of the
deck placing them face up and simultaneously counting
backward from 10 to 1 checking to see if the number on any
49
card is the same as the number being counted (aces coun"t
as one while face cards count as ten).
If there's a
match (a 7 turns up as you say 7), stop dealing on that
pile and begin another pile, repeating the process.
If
there is no match, "kill" the pile by placing a card face
down on top of it.
piles.
Do this procedure until you have 4
Finally, add the numbers showing on the piles and
count out that additional number of cards from the deck.
The last one will be the original selected card.
How does
this card trick work?
44.
Crossing Tiles (8)
A rectangular floor is composed of square tiles of
the same size, 81 along one side, 63 along the other.
If
a straight line is drawn diagonally across the floor from
corner to corner, how many tiles will it cross?
How many
if there were 472 tiles along one side and 296 along the
other?
45.
The Refreshed Runner (6)
A man runs
radius is
t
miles.
n
times around a circular track whose
He drinks
every mile that he runs.
quart!
s
quarts of water for
Prove that he will need only one
50
solutions to Algebra
Puzzles
23.
x2
1800 s x
must be such that:
2
S
1899
since
he lived in the nineteenth century.
11800 s x s 11899, i.e.,
44
2
=
1936) .
x
=
Therefore,
2
(since 42 = 1764
43
and
Thus, Augustus was born in 1806 (1849-43).
24.
You sent out 6 letters with your name in the 6th
position on the list.
sends out 6 more.
Each person receiving a letter
This yields
your name in the 5th position.
36 x 6
=
6 x 6
=
36
letters with
These 36 letters yield
216
letters with your name in the 4th position.
4
216 x 6 = 1,296 = 6
3rd position
5
1,296 x 6 = 7,776 = 6
2nd position
7,776 x 6 = 46,656 = 66 1st position
6
Hence, you receive 6
mathematics teachers.
In the general case, if you send out "x" letters
to your friends rather than 6, and if the list contains
"n" names rather than 6, you would receive
matics teachers for your troubles.
x
n
mathe-
51
25.
1
2" of 5 =
1
2
2"
1
_.- 3 1
3" of 10
3"
Thus, if 2 1
2" = 3,
then
3 3"
1 :::; x
1
1
2
3
2"
3"
-3- -" -x5
2" x = 10
5x :::; 20
x
=
4
the desired number.
26.
x = the cost of the cork
Let
then,
x
2x + 1.00
x
1.00
=
the cost of the bottle
x + (x + 1.00)
so,
2x
=
=
=
=
=
1.10
1.10
.10
$.05
x + 1.00
(the cost of the cork)
=
$1.05
(the cost of the bottle)
27.
4 brothers and 3 sisters.
A boy having as many brothers as sisters implies
one more boy than girl.
So, let
n
=
the number of girls
n+l
=
the number of boys
52
Each sister has half as many sisters as brothers
implies:
2 (n-l)
=
n+l
2n-2
=
n+l
n
=
3
n+l
=
4
28.
1\
Let
x
=
the weight of the brick
1
2" x = half of the brick's weight
x
1
= 2"
x + 6
2x
=
x + 12
x
=
12
The brick weighs 12 pounds.
29.
x
=
number of pancakes Janice ate
x+6
=
number of pancakes vicki ate
x+12
=
number of pancakes Cathy ate
x+18
=
number of pancakes Rich ate
x+24
=
number of pancakes Bill ate
Let
x + (x+6) + (x+12 ) + (x+18) + (x+24 )
=
100
53
5x + 60 -- lOa
30.
5x
=
40
x
=
8
x+6
=
14
x+12
=
20
x+18
=
26
x+24
=
32
x
=
number of problems correct
26-x
=
number of problems incorrect
Let
.08x
8x - 130 + 5x
=
l3x
=
.05(26-x)
:. x
(26-x
=
0
0
130
=
10
=
16
correct problems
incorrect problems)
31.
21
are the
studied mathematics (included in this quantity
31
that studied geometry since geometry is a
71
branch of mathematics) ,
studied chemistry and
didn't study at all, so
Let
x
1
1
=
number of students
2"x + 7x + 20
7x + 2x + 280
280
=
5x
56
=
x
=
x
=
l4x
20
54
1
-x
28
2 =
~x
--
7
study mathematics
study chemistry
8
students in all
56
32.
Let
with.
=
x
the number of eggs the farmer started
To his first customer he sold
1
1
2x + 2"
e.ggs;
to his second customer he sold
1
2
1
2
+ 1:.)
2
-(x-(-x
)
1
+ 2"
=
1
:rx
1
+ "4
eggs;
to his third customer he sold
1
2
1
2
1
+ -x
+ 1:.
+ 1:.)
2
4
4
-(x- (-x
)
1
+ 2"
=
1
8
-x
1
+ 8"
eggs.
Since he sold all his eggs
I
x
-
( (-x
2
x
-
(-x
7
8
8
x
=
=
=
x - 7
7
!)
2
I
I
+ !) + (ax
+ ~»
+ (-x
8
4
4
= 0
7
+ -)
= 0
8
8
1
7
-x - -
+
0
0
eggs the farmer started with.
33.
Let
h
=
d
=
the distance along the level road, and
the length of the hill.
we get:
Using distance
=
rate x time
55
d
d
4" + 4"
time walking on level road;
::::
h
3' = t.ime walking uphill;
h
time walking downhill;
::::
6"
thus,
~ + ~ + h + ~ = 2
436
hours or,
4
3d + 4h + 2h + 3d :::: 24
=
d + h
4
or,
and,
2(d+h) :::: 8
miles is the total distance traveled.
34.
Let
girls where
b
=
the number of boys,
= 2,240.
b + g
Phil gave away.
6(b+g)
= the
number of
x:::: total amount
Thus,
(.6) (10)b + 6g
6b + 6g
And let
g
=
=
x
x
= x
6(2,240) :::: x
Therefore, Mr. Phil Anthrope spent $13,440.
35.
Since the two pitchers contain equal quantities
of liquid at the beginning and at the end of the
interchanges, the amount of milk in pitcher "A" must
exactly equal the amount of water in pitcher "B".
56
1st:
from "Aft, leaving
.1 t
Remove
water in "A"
1 t
2nd:
I
.1 t
of
add this to "B" leaving
milk + . 1 t
Remove
.9 t
water in "B".
of mixture containing
xt
of water from "B", add this to "A" leaving
(. 9+x) t
of water and
(1- ( .l-x) )
in "A" , and leaving
of milk and
That is, they each have
introduced.
(Try
=
x
of milk
=
(.9+x)f
of water in "B".
(.l-x)t
( . I-x) t
(.l-x)t
of foreign liquid
centilitre
=
.01.)
36.
Let
C
=
C
=
circumference of the ball + 10', so
c + 10
=
2nr + 10
=
2n(r + b)
2n (r + b)
=
2nr + 10
2nr + 2nb
=
2nr + 10
2nb
=
10
b
=
5
feet
57
Now, let
C
=
eart~
circumference of the
the same argument holds, i.e.,
b
= -1T5
+ 10'.
feet.
Clearly,
Therefore,
both tapes are away from each sphere exactly the same
distance--~
feet.
1T
37.
Let
x
=
y
= number of chickens
number of cows
thus,
x + y
=
4x + 26
35
and,
= 78
solving simultaneously yields:
x
=
4
and
y
=
31
38.
The flaw occurs from step 4 to 5.
(a-b)
=
0,
and
=
-4 + 6.
division by zero is not defined.
39.
(4_6)2
Similarly,
has two roots:
(8-6) 2
4 - 6
has two roots:
and
8 - 6
Hence, true equations:
4
-
6
-4 + 6
Note,
a
2
=b 2
= -8 + 6
=8 - 6
does not imply
o =N
-(4-6)
and
-8 + 6.
58
40.
Mathematical operat.ions can be performed on numbers
only, not on units of different kinds such as various
units of measure.
41.
If you answered 7, bearing in mind the ships that
haven't started yet, you forgot about the ships already
enroute.
A convincing solution is shown in the
diagram.
AB
is the ship's route leaving Le Havre today.
It will meet 13 ships at sea and 1 in each harbor, a
total of 15.
The meetings are daily, at noon and midnight.
59
42.
PEAk
•\
UPHILl..
.- '- .,
DOWNHilL:
\.
I
~_
:
-_-0
·---e__,
....x..;::~~
_--L
llME
-e,
The above graph clearly shows that the mountain
climber's recall could be correct since the paths
intersect at
(t , d ),
l
l
and
S
d:
Base
d
l
for any
t:
6 am
t
S
l
S
6 pm
sPeak.
43.
The maximum number of cards occurring when there
is no match is
x, y, z, w
=
lO(cards up) + l(card down)
equal the "number" in each pile.
11.
Let
The "number"
in this case refers to the value of the matched card
(for example, if no match occurred in the far left pile,
=
then we have
11 - x
piles we use
(II-x) + (ll-y) + (ll-z) + (11-w)
44 - x - y - z - w
cards from the deck.
11
cards.
cards used).
Now count out
In the four
=
x + y + z + w
Thus, we have now used
60
=
(44-x-y-z-w) + (x+y+z+w)
deck.
44
cards from the top of the
Note that the 44th card from the top is the 9th
card from the bottom which is always the placement of the
originally selected card.
44.
A
In going from
EHE
B
.-
A
to
B
the diagonal line
crosses 4 vertical lines and 2 horizontal lines
(including the line at
A).
B,
and excluding the line at
Each time the diagonal crosses a horizontal or
vertical line it passes through 1 square, except when it
crosses them simultaneously it passes through a point.
The answer with respect to the number of tiles it crosses
will be:
(number of vertical lines crossed) + (number
of horizontal lines crossed) -
(number of occasions when
a vertical and horizontal line are crossed simultaneously) .
This last bracket will be the greatest common factor of
the number of vertical and horizontal lines.
above diagram:
(4+2) -
(2) = 4.
the greatest common factor of
(81+63-9)
=
135
tiles crossed.
In the
In the first question:
81, 63
=
9.
Therefore,
In the second question:
61
the greatest common factor
(472+296-8)
=
=
Therefore,
8.
760 tiles crossed.
45.
To determine the amount of water needed, we
must calculate the number of miles run, and in order to
do this we need to find the circumference of the track.
As the radius is
t,
As he goes around
n
the circumference is
times,
of miles run, and as he drinks
consumes
2TInts
2TItns.
2TIt
miles.
2TItn
equa.ls the number
s
quarts per mile, he
Interhcanging the
nand
s
yields
(two "pints") or one quart of water.
GEOMETRY PUZZLES
46.
Finding a Proof (17)
Place 2 matches side by side so they lie on a
straight line.
Prove they do so.
(You may use extra
matches for the proof.)
47.
Irish Friends (14)
Prove that at least two Irishmen have the same
number of Irish friends.
62
48.
The Game o.f "St.ogey"
(14)
In the game of "Stogey", two players alternately
place cigars on a rectangular table with the restriction
that each new cigar must not touch any of the previously
placed cigars.
Can the first player assure himself of
victory, if we define the loser as the first player who
finds himself without sufficient room to place a cigar?
49.
Toe-Tac-Tic (14)
The game of reverse Tic-Tac-Toe (known to some
as Toe-Tac-Tic) has the rules as the standard game with
one exception.
row loses.
The first player with three markers in a
Can the player with the first move avoid being
beaten?
50.
Medians of a Triangle (14)
Prove that each median of a triangle is shorter
than the average of the two adjacent sides.
63
51.
A Paradox (1)
J
J
II
~
,
, ..
J
J
I
8
~
~
I
~
J
~
(3
~
,l
~~
13
J
II
r
J
I
~
s
The square above contains 64 square units (8 x 8),
yet when its parts are rearranged, as shown in the
rectangle, it appears to contain 65 square units (5 x 13) .
Can you determine where the
from?
~extra"
square unit has come
64
52.
What is the Length? (10)
A
Point B is the center of a circle
is 2 inches.
A rectangle is formed by
What is the length of
53.
AB,
E, F, G
the radius
and
GF?
To Find the Center of a Circle (17)
Find the center of the circle using only the
drafting triangle and pencil as shown.
B.
65
54.
The Geomet.ry Chlb I
S
Badg~_
(21)
The geometry club of our high school designed
for itself a membership pin in the form shown by the
diagram.
Our president took the design to the local
jeweler who asked,
"How large do you want this pin?"
Archie, our president replied, "We would like it to be
just two-thirds of an inch in diameter.
of course is the outside edge."
The larger circle
"Hum," said the
jeweler, "that's going to make the letters pretty small.
What size do you expect them to be?"
Archie answered,
"Of course that depends on how much margin is left
between the letters and the triangle.
ought to do whatever you think best.
that the side of the triangle will be
you are as bright as Archie.
the triangle?
I suppose you
But I can tell you
"
Surely
How large is the side of
66
55.
Remaining Metal (11)
A metal sheet has the shape of a two-foot square
with semicircles on two opposite sides.
If a disk with
a diameter of two feet is removed from the center as shown,
what is the area of the remaining metal?
56.
Yang, Ying and Yung (14)
A Yang, Ying and Yung is constructed by dividing
a diameter of a circle,
points
C
and
D.
semicircles having
other side of
diameters.
AB
AB,
into three equal parts by
Then describing on one side of
AC
and
AD
AB
as diameters, and on the
semicircles having
BD
and
BC
as
Which is larger, the central portion or one
of the outside pieces?
67
57.
A Grazing Goat (4)
A square shack 30 feet by 30 feet is in the middle
of an open field.
A goat is tethered to one corner of the
shack by a chain 60 feet long.
She can not get under the
shack but can graze anywhere else she can reach on her
chain.
What is the area of the portion of the field she
can graze?
58.
Sepa~ating_the ~l1eep'
(7)
A one-acre field in the shape of a right
triangle has a post at the midpoint of each side.
A
sheep is tethered to each of the side posts, and a goat
to the post on the hypotenuse.
The ropes are just long
enough to let each animal reach the two adjacent vertices.
What is the total area the two sheep have to themselves,
i.e., the area the goat cannot reach?
68
59.
.
Koch's Triangles (27)
S~P3
In the sequence of figures shown above the first
is an equilateral triangle with side of unit length.
At
each step an equilateral triangle is constructed on each
side of the proceding figure with length equal to onethird of the side.
sides of the
Find the following:
(1) the number of
n-th figure; and (2) the perimeter of the
n-th figure.
60.
Overlapping Figures (14)
7
7
The isosceles right triangle shown above has a
vertex at the center of the square.
the common quadrilateral?
What is the area of
69
61.
A Square Peg in a Round Hole (4)
What is the maximum size for a square peg which
can be inserted in a round hole 2 inches in diameter?
62.
The Carpenter's Rope (26)
A carpenter, needing to form the corners of a
rectangular patio, discovered he had forgotten his
trisquare.
Instead, tying a couple of quick knots in a
rope from his tool box enabled him to form the corners and
complete the job.
63.
How was this done?
The Spider and the Fly (4)
a'
It'
A room is 16 feet long, 12 feet wide, and 8 feet
high.
A spider on the ceiling of the room, 2 feet from the
70
right wall and 2 feet from the front wall f
sees a fly
sleeping on the left wall, 2 feet from the front wall and
1 foot from the floor.
If the spider crawls across the
ceiling parallel to the front wall, and then crawls down
the north wall to the fly, he will travel 10 feet across
the ceiling and 7 feet down the wall, making 17 feet in
all.
There is a shorter route the spider can crawl to
reach the fly.
64.
Can you find it, and calculate its length?
The Hunter's Dilemma (14)
A hunter wished to take his one-piece rifle on a
train but the conductor refused to permit it in the coach,
and the baggage man could not take any article whose
greatest dimension exceeded 1 yard.
rifle was 1.7 yards.
65.
The Fly and
The length of the
What could the hunter do?
Hone~
(12)
On the inside, 1 inch from the top of a cylindrical
glass 4 inches high and 6 inches in circumference, is a
71
drop of honey.
On the outsider 1 inch frrnn the bottom,
and directly opposite, is a fly.
What is the shortest path
by which the fly can walk to the honey, and exactly how
far does the fly walk?
66.
A Familiar Object (21)
I have here a familiar object.
If I hold a candle
under it, the shadow it casts on the ceiling is circular.
If I hold the candle due south of it, the shadow it casts
on the north wall is a square.
If I hold the candle due
east, the shadow it casts on the west wall is triangular.
What is the object?
67.
A Bottle's Volume (27)
_.--~
Given a partially filled bottle with a rectangular
or circular base can you find its volume using only a
ruler?
72
~_A
Curio_us
SphE.:~
(7)
The area and volume of a certain sphere are both
four-digit integers times
TI.
What is the radius of the
sphere?
69.
A Geometrical "Vanish" (17)
Here is an amusing paradox that most people find
hard to explain.
If you move part of a diagram, a line
segment will disappear before your very eyes.
Draw the
13 line segments shown in the figure at the left.
along
MN.
Slide top piece to the left 1 space (figure
at the right).
segment go?
Cut
But wait!
Where did the thirteenth
73
70.
What's My Angle?
(4)
o
T
L
If
L
E
L
= 20°,
OTH
OEM
= 30°,
L
then what is the measure of
HTE = 60°,
MET
L
= 50°,
and
THM?
Planting Trees (13 )
71-
How can you plant ten trees in ten straight rows
with three trees in each row?
Solutions to Geometrv
+
Puzzles
46.
\
\
I
\
I
\
I
,
I
\
I
\
I
\1-
I'
}
,
I
(b)
I
\
I
\
\
I
J
\
\
,I
I
First solution:
build three adjacent equilateral
triangles as shown in (a).
The two solid matches make an
74
angle of 3 x 60°
=
180°, so they fonn a straight line.
For a second solution see (b).
48.
Yes.
The first player should appropriate the only
unique point by placing the first cigar vertically on its
flat end over the center of the table.
From then on, he
can counter each of his opponent's moves by "reflecting"
them
t~rough
the center (reflection through a point) of
the table.
49.
Very easily.
He takes the center square, and then
counters each of his opponent's moves by taking the
diametrically opposite square.
That is, reflection
through a point (the center) .
50.
- - - - - -- - - ,',
- ."./ 8'
0........... '
,
I
I
/
/
B ..:;..
.;a./
c..
Complete the parallelogram as shown in the diagram.
The problem now reduces to proving a diagonal of the
75
resulting parallelogram is shorter Ulan the sum of two
adjacent sides.
This follows from the triangle inequality
theorem.
51.
.........
r--...
~
---.
~
.~
. .·.L..
......... -...
.
~
.,-.- '--~ ....
'" .......
When the four pieces are put together to make
the rectangle, they do not fit perfectly.
There is space
in the middle of the rectangle that accounts for the extra
square.
It is barely noticable because it is spread out
in a long and narrow parallelogram.
(This may be
confirmed by calculating the different slopes of the
pieces. )
52.
EB
=
AB
since radii of a circle are equal.
EB
=
2"
since it was given that the radius equals 2".
EB
=
GF
since the diagonals of a rectangle are equal.
Therefore,
GF
= 2".
76
53.
A
Place
C,
the right angle of the drafting
triangle on the circumference, as shown.
D
and
E,
where the triangle's legs cross the circurnference, are the
and get a second
endpoints of a diameter.
Draw DE,
diameter the same way.
Their intersection is the center of
the circle.
54.
AC
= "31" (1:.2
AD
= /2""
6
AE =
/6""
IS '
diameter)
(tACD:
DE
=
(L.ADE:
30° -
60° -
90°)
77
BE
= 3/2 - /6""
EF
=
(AB - AE)
18
312 -16"
9
DF
=
(6BEF :
30 0
60 0
-
--
90 0
)
12
3 - .4714 (DE + EF)
55.
The two semicircles together form a circle that
fits the hole.
The remaining metal, therefore, has a
total area of four square feet.
56.
B
A
1 2
-TIr
Area (semicircle AC)
=
Area (semicircle AD)
= 31 TIr
Area (semicircle AB)
=
Area (region 1)
=
1
2"TIr
2
6
2
1 2
-TIr
2
1
3
-ITr
2
= 61 ITr
2
78
1 2
.- -TIr
Area
6
Area (outside piece)
=
Area (region 1) +
Area (semicircle AC)
Area (central piece)
=
1 2
1 2
1 2
-TIr + -TIr = 3 1Tr
6
6
=
2 . Area (region 2)
1 2
- -TIr
3
Both sections have the same area.
57.
If we assume the goat tethered to the S.E. corner
of the shack, she can graze an area
43
of a circle 60 feet
in radius from the north, through the east and the south
t
to t h ewes.
T h e area
0
'
. .1S 4
3 TI 60 2 sq. f t.
f th 1S
reglon
When the goat tries to graze on the other side of the
shack, her chain will have to go around the N.E. or S.W.
corner of the shack.
She can only graze two quarters of
a circle, each 30 feet in radius with centers at the N.E.
79
and S,
1
w.
is
'21f
450
'IT
cornE~rs.
30
2
The area of these two combined regions
sq. ft.
= 9,900
The total area grazed is 2,7007T+
sq. ft.
58.
The sheep have to themselves the two crescents
cut from the semicircles on the sides by the semicircle on
the hypotenuse.
The area of these crescents is equal to
the sum of the triangle and the two smaller semicircles
minus the area of the semicircle on the hypotenuse.
Since
the sum of the two smaller semicircles is equal to the
area of the semicircle on the hypotenuse (Pythagorean
theorem extended) the area of the two crescents is equal
to the area of the triangle.
Hence, the sheep have
exactly one acre to themselves.
80
59.
A.)
Each time a triangle is attached to a side, one
side becomes four sides.
Thus, with each step the number
of sides in the entire figure increases four fold.
Hence,
S
on the
nth
n
=
3 x 4(n-l),
where
S
n
:: the number of sides
step.
8.)
•a. •4. • a. •
31
Each side of an attached triangle has a length
as great as the side to which it was attached.
divide a side into 3 equal segments of length
If we
a,
we see
that after a triangle is attached to that side there is
one more segment of length
the original side.
a
between the midpoints of
Thus, with each step, the perimeter
of the figure increases by a factor of
where
step.
P
n
=
4
3.
Hence,
the perimeter on the
nth
81
60.
Rotating the triangle about the square's center
does not change the common area; what is lost in one
quadrant is added to an adjacent quadrant.
Therefore,
rotate so that the two legs of the triangle are flush
with the square's diagonals.
readily seen to be
41
Then the common area is
that of the square or 12.25 square
units.
61.
Making use of the pythagorean theorem:
x
x
x
222
+ x = 2
2
=
2
= 12 -
1.414 inches
82
62.
Suppose the length of the carpenter's rope is
t.
The carpenter first folds the rope in half, and then in
half again leaving four equal lengths.
Tying a knot
between the first two segments separates the rope into two
lengths
3
and
Stretching the knotted rope out,
4" t.
and folding it a second time into three equal lengths, and
again tying a knot between the first two segments
(starting from the opposite end of the rope from the
previously existing knot)
segments
- (1:.3
t
l
1
4
1
-2.,
t
3"
1
+ 4" t)
separates the rope into three
and the middle section being
5
= 12 t.
Now, joining the t'.>JO ends such
that it forms with the two knots the three vertices of a
3
-
TIt,
4
12(,
5
()
TIt-
right triangle enabling him to form
the corners and complete the job.
63.
C.E"ILIN6-
5
1'2.'
I
LEFT
,
'I' :
WAll
t
FRONT
WALL
F
The spider can reach the fly by crawling IS feet
if he chooses a route across the west wall.
To determine
83
the route we consider the room to be like a cardboard box
cut open along the edge joining the north wall and
ceiling so that the pieces representing the north wall,
west wall, and the ceiling can be laid out flat.
The
route taken by the spider is now a straight line, and its
length is:
/12
2
+ 9
2
=
15
feet.
64.
l~
He could put his gun diagonally in a cubical box,
1 yard on a side.
(i)
( ii)
2
2
=
x
2
= 12
2
+ (12)2
Y
= 13
> 1. 73
x
Y
=
1
1
2
+ 1
84
65.
The fly reaches the honey along the 5 inch path
drawn on the unrolled cylinder depicted in the above
figure.
This is the path that would be taken by an
imaginary beam of light moving across the rectangle's
upper boundary.
The length of the path described has the
same length as the hypotenuse of a right triangle with
sides of 3 and 4.
Hence, the fly walks 5 inches.
66.
The object is a type of paper drinking cup (or
French fries container).
The cup folds flat,
condition its shape is the frustum of a cone.
in which
When it is
85
opened, with the opening made circular, the creases at
the sides become parallel.
It might be described as a
kind of cone with a circular base whose elements do not
meet in a single point but which all intersect a line
segment parallel to the base and equal to its diameter.
67.
b
h~
(i)
Measure the height,
h ,
l
of the liquid and
the base of the bottle giving enough information to find
the volume of the liquid
(ii)
height,
h ,
2
(V
l
=
b x h ).
l
Tip the bottle upside down.
of the void space giving enough information
to find the volume of the void space
(iii)
Measure the
(V
2
=
b x h )•
2
Adding the volume of the liquid to the
volume of the void space yields the total volume of the
bottle, i.e., total
V
=
b x (h
l
+ h ).
2
86
68.
The surface area of a sphere
=
2
4nr .
The
4r 2 and
4
3
However, both
volume of a sphere = 3nr .
4 3 must
lie between 1,000 and 9,999. Thus,
-r
3
2
1,000 s 4r s 9,999 implies 15 < r < 50;
3
1,000 s 4r s 9,999 implies 9 < r < 20.
Therefore,
4 3
3r
r
lies between 16 and 19 inclusive.
to be an integer,
Therefore,
r
=
18,
r
area
But for
must be divisible by 3.
=
1,296n,
and volume
=
7,776n .
69.
No line vanished.
The 13 line segments were
1
replaced by 12 segments that are each 12
the old ones.
longer than
If you draw them long enough, you can
measure the difference with a ruler.
70.
o
T
Draw
Now,
L TRE =
TR
~
such that
180° -
L TER
/
-
RTE
=
L RTE =
20°.
Join
MR.
180° - 80° - 20°
=
80°;
87
L
L
TRE
TME =
TM
=
TR;
Again,
= L
L
TER
and
TEM = 50°;
L THR
TE.
Similarly,
TM = TE = TR.
=
and
70°
L HTE
= 180° -
:.
MHR
=
L
MTR = 60°
:. t::. TMR is an equilateral triangle;
HR = 'I'R = MR;
L
TR
L
RMH =
L
MHT
L HET
-
L
RHM.
=L
THM
= 40° =
Now,
=
L
and
:. MR == TR.
L HTR;
HRM = 40°;
30°.
71.
o
A"
Making use of Desargues Theorem:
triangles
(t::.ABC, t::.A'B'C')
if two
are in perspective from a
point (0), then they are in perspective from a line
(A"C"B"), and conversely.
Hence, the ten trees:
0, A, B, C, AI, B ' , C', A", B", C"
three trees in each row.
are in ten rows of
88
IN'l'EHMEDIATE ALGEBRA J?UZZLES
72.
Three Daughters (15)
"What are the ages of your three daughters?" I
asked.
"The product of their ages is 36."
"Tell me more," I requested.
"I would tell you the sum of their ages but that
wouldn't help you."
"I need help," I confessed.
"Okay, my oldest daughter has brown hair."
What are the ages of the three daughters?"
73.
How Many Pages? (20)
To number the pages of a bulky volume, the printer
used 2,989 digits.
74.
How many pages has the volume?
The Evasive Engineer (7)
While visiting Cape Kennedy, we carne upon an
engineer digging a hole.
asked.
"How deep is that hole?" we
"Guess," said the engineer, being evasive.
height is exactly 5 feet 10 inches."
are you going?" we inquired.
the answer,
"My
"How much deeper
"I am one-third done," was
"and then my head will be twice as far below
ground as it is now above ground."
hole be when finished?
How deep will that
89
75.
Watches (17)
My watch is 1 second fast per hour, and Janice's
is 1
!2
seconds slow per hour.
same time.
Right now they show the
When will they show the same time again?
When
will they show the same correct time again?
76.
Two Candles (17)
Two candles have different lengths and thicknesses.
The long one can burn 3 } hours; the short one, 5 hours.
After 2 hours burning, the candles are equal in length.
Two hours ago, what fraction of the long candle's height
gave the short candle's height?
77.
Two Horses and a Fly (4)
Two horses are running toward one another, each
traveling at 30 miles per hour.
A fly, traveling at 60
miles per hour, starts from the nose of the Pinto, when the
horses are 10 miles apart, and flies until it reaches the
Arabian.
It turns around without loss of time, and flies
back to the Pinto.
Here it again turns around, and flies
back towards the Arabian.
The fly continues to travel
back and forth between the horses, maintaining its speed
of 60 miles per hour, and flying a shorter journey on each
successive trip.
Finally the fly is killed when the
horses collide nose to nose.
travel?
How far does the fly
90
78.
Draining a Water rrank (10) ,
A water tank has three drains.
If number 1 drain
is open, the water drains in 15 minutes.
If number 2
drain is open, the water drains in 30 minutes.
If number
3 drain is open, the water drains in 45 minutes.
How
long will it take to drain the water out of the tank if
all three drains are opened at the same time?
79.
T~.e
Physici...§_t
an~_
the Escalator (14)
A famous physicist who is always in a hurry,
walks up an up-going escalator at the rate of one step per
second.
Twenty steps bring him to the top.
Next day he
goes up at two steps per second, reaching the top in 32
steps.
80.
How many steps are there in the escalator?
Average Speed (16)
One Friday afternoon, Dr. Lee left his office
early to make a trip to his beach house.
He was able to
beat the traffic, and made the trip at an average speed
of 60 miles an hour.
Wishing to return Monday morning,
he unfortunately overslept, and hit the city traffic
reaching his office weary having only averaged 40 miles an
hour.
What was his average speed for the round trip?
91
81.
The Confused Teller (27)
Dr. Brown cashed a certain check; however, the
teller became confused, and interchanged the dollars and
cents giving him the same number of dollars as was cents
on the check, and the same number of cents as was dollars
on the check.
The doctor then spent $9.32, and
discovered he had twice the amount of the check.
What
was the original amount of the check?
82.
How Long was the Vacation? (13)
During a vacation it rained on thirteen days, but
when it rained in the morning the afternoon was fine, and
every rainy afternoon was preceded by a fine morning.
There were eleven fine mornings, and twelve fine
afternoons.
83.
How long was the vacation?
The Restaurant Bill (21)
After the boxing matches a group of friends went
into a restaurant for a midnight snack.
bill," they told the waiter.
"Put it all on one
The bill amounted to $6.00,
and the men agreed to split it equally.
Then it was
discovered that two of their members had slipped away
without paying, so that each of the remaining men were
assessed 2S cents more.
originally?
How many men were In the party
92
84.
Scales (18)
0
6
0
J
[
I
/\
00
D
( l)
/\
l
(it)
DOD
LL
11---_/\
__1
o[
OU)
7\
( tV)
How many circles will balance the square?
85.
Bank Shot (14)
y
"i I - - - - r - - - _
3
Z
I
Draw the pool table with vertices at (0, 0),
(0, 4),
(4, 4) and (4, 0).
A broken line is to be drawn,
93
consisting of three segments, starting at (0, 1), angling
successively off the top and bottom sides of the table,
and terminating at (4, 2).
At what points will it meet
the top and bottom sides?
86.
An Infinitude of Twos (7)
+ ...
Assuming that the expression
converges to a finite limit, evaluate it.
87.
The Golden Ratio (4)
The golden ratio is considered to have special
artistic significance.
There are many expressions for it,
one of which is the simple continued fraction:
1 +
1
1 +
1
1
I + 1 + e t c.
While this has attractive simplicity, it can be expressed
in more practical forms.
(1) in a simple formula;
88.
Can you express the golden ratio
(2) as a decimal?
A Swarm of Bees (9)
Bhaskara, a Hindu mathematician in the twelfth
century, considered the following problem:
root of half the
n~~er
upon a jessamine bush,
The square
of bees in a swarm has flown out
98 of the swarm has remained behind;
one female bee flies about a male that is buzzing within a
94
lotus flower into which he was allured in the night by
its sweet odor, but is now imprisoned in it.
Tell me,
most enchanting lad, the number of bees.
89.
An Algebra Error (9)
Students of algebra will agree to the following
theorem:
"If two fractions are equal and have equal
numerators, then they also have equal denominators."
Consider the following problem.
We wish to solve the
equation
X + 5
- 5
x - 7
4x - 40
13 - x
=
Combining the terms on the left side, we find
(x+5) - 5(x-7)
x - x
=
4x - 40
13 - x
or
4x - 40
7 - x
=
4x - 40
13 - x
By the above theorem, it follows that
upon adding
wrong?
x
to both sides, that
7 - x -- 13 - x,
7
=
13.
What is
or,
95
90.
An Elephant and a Mosquito (17)
Does the weight of an elephant "equal the weight
of a mosquito?
and
Y
Let
x
be the weight of an elephant,
that of a mosquito.
weights
2v,
=
2v.
=-
Yi
x + Y
then
Call the sum of the two
From this equation we
can obtain two more:
x - 2v
x
=
-y + 2v
multiply:
X
add
v
2
2 _ 2vx _- 2
Y - 2 vY
:
v
2
,
or
(x-v)
2
=
(y-v)
2
take square roots:
x - v
x
=
Y -
v
=Y
That is, the elephant's weight (x) equals the mosquito's
weight (Y).
What is wrong here?
96
91.
Is
1
=
-l?
(9)
Explain the following paradox:
we have
!=TvCI = -1.
92.
=
l-lvCI
1(-1) (-1)
Hence,
-1
=
= II =
1.
/alb
=
ab,
But by definition
1.
Explain the Paradox (9)
Explain the following paradox:
1-1
= I=T,
and
~
=If,
and
=
we have,
and
/III = 1-1 1-1,
1
93.
since
=
and
-1 .
Another Paradox (9)
Explain the following paradox:
Multiplying both sides by
1
1
3 log (2) > 2 log (2)
log (!)3 > log (!)2,
(~)3
>
1 > 1
8"
4'
(~}2,
or
certainly
we find
,
or
hence
3 > 2.
97
solutions to Intermediate
Algebra Puzzles
72.
Let
x, y, z
Clue # 1 :
factors of 36;
and
x· y . z
= 36,
i .e . ,
all factors of 36 are:
6, 9, 12, 18, 36.
x, y,
be the ages of the 3 daughters.
x, y, z
are
1, 2, 3, 4,
Form all possible cOmbinations of
z:
x
Y..
-z
x + y + z
1
1
1
1
1
2
2
3
1
2
3
4
36
18
12
38
21
16
14
13*
13*
11
10
6
2
3
3
Clue #2:
9
6
9
6
4
sums wouldn't help implies at least
~wo
sums are equal (since, if sums were distinct, this would
be an obvious clue for the solution of the problem) ;
consider those sums that are not unique (*).
Clue #3:
oldest has brown hair implies unique-
ness of oldest daughter.
Therefore, the ages of the daughters are 2, 2 and
9.
98
73.
This problem requires that we make a preliminary
estimate of the unknown.
(i)
A volume of 99 pages needs:
=
9 + 2(90)
189
digits;
(ii)
3(900)
=
a volume of 999 pages needs:
2,889
9 + 2(90) +
digits.
Thus, if the bulky volume in question has
2,889 + 4(x-999)
pages, then:
yields
x = 1,024
=
2,989.
x
Solving for
x
pages.
74.
Let
(!3
x
=
depth now
=
height above now (since 5'10"
70
-
x
3x
=
finished depth
=
finished)
=
70" )
70 + 2 (70-x) , since the
engineer's head will be twice as far below the ground when
finished.
Therefore,
3x
=
210
5x
=
210
x
=
-
2x
42
Therefore,
3x
=
126"
=
10 feet 6 inches.
75.
The watches will show the same time again when the
gain of mine plus the lag of Janice's equals 12 hours
(43,200 seconds).
In
x
hours my watch will be
x
99
seconds fast, and Janice's watch will be
slow.
3
2"
x
seconds
Then:
3
x + 2" x
x
=
43,200
= 17,280
hours
= 720 days.
To show the same correct time will take even longer-until my watch is a multiple of 12 hours fast, and
Janice's is a multiple of 12 hours slow.
This will happen
to my watch every 43,200 hours (1,800 days), and to
Janice's watch every 1,200 days.
The lowest common
multiple of 1,800 and 1,200 days is 3,600 days
(almost
10 years), which is the second answer.
76.
x
Let
and
y
the short candle.
has burned, and
lengths of
75
be the original length of the long candle,
3
7" x
2
5"
y
and
After 2 hours,
2 7
3
4
~ = -x
7
has burned, leaving the equal
3
5" y.
Then the short candle had
the height of the long one.
77.
r------»«~--J
5 1I'\i..
5 P\ t.
Each horse travels 5 miles before colliding, taking
10 minutes since each is running at 30 miles per hour.
Since the fly travels at 60 miles per hour it will travel
100
10 miles in the 10 minutes.
The distance traveled by the
fly is 10 miles.
78.
Let
=
x
number of minutes to drain the tank if
all drains are open.
x
x
x
15 + 30 + 45
=
1
6x + 3x + 2x
=
90
llx
x
=
=
90
2
8 11 minutes -
8 minutes 11 seconds.
79.
Let
s
=
the number of steps, and
of the escalator (steps/seconds), then
s - 20
=
20 rand,
s - 32
=
16 r.
Solving simultaneously,
s
=
80.
r
=
the rate
101
80.
Assume the trip from the office to the beach house
covered
x
miles.
On the trip to the beach house Dr. Lee
averaged 60 miles an hour, thus taking :0 hours (since
distance
=
rate x time).
Since he made the trip back to
the office at an average speed of 40 miles an hour, he
took
x
40
hours.
In all, the round trip of
x
x
65" + 40
took him
hours.
2x
miles
Hence
2x
- - - - = 480 x
x
x
10 £
65"+4"0
= 48
miles an hour.
Forty-eight miles an hour is the average round trip speed.
81.
Let
check, and
d
c
=
=
the number of dollars in the original
the number of cents in the original check.
Then,
100c + d - 932
=
2(100d+c)
100c + d - 932
=
200d + 2c
98c
c
=
=
199d + 932
199d + 932
98
omitting decimal points
102
Searching for values of
d
which will make
c
(and needing only to check even values for
smallest value of
d
is 16 and
c
integral
d),
becomes 42.
the
Hence,
the amount of the original check was $16.42.
82.
There are three possible types of day:
in the morning and fine in the afternoon,
(a) rain
(b) fine in the
morning and rain in the afternoon, and (c) fine in the
morning and fine in the afternoon.
such days in each category be
tively.
=
=
c,
respec-
number of days on which rain falls
=
=
number of days having fine mornings
=
11;
(iii)
a + c
and
13;
( ii)
b + c
b,
That is,
( i)
a + b
a,
Let the number of
number of days having fine afternoons
=
12.
Solving the first equation for
b,
and substituting into
the second equation yields the following equations:
a
-
c
=
2
derive that
and
a
=
a + c
=
7,
b
12.
=
6
From these equations, we
and
number of days on vacation is
c
=
5.
7 + 6 + 5
Therefore, the
=
18.
103
83.
Let
y
=
=
x
original number in party and
amount each original member owed.
§.. =
(i)
y
x
;;)
( ......
6
x - 2
Substituting (i)
(iii)
25 x
=
6
x
2
Y + . 25
into (ii), yields
x - 2
Solving for
=
6
+ .25
x
in (iii),
- SOx - 1200
(5x-40) (5x+30)
x
=
x I -6,
But
So,
=
=
0
0
8, -6
thus
= 8
x
members in original party.
84.
Let the weights of the square, circle, triangle,
and rectangle be
a, b, c,
and
d,
respectively.
third illustration yields the equation
( ...;)
d
= 3"2
c.
=
b + d.
into (ii), and solving for
=
=
3d
implies
The second illustration yields the
equation (ii) a
... ) c
( ~~~
2c
The
3 (a-b)
2
equation (iv) a + b
Substituting equation (i)
c
yields the equation
'
T h e f'~rst ~'11 ustrat~on
=
c.
into (iv), and solving for
, ld s t h e
y~e
Substituting equation (iii)
a
yields the equation a - 5b.
Thus, to balance the square in the last illustration we
need 5 circles.
104
85.
y
If t-+.---t
(OJ I )
--~~-~X
Draw 2 additional 4 x 4 squares atop the one
already drawn.
Twice using the principle that the angle
of incidence equals the angle of reflection implies that
the initial path of the ball will be parallel to the path
of the ball after its second rebound.
This implies that
the two right triangles in the lower right corners of the
bottom and top squares are congruent (ASA).
longer leg must be 2.
at (4, 10).
Thus, the
Therefore, we must aim the shot
Connecting the points
(0, 1)
and
(4, 10)
with a straight line, and using the 2-point form, the
equation of this line is
equation (i) with the line
4
(3' 4)
(i)
y
y
=
=
4
i
x + 1.
Solving
yields the point
where the ball meets the top of the table.
Again
considering the angle of incidence-reflection principle
leaves us with the rebounding line having slope of
-9
L1.
105
Using the point-slope form, the equation of this line is
-9
Y = -~
x + 3.
(ii)
y
=
0,
Solving equation (ii) with the line
yields the point.
(2 8, 0)
9
where the ball meets the
bottom of the table.
86.
Let
x
+/2 + 12+~,
/2
=
so,
or,
x
2
- x - 2
=
=
(x - 2) (x+l)
=
x
2
or
0
then,
0
thus,
-1.
Rejecting negative roots, we conclude that
/2
+
/2
+ IT + ...
= 2.
87.
Let
x
=
the golden ratio, then
As a decimal:
=
x
x
x
x
2
2
1
1 + x
= x +
1
- x - 1
=
=
0
1 + 11 + 4 - 1. 618
2
106
88.
=number of bees,
;1+ g-8 x +2 _. x
x - g-x - 2 =;1
g-x - 2=;1
x - 18 = 9;1
Let
x
then
2
8
1
x
2
2x
2
- 153x + 648
=
(2x-9) (x-72)
x
= 31 (~)
36x + 324
= 72
= 0
0
bees.
89.
If two fractions are equal, and have equal
nonzero numerators, then they also have equal denominators.
90.
The wrong square root of
(y-v)
2
was used.
According to the conditions of the problem, it should
have been
-(y-v);
not
x - v
= - (y-v)
x + y
=
Note that
(x-v)
(y-v):
;
2v.
(an elephant minus a half-elephant,
half mosquito) is positive, while
(y-v)
is negative.
If
1.07
numbers had been used, you would have seen the fallacy.
For example,
81
= 81
implies
9
=
-9.
91.
lalb = lab
iff
la, Ib
are real numbers.
92.
rya
iff
la,
are real numbers.
93.
Hence, reversing the
relation, and following, therefore,
ADVk~CED
94.
~~THE~ffiTICS
1
.'>'1
1
- < 8
4
PUZZLES
Two Trees with the Same Number of Leaves? (25)
If there are more trees in the world than there are
leaves on anyone tree, then granting that each tree has
at least one or more leaves, there must be at least two
trees with the same number of leaves.
True or false?
108
95.
Slicing a Cube (21)
A wooden cube is painted black on all faces.
It is then cut by two parallel planes in each of its three
dimensions (i.e., cut into 27 smaller cubes).
How many
of the smaller cubes are found to be painted on three
faces, two faces, one face, and no faces?
What if the cube is cut by
in each of its three dimensions.
n
parallel planes
How many of the smaller
cubes are found to be painted on three sides, two sides,
one side, and no sides?
96.
Tower of Hanoi (27)
The object of the game, Tower of Hanoi, is to
transfer the entire tower from one peg to either of the two
109
vacant pegs in the least possible moves.
Each move
consists of moving a disc from one peg to another.
while, these rules must be observed:
disc at a time;
smaller disc.
Mean-
(1) move only one
(2) never put a large disc on top of a
If the tower consists of
n
discs, what is
the number of the fewest possible moves to complete the
game?
97.
A Deceitful Proof by Hathematical
I~duction
(27)
Find the fallacy in the following proof by
mathematical induction:
Claim:
P(n):
any set of
n
people are the same age.
all people in a set of
1.
P(l) is obviously true.
2.
Suppose
k
n
people are the same age.
is a natural number for which
P(k)
is
true.
Let
aI' a 2 , . . . , a k , a k + l
Then, by the supposition,
a2
=
a3
= .
=
ak
~
a
a k +l ·
l
be any set of
k + 1
=
a
a
2
= ... =
Therefore,
al
k
=
people.
and
a2
= ... =
= a k + l , and P(k+l) is true. It follows that P(n)
k
is true for all natural numbers n, i.e., any set of n
a
people are the same age.
110
98.
Glasses (27)
Nine glasses are positioned upright.
Define a
"move" to be reversing the position of six glasses (i.e.,
up position
~
down position; down position
~
up position).
How many moves will it take to get all glasses in the down
position?
99.
Fa~l~y
Scale (27)
A butcher weighs meat on a balance.
In order to
compensate for a misplaced fulcrum he weighs first on one
side and then on the other.
and charges accordingly.
A consumer advocate group claims
the butcher is overcharging.
100.
He averages these two weights
Do you think they're right?
Time to Trisect any Angle (5)
Using only a straight edge and a compass and all
the time in the world, how can you tLisect any angle?
101.
The Triangle is Equilateral (22)
that
If a, band c are sides of a triangle such
a2
+ 2
b +2
c = ab + bc + ca, show that the triangle
must be equilateral.
102.
Four Bugs (24)
Four bugs,
A, B, C,
and
of a square 10 inches on a side.
Band
D
are female.
D
A
occupy the corners
and
Simultaneously
C
A
are male,
crawls directly
III
toward
A.
B,
B
toward
C,
C
toward
D,
and
D
toward
If all four bugs crawl at a constant rate, they will
describe four congruent logarithmic spirals \vhich meet
at the center of the square.
How far does each bug travel
before they meet?
103 ~_._House Number (14)
My house is on a road where the numbers run
1, 2, 3, 4,
consecutively.
My number is a 3-digit
one, and by a strange coincidence the sum of the numbers
less than mine is the same as the sum of the numbers
greater than mine.
What is my number?
How many houses
are there on the road?
104.
A Calculus Paradox (9)
c.
A
--aB
-.L.._ _
Consider the isosceles triangle
base
point
AB
P
minimum.
=
12
on
and altitude
CD
such that
CD
S
=
=
3.
ABC
in which
Surely there is a
PC + PA + PB
Let us try to locate this point
P.
is a
Denote
DP
112
1
by
x.
Then
S
fore,
Setting
=
ds
PC
=
3 - x
and
3 - x + 2(x 2 +36)2
dx = 0,
point on the segment
and
PB = (x +36)2.
There1
ds _
2dx - -1 + 2x(x +36)-2 .
=
x = 2/3 > 3,
we find
outside the triangle on
PA
1
2
DC
CD
produced.
for which
S
·and thus
P
lies
Hence, there is no
is a minimum.
What is wrong here?
105.
How Many_Handshakes?
(25)
Ten friends met and each shook hands with every
other.
106.
How many handshakes were there?
Triangles (5)
How many triangles are formed when six lines are
drawn on a piece of paper such that each line intersects
each other and no three intersect in the same point?
107.
Thirty-one Flavors (3)
Baskin-Robins boast of having 31 famous flavors of
ice cream to choose from.
Selecting your 3 favorite
flavors, how many different double scoop cones are
113
possible (it doesn't matter which flavor is on the top or
bottom)?
How many different
poss~bilities
are there
for all flavors they carry?
108.
. How Many Routes? (11)
1---+--.. -+--t-----,1-+--1I---I--f-"--
A man who lives at the top left corner of a
rectangular gride of city blocks works in an office
building at the bottom right corner.
It is clear that
the shortest path along which he can walk to work is
10 blocks long.
Bored with walking the same route every
day, he begins to vary it.
How many different 10-block
routes are there connecting the two spots?
109.
The Prize Contest (22)
Instructor Rubio is trying to supplement her
meager academic salary by entering soap contests.
One
such contest requires the contestants to find the number
of paths in the following array which spell out the word
mathematician:
114
M
M A M
M A T A M
M A T H T A M
M A T
M A T
M A T
M A T
M A T
H E M E H T A M
H E M A M E H T A 1-1
H E M A T A M E H T A M
H E M A T I T A M E H T A M
M A T H E M A T
H A T H E
M A T
H E H T A M
]\1
A T I
I C I T A M E H T A M
C I
C I T A M E H T A M
H E M A T I C I A I C I T A M E H T A M
M A T H E M A T I
C I A N A I
C I T A M E H T A H
Rubio has counted 1,587 paths which originate from
one of the first five rows.
With the deadline for sub-
mitting entries approaching, she is distraught, to say the
least.
Help instructor
Rubio
out by finding the number of
paths with a minimum of computation.
110.
A Fast Deal
(22)
'"
Five cards are drawn at random from a pack of
cards which have been numbered consecutively from
97, and thoroughly shuffled.
1
to
What is the probability that
the numbers on the cards as drawn are in increasing order
of magnitude?
Ill. Four
L~tters
(11)
A secretary types four letters to four people and
addresses the four envelopes.
If she inserts the letters
115
at random, each in a different envelope, what is the
probability that exactly three letters will go into the
right envelopes?
112.
The Pentagon Building (14)
While still at a sizable distance from the
pentagon building, a man first catches sight of it.
Is
he more likely to be able to see two sides or three?
113.
The Same Birthday (5)
Thirty people gather at random in a room.
How
likely is it that among them any two people will share the
same birthday (month and day)?
114.
Three Darts (14)
Three dart players threw simultaneously at a
Tic-Tac-Toe board, each hitting a different square.
What
is the probability that the three hits constituted a win
at Tic-Tac-Toe?
115.
Professor of Ancient History (16)
As the professors were taking their seats on the
platform, a freshman asked, "Who is the man with the white
beard?"
"That's the professor of ancient history,"
answered his right-hand neighbor.
next man.
"Ugly old boy isn't he?"
"So it is," said the
Assuming that one of
these speakers makes a point of telling the truth three
116
times out of four, and the other tells it four times out
of five, what is the chance that the white beard belongs
to the professor of ancient history?
116.
Are You Certain? (5)
Asked Johann Bernoulli, "If a certain missile will
hit its target one out of four times, and four such
missiles are fired at one target, what is the probability
the target will be hit?"
"That's easy," answered one of
his students, "It's certain that one missile will land
on the target."
117.
Are you certain of his answer?
A Chance for Survival (14)
A prisoner is given 10 white balls, 10 black balls,
and two boxes.
He is told that an executioner will draw
one ball from one of the two boxes.
If it is white, the
prisoner will go free; if it is black, he will die.
How
should the prisoner arrange the balls in the boxes to
give himself the best chance for survival?
Solutions to Advanced Mathematics Puzzles
94.
Since each tree has at least one or more leaves,
we must consider starting with a minimum of 2 trees.
117
# of trees
leaves on each tree
2
Each tree may have only 1 leaf.
3
Each tree may have 1 or 2 leaves.
Thus,
two trees must have same number of leaves.
Each tree may have 1, 2 or 3 leaves.
4
Thus,
at least two trees must have same number
of leaves.
Each tree may have 1 , 2, 3,
n
leaves.
•
•
•
I
or n-l
Thus, at least two trees must have
same number of leaves.
n
+ 1
Each tree may have 1, 2, 3,
leaves.
...,
or
n
Thus, at least two trees must have
same number of leaves.
Therefore, by induction, there must always be at least two
trees with the same number of leaves.
95.
(i)
8 are painted on 3 sides
12 are painted on 2 sides
6 are painted on 1 side
1 cube is unpainted.
118
(ii)
parallel
cuts
faces painted black
3
2
1
cubes
0
1
8
8
0
0
0
2
27
8
12
6
1
3
64
8
24
24
8
4
125
8
36
54
27
8
12 (n-l)
(n+l)3
n
6 (n-l) 2
(n-l)3
96.
By induction we conjecture the following formula:
number of discs
minimum number of moves
=
21
-
1
2
-
1
23
-
1
1
1
2
3 = 2
3
7
4
=
15 = 2
4
1
n
97.
Examine step 2 for
a
l
and
a
2
and
B'
= a2·
a
2
k
=
2.
be any set of 2 people.
= r,a 2J1-
Since
P (1)
P(l)
is true.
Then, let
is true,
a
But we can not conclude that
a
l
l
=
a
=
a
B =
l
2
Let
{a }
l
and
since
119
B
n
B'
¢.
:=
for
Hence, the inductive step does not work
to
98.
Label the position of a glass as
reversing its position as
positioned upright
:=
(1)9
down position
:=
(_1)9
:=
-l.
six positions
:=
(-1) 6
:=
1.
where
( -1) n
position, and
vice versa.
1,
and nine reversed to the
For any "move":
reverse
In general,
.
represen t s reverslng
( 6-n)
and
Then, nine initially
(-1) .
=
(+1)
n
glasses to the down
glasses to the up position or
Hence, each "move" yields
it is impossible to produce a product of
+1.
-1,
Therefore,
i.e., it is
impossible to reverse all nine glasses to the down
position.
Notice that it is impossible to reverse the
direction of any odd number of objects, when required to
"move" an even number each time.
Similarly, it is
impossible to reverse the direction of any even number of"
objects, when required to "move" an odd number (greater
than 1) each time.
120
99.
s;:-x
Let
pan
A,
w
=
l\
weight of the meat.
With the meat in
the butcher obtains a reading of
(equal to the total weight in pan
(1)
wx
=
B
wI
pounds
to balance), thus,
wI
With the meat in pan
B,
the butcher obtains a reading
pounds, thus,
of
(2)
w2 x
Solving (1) for
=w
x
and substituting in (2) yields the
correct weight
(3)
w
= /w l w2
(the geometric mean)
(wI + w )
2
2
~
It remains to be observed that
equality if and only if
x
=
1.
w
1
=
w
2
=w
__
IW l w2
with
if and only if
So the butcher is overcharging, and hence, the
consumer advocate group is right.
121
100.
A
O
~8
The catch is "all the time in the world".
1
Starting with any angle, if you take away
1
4" of it, take away
add back
1
1
. . . =
L
1
1
BOY
=
1
2
-
(- - )
= L AOB
-
= L AOB
-
L
AOB (1 -
=L
AOB
1 +
2"
BOe
+
L
+
1
COD -
L
1
= L AOB (~)
AOB
1
1
2" + 4" - "8 + 16 - ... )
1
(
1)
+ 1
!. +
4" - 8
L
FOE -
+ 4" L AOB - "8 L AOB +
AOB
1 + 2"
tL
DOE
etc.
= 3"
1
AOB -
= L
=
- !.2
,
2
1
2" L
1
BOY
1
I I I
16 L
L
1
16
of it, add
"8
you \,vil1 end up with a geometric series:
16-
of it,
2"
122
101.
a2 + b
The equation
= b = c,
+ c2
=
ab + bc + ca
(a-b)2 + (b-c) 2 + (c-a)2
equivalent to
a
2
= o.
is
Hence,
since each term vanishes.
102.
At any given instant the four bugs form the
corners of a square which shrinks and rotates as the bugs
move closer together.
The path of each pursuer will,
therefore, at all times be perpendicular to the path of
the pursued.
approaches
This tells us that as
B,
B
will capture
if
B
for example,
there is no component in
which carries
A
A,
toward or away from
B
B's
A.
motion
Consequently
in the same time that it would take
had remained stationary.
The length of each spiral
path will be the same as the side of the square:
10
inches.
103.
Let
x
=
the number of my house.
.
num b ers I ess th an mlne
=I
+ 2 + 3 +
..•
The sum of the numbers greater than mine
2(x;1)X + x
m
=
x2
=
n(n;l)
The sum of the
+ (x-I)
=
(x-l)x
2
=
Since the sum of the first
~ (the first
n(n+l)
is a
Recognizing that
2
terms of an arithmetic progression is
term + the
mth
term).
triangular number we have a sum which is simultaneously a
123
square and triangular number.
Searching for such a sum we
develop the following table:
x
2
n(n+l)
- -"22
1
2
6
:::
x
1
2
1
= 36
6
2
2
2a·b
-2-
a 2 . b2
:::
= 1'1
2·1
-2-
4·9
8·9
-2-
=:
va+1b
n
l~l
1
1j
2 --;;.3
8
Using the pattern in the fourth column, we derive the next
pair of numbers
and
b = 7
2 + 3
yielding
=
and 2 + 5 = 7, i.e.,
2
2
x = 35
or x = 35.
But
5
a
~
x
5
must
be a 3-digit number, so we derive the next pair:
a
= 5 + 7 = 12
and
b
=:
5 + 12
=
17; yielding
x
=
204
and there are 288 houses on the road.
104.
Examine for endpoint maxima and minima.
If
P = D,
then
S = PC + PA + PB -- 3 + 6 + 6 = 15.
P = C,
then
S = PC + PA + PB = 0 + 3/5 + 3/5 = 6/5 < 15.
Hence, the desired point
P
on
CD
is the endpoint
105.
This is a combination problem:
lor
2! (10-2)! = 45
If
handshakes.
Another way of viewing the problem is as follows:
C.
124
1st person shakes with 9 others
2nd person shakes with 8 others
3rd person shakes with 7 others
ith
1 < i < 10.
person shakes with
(lO-i)
others,
n-l=9
Thus,
L
i = 45 where n = 10.
i=O
106.
This is a combination problem; 6 lines taken 3 at
a time:
6!
3! (6-3)
= 20
107.
(i)
3!
3!
3C 2 + 3C 1 = 2!!! + 2111 = 3 + 3 = 6
different
possibilities from 3 flavors using double scoops.
108.
The numbers of different arrangements, or permutations, of
n
objects of which
a
objects are identical
125
and the remaining
n!
b
objects are also identical is
The rectangle is 6 blocks long by 4 blocks wide,
a!bT
thus, the number of different routes is equivalent. to the
problem of finding the number of different ways 6 pennies
and 4 dimes can be placed in a row.
is
lor
6!4!
=
Therefore, the answer
routes.
210
109.
One may count paths "backwards" from the
N.
In counting the left half of the array, including the
center column, there are two choices for each backward
step.
Thus, this portion yields
2
12
(power set) paths.
Doubling this number and subtracting the center column
to keep from counting it twice, yields
or
8,191 paths.
110.
We need be concerned only with the order of the
cards drawn.
numbers is
The number of permutations of any five
so the probability is
5! ,
1
120 .
Ill.
Zero.
If three letters match the envelopes, so
will the fourth.
126
112.
Assume another man is also approaching the
building from the diametrically opposite direction.
If
the first man can see two sides, then the second man can
see three sides, and vice versa.
Therefore, the chance of
seeing two sides must be the same as the chance of seeing
three sides, and since only two or three sides can be
1
seen at once each of these probabilities must be
2'
113.
The probability that no two people have same
birthday:
365 x 364 x .•• x 336
30
365
Therefore,
=
.30
P(two people share same birthday)
=
1 -
.30
=
.70.
114.
Of the
9!
9 C 3 -- 3!6!
=
84
possibilities, only the
3 rows, 3 columns, and 2 diagonals constitute a win.
Therefore, the probability of a Tic-Tac-Toe is
8
2
84 = 21 .
127
115.
The simplest way to approach this problem is to
consider all possibilities.
A
Answers
B
Truthful
rrruthful
Truthful
Lie
Lie
Truthful
Lie
Lie
Probability
Same
4
12
1
3
4
4
1
1
3
5" = 20
4"
Different
"4
3
5" = 20
Different
1
4"
5" = 20
1
Same
"4
5" = 20
Given the fact that the two men made the same response,
the probability that they are telling the truth is:
12
20
12
1
20+20
=
12
13
116.
Each missile has 3 chances in 4 of missing.
probability,
P(of 4 missiles missing)
3
= "4
The
3
3
3
x "4 x "4 x "4
=
81
256'
T h e pro b a b'l'
1 lty the target will not be missed is
81
175
1 - 256 = 256 .
117.
If the prisoner places one white ball in one box
and the remaining balls (9 white and 10 black) in the other
box, his chance of survival would be:
( ~.l) + (~.9)
2
2
1
9
28
19 = 2 + 38 = 38 = .737 = 73.7%
128
PUZZLE SOURCES
Books
1.
Adler, Irving. !.'i.agic _}~g_llse ~!f Numbers.
New American Library, 1957.
New York:
2.
Brandes, Louis Grant. Math Can Be Fun.
J. Weston Walch, 1975.
Portland:
3.
Burns, Marilyn. The I Hate Mathematics Books.
Little, Brown, and Co., 1975.
4.
Cook, L. H. Work This One Out.
World Librai~~, 196-b-.
5.
Dinesman, Howard P. Superior Mathematical Puzzles.
New York: Simon and Schuster, 1968.
6.
Dudeney, Henry Ernest.
536 Puzzles and Curious
Problems, ed. Martin Gardner. New York: Charles
Scribner's Sons, 1967.
7.
Dunn, Angela. Mathematical Bafflers.
McGraw-Hill Book Co., 1964.
8.
Emmet, E. R. Brain Puzzler's Delight.
Emerson Books, Inc., 1968.
9.
Eves, Howard. An Introduction to the History of
Mathematics.
3rd ed. New York: Holt, Rinehart,
and Winston, 1969.
Greenwich:
Boston:
Fawcett
New York:
New York:
10.
Frohlichstein, Jack. Mathematical Fun, Games and
Puzzles. New York: Dover Publications, Inc.,
1962.
11.
Gardner, Martin. Mathematical Magic Show.
Random House, Inc., 1978.
12.
13.
New York:
Scientific American Book of Mathematical
Puzzles and Diversions.
2nd ed. New York: Simon
and Schuster, 1961.
Heafford, Philip. The Math Entertainer.
Harper and Row, 1959.
New York:
129
14.
Hurley, James F. Litton's Problematical Recreations.
New York: Von Nostrand Reinhold Co., 1971.
15.
Jacobs, Harold. A Teacher's
Algebra. San Francisco:
1979.
G~jde
~.~.
to Element~£z
H. Freeman and Co.,
16.
Jacoby, Oswald. Mathematics for Pleasure.
McGraw Hill Book Company, 1962.
17.
Kordemsky, Boris A. The Moscow Puzzles, trans.
Albert Parry, ed. Martin Gardner. New York:
Charles Scribner's Sons, 1972.
18.
Loyd, Sam. Mathe:~~tica~ puzz~.~s__.of Sam Loyd.
York: Dover publications, Inc., 1960.
19.
Meyer, Jerome S. Arithrnetricks.
Book Service, 1965.
20.
Polya, G. How to Solve It.
2nd ed. Princeton:
Princeton University Press, 1957.
21.
Smith, Geoffrey. Mathematical Puzzles for Beginners
and Enthusiasts.
2nd ed. rev. New York: Dover
Publications, Inc., 1954.
22.
Trigg, Charles W. Mathematical Quickies.
McGraw-Hill Book Company, 1967.
23.
Wylie, C. R., Jr. 101 Puzzles in Thought and Logic.
New York: Dover Publications, Inc., 1957.
New York:
New York:
New
Scholastic
New York:
Other
24.
Gardner, Hartin.
"Mathematical Games." Scientific
American, MCXXXXVII (November, 1957).
25.
Taylor, Harold. A Think Twice Quiz for a Cold Night.
California Mathematics Council: pamphlet of
mathematical puzzles, (date not given) .
26.
Original by author.
27.
Author unknown.
Chapter 4
RESULTS
A collection of mathematical puzzles,
(presented
in Chapter 3), was compiled for secondary mathematics
teachers.
An evaluation of the puzzles was made by five
secondary mathematics teachers.
Each teacher rated each
puzzle with respect to four criteria on a four point scale.
This chapter is a presentation of the results of the
evaluation.
Individual evaluations made by the teachers were
compiled, and converted to numerical values.
Summary
evaluations for each puzzle are given in table 1 (Appendix
II, p. 154), and in Tables 2 and 3 in this chapter.
The distribution of the five respondents' ratings
of each criterion for each puzzle is presented in Table 1.
For example, in evaluating puzzle 1 on criterion I
(clarity
and understandability), all five respondents gave a rating
of "excellent."
For criterion IV (illustration of a
mathematical concept), four respondents rated it "excellent"
and one gave a rating of "good."
The ratings from Table 1 were converted to
numerical
values based on the following scale:
130
131
These
n~~erical
Excellent
=
3
Good
=
2
Fair
=
1
Poor
=
0
values for all respondent ratings were
then totalled for each criterion per puzzle.
These
summarized values for each puzzle, including an overall
rating, are summarized in Table 2.
For example, the ratings
from Table 1 for puzzle 1, criterion I--five ratings of
"excellent"--were converted to a value of 15 in Table
2~
for ratings of puzzle 1, criterion IV--four "excellent"
and one "good"--were converted to a value of 14.
The summarized values for each criterion in Table 2
can be interpreted as follows:
13-15
Excellent
8-12
Good
3-7
Fair
0-2
Poor
An overall rating for each puzzle was achieved by taking
the sum of the values of the four criteria per puzzle, and
can be evaluated as follows:
51-60
Excellent
31-50
Good
11-30
Fair
0-10
Poor
132
TABLE 2. Puzzle Ratings:
Surmnarized Ratings by Respondents
Criterion
Puzzle
No.
I
II
III
IV
Overall
1
2
3
4
5
15
11
14
13
12
15
13
14
11
13
13
12
13
12
12
14
9
14
13
7
57
45
55
49
44
6
7
8
9
10
10
13
15
13
15
11
13
15
13
14
12
13
15
13
14
9
14
15
10
12
42
53
60
49
55
11
12
13
14
15
13
12
15
11
14
9
8
8
12
14
12
10
13
12
14
15
12
11
10
11
49
42
47
45
53
16
17
18
19
20
12
13
14
13
12
13
14
13
13
14
13
13
14
15
13
12
12
13
14
12
50
52
54
55
51
21
22
23
24
25
15
13
14
13
14
15
14
14
14
15
15
12
13
14
15
15
12
14
15
15
60
49
55
56
59
KEY:
Criterion
Clarity and understandability.
I I Time factor practicality.
Motivational value.
.J.~J.
IV Illustration of a
mathematical concept.
Summarized Ratings Scale
Criterion I Overall
I
~T~
Excellent
Good
Fair
Poor
13-15
8-12
3-7
0-2
51-60
31-50
11-30
0-10
133
,]~ABLE
2
(Continued)
-----
(~riterion
Puzzle
.0_ _IV
_ _ _·_. Overall
No.
I
II
III
26
27
28
29
30
14
11
14
15
15
14
15
14
13
14
13
14
13
14
14
14
15
14
14
14
55
55
55
56
57
31
32
33
34
35
10
14
15
14
8
13
13
12
10
10
11
13
13
12
9
14
15
15
13
12
48
55
55
49
39
36
37
38
39
40
12
13
14
12
14
13
14
14
14
14
14
14
15
15
15
15
14
15
14
14
54
55
58
55
57
41
42
43
44
45
12
11
10
12
12
9
11
8
10
12
11
8
10
11
8
13
10
6
43
38
40
41
40
46
47
48
49
50
13
12
9
13
14
13
11
8
10
14
14
10
9
12
14
14
11
10
11
14
54
44
36
46
56
9
9
KEY:
Criterion
Clarity and understandabi1ity.
II Time factor practicality.
III Motivational value.
IV Illustration of a
mathematical concept.
Summarized Ratings Scale
Criterion IOverall
I
Excellent
Good
Fair
Poor
13-15
8-12
3-7
0-2
51-60
31-50
11-30
0-10
134
TABLE 2 (Continued)
Criterion
Puzzle
No.
I
II
III
IV
51
52
53
54
55
12
15
13
12
14
13
14
12
8
14
14
14
14
10
13
13
15
14
12
14
52
58
53
42
55
56
57
58
59
60
12
14
14
13
13
10
14
10
9
12
12
14
14
11
14
14
15
1.3
12
12
48
57
51
45
51
61
62
63
64
65
15
10
14
13
13
15
11
14
14
13
14
10
15
15
13
15
12
15
15
14
59
43
58
57
53
66
67
68
69
70
13
11
15
11
14
12
12
11
11
12
14
13
11
9
11
14
15
9
14
48
51
54
40
48
71
72
73
74
75
13
14
15
13
13
10
13
13
15
11
13
13
12
14
14
13
12
13
15
15
49
52
53
57
53
9
Overall
KEY:
Criterion
I Clarity and understandabi1ity.
II Time factor practicality.
III Motivational value.
IV Illustration of a
mathematical concept.
Summarized Ratings Scale
Criterion IOverall
Excellent
Good
Fair
Poor
13-15
8-12
3-7
0-2
51-60
31-50
11-30
0-10
135
TABLE 2 (Cont;inued)
Criterion
Puzzle
No.
I
II
III
IV
76
77
78
79
80
13
14
15
11
13
12
13
15
13
12
14
13
14
12
11
14
12
15
15
14
53
52
59
51
50
81
82
83
84
85
13
15
15
11
13
8
14
15
12
8
10
14
15
14
12
10
15
15
15
14
41
58
60
52
47
86
87
88
89
90
14
14
14
14
10
12
12
15
14
10
13
11
15
14
11
15
14
15
13
14
54
51
59
55
45
91
92
93
94
95
14
15
15
13
12
14
14
14
11
12
14
14
14
11
14
14
14
15
14
13
56
57
58
49
51
96
97
98
99
100
12
12
13
11
14
11
12
12
13
14
14
11
14
12
12
12
12
12
12
14
49
47
51
48
54
--------
Overall
KEY:
Criterion
I Clarity and understandability.
II Time factor practicality.
III Motivational value.
IV Illustration of a
mathematical concept.
Summarized Ratings Scale
Criterion I Overall
Excellent
Good
Fair
Poor
13-15
8-12
3-7
0-2
51-60
31-50
11-30
0-10
136
TJl.BLE 2 (Continued)
-----··--·Cr iter ion
Puzzle
No.
..
.
__
.I
II
III
IV
Overall
--._0___'-
101
102
103
104
105
14
13
13
13
14
10
11
11
13
15
9
10
13
13
15
10
11
13
14
15
43
45
50
53
59
106
107
108
109
110
13
15
13
12
14
15
15
13
11
13
13
15
13
12
13
15
15
14
13
15
56
60
53
48
55
111
112
113
114
115
14
15
13
14
13
14
15
10
13
13
12
15
11
14
13
12
15
10
14
15
52
60
44
55
54
116
117
15
12
13
13
14
14
15
13
57
52
KEY:
Criterion
I Clarity and understandability.
II Time factor practicality.
III Motivational value.
IV Illustration of a
mathematical concept.
Summarized Ratings Scale
CriterionlOverall
Excellent
Good
Fair
Poor
13-15
8-12
3-7
0-2
51-60
31-50
11-30
0-10
137
In Table 3, a summary of the data from 'l'able 2 is
presented to illustrate the distribution of the ratings
with respect to the four criteria and an overall evaluation.
For example, 83 puzzles were rat.ed as "excellent"
and 34 were rated as "good" with respect to criterion I.
Table 3 summarizes the ratings of the puzzles
presented in Tables 1 and 2.
Overall, the ratings
indicate that the puzzles were judged to meet the selected
criteria.
138
Summary of Ratings
of Puzzles
TABLE 3.
Summarized
Rating
Excellent
Good
Fair
Poor
Criterion
13-15
8-12
3-7
0-2
Total
Rated
I
83
34
0
0
117
II
68
49
0
0
117
III
75
42
0
0
117
IV
79
36
2
.-0
117
74
43
0
0
117
Overall
KEY:
Criterion
I Clarity and understandability.
II Time factor practicality.
III Motivational value.
IV Illustration of a mathematical concept.
Chapter 5
ANALYSIS, SUHMARY, AND
RECOMl'{ENDATI ON S
A collection of mathematical puzzles was compiled
for use in secondary mathematics classes.
An evaluation
of the puzzles was made, and an analysis of the data was
performed.
This Chapter is a presentation of the analysis
of the results, a summary of the study, and recommendations
for further investigations.
ANALYSIS OF EVALUATIONS
As Table 3 (in Chapter 4)
indicates, all summarized
ratings of the criteria were shown to be very high.
From
a total of 468 ratings, there are 305 "excellent", 161
"good," 2 "fair," and 0 "poor."
Hence, nearly all of the
puzzles, 74 out of 117, have an overall rating of "excellent," and the remaining 43 rate as "good."
No puzzle had
an overall rating of "fair" or "poor."
In Table 2, results of the summarized values for
each criterion indicate that the following puzzles were
rated "excellent" by all respondents for each criterion:
8, 21, 83, 107 and 112.
Furthermore, analysis of Table 1,
a distribution of evaluations for each criterion per puzzle,
139
140
shows that 18 other puzzles were rated "excellent" by at
least four of the five respondents for every criterion,
and they are:
25, 29, 30, 38, 40, 50, 52, 57, 61, 63,
78, 82, 86, 88, 91, 93 and 105.
Puzzles 42 and 48, although considered "good, "
"Nere given the lowest overall ratings.
They were rated
down on criterion III (motivational value) and criterion
II (time factor practicality), respectively.
Perhaps an
especially well-planned presentation of these puzzles is
needed.
SUMMARY
The purpose of this study was to provide secondary
mathematics teachers with a collection of mathematical
puzzles classified according to content area.
A collection
of 117 puzzles was compiled, and each puzzle rated,
(based
on a four-point scale), on four established criteria by
five teaching professionals.
An evaluation system was
designed, and an analysis was made.
Overall, the findings
of this study indicate that the puzzles were generally
I
I
considered an "excellent" resource for secondary mathematics teachers.
141
RECO~1ENDATIONS
The author proposes three suggestions for further
study:
1.
That the puzzles presented in this study be
evaluated in controlled classes with an instrument designed
to measure their effect on mathematics achievement;
2.
That a similar study be conducted where
students rate the puzzles for motivational value;
3.
That a collection of mathematical puzzles
be compiled for use in the secondary science classroom to
illustrate the relationship between the mathematical,
physical, and natural sciences.
BIBLIOGRAPHY
142
143
BIBLIOGRAPHY
Books
Adler, Irving. Magic House of Numbers.
American Library;-1957.
New York:
New
Bellman, Richard, Kenneth L. Cooke, and Jo Ann Lockett.
Algorithms, Graphs, and Computers. New York:
Academic Press, 1970.
Brandes, Louis Grant. Math_~~n Be Fun.
Weston Walch, 1975.
Portland:
Burns, Marilyn. The I Hate Mathematics Book.
Little, Brown and Co., 1975.
J.
Boston:
Butler, Charles H., Flynwood Wren, and J. Houston Banks.
The Teaching of Secondary Mathematics. 5th ed. New
York: McGraw-Hill Book Co., 1970.
Cook, L.H. Work This One Out.
Library, 1960.
Greenwich:
Fawcett World
Dinesman, Howard P. Superior Mathematical Puzzles.
York: Simon and Schuster, 1968.
New
Dodgson, Charles Luturdge. Mathematical Recreations of
Lewis Carroll.
2nd rev. New York: Dover Publications,
1958.
Dudeney, Henry Ernest. 536 Puzzles and Curios Problems,
ed. Martin Gardner. New-'York: Charles Scribners
Sons, 1967
Dunn, Angela. Mathemtical Bafflers.
Hill Book Co., 1964.
Emmet, E.R. Brain Puzzler's Delight.
Books, Inc., 1968.
New York:
New York:
McGrawEmerson
Eves, Howard. An Introduction to the History of Mathematics. 3rd ed. New York: Holt, Rinehart, and
Winston, 1969.
Frohlichstein, Jack. Mathematical Fun, Games and Puzzles.
New York: Dover Publications, Inc., 1962.
144
Gardner, Hartin. Hathematical Carnival.
A. Knopf, Inc., 1975.
Hathematical Bagi.c Show.
House, Inc., 1978.
Nevi York:
New York:
Mathematical Puzzles & Diversions.
Simon and Schuster, Inc., 1959.
Alfred
Random
New York:
Scientific American Book of Mathematical
Puzzles and Diversions.
2nd ed. New York:' Simon and
Schuster, 1961.
Graham, L.A.
Ingenious Hathe~atical Problems and Methods.
New York: Dover Publications, Inc., 1959.
Huber, Philip. Mathematical Puzzles and Pastimes.
Vernon: The Peter Pauper Press, 1957.
Heafford, Philip. The Hath Entertainer.
and Row, 1959.
Mount
New York:
Harper
Hurley, James F. Litton's Problematical Recreations.
York: Von Nostrand Reinhold Co., 1971.
New
Hurwity, Abraham B., Arthur Goddard, and David T. Epstein.
Number Games to Improve Your Child's Arithmetic.
New York: Funk and Wagnals, 1975.
Jacobs, Harold. A Teacher's Guide to Elementary Algebra.
San Francisco: W.H. Freeman and Co., 1979.
Jacoby, Oswald. Mathematics for Pleasure.
NcGraw-Hill Book Company, 1962.
New York:
Johnson, Donavan A. and Gerald Rising. Guidelines for
Teaching Mathematics.
2nd ed.
Belmont: Wadsworth
Publishing Co., Inc., 1972.
Jones, Samuel I. Hathematical Wrinkles.
Press, Inc., 1929.
Kingsport:
Kordensky, Boris A. The Hoscow Puzzles, trans. Albert
Parry, ed. Martin Gardner. New York: Charles
Scribner's Son, 1972.
Loyd, Sam. Mathematical Puzzles of Sam Loyd.
Dover Publications, Inc., 1960.
New York:
145
Meyer, Jerome S. Arithmetricks.
Book Service, 1965.
Polya, G. How to Solve It.
University Press, 1957.
New York:
2nd ed.
Scholastic
Princeton:
Princeton
Ransom, William R. One Hundred Mathematical Curiosities.
Portland: J. Weston Walch, 1955.
Schaaf, William L. Recreational Mathematics: A Guide to
the Literature. 4th ed. Washington D.C.: The
National Council of Teachers of Mathematics, 1970.
Schuh, Fed. The ~1aster Book of Mathematical Recreations,
trans. F. Gobel. New York: Dover Publications, Inc.,
1968.
Smith, Geoffrey. Mathematical Puzzles for Beginners and
Enthusiasts. 2nd ed. rev. New York: Dover Publications, Inc., 1954.
Smith, Seaton E., Jr. Games and Puzzles for Elementary
and Middle School Mathematics, ed. Carl A. Backman.
Washington D.C.: National Council of Teachers of
Mathematics, 1975.
Trigg, Charles W. Mathematical Quickies.
McGraw-Hill Book Company, 1967.
New York:
White, William F. A Scrap-Book of.Elementary Mathematics.
Chicago: The Open Court Publishing Co., 1908.
Wylie, C.R., Jr. 101 Puzzles in Thought and Logic.
New York: Dover Publications, Inc., 1957.
Journals
Allen, Layman E., Gloria Jackson, Joan Ross, and Stuart
White.
"What Counts is How the Game is Scored. One
Way to Increase Achievement in Learning Mathematics."
Simulation and Games, IV (December, 1978), 371-389.
Bradfield, Donald L.
"Sparking Interest in the Hathematics
Classroom." The Arithmetic Teacher, XVII (March,
1970)
f
239-242.
Brandes, Louis Grant.
"Using Recreational Mathematics in
the Classroom." The Mathematics Teacher, XLVI (April,
1953), 326-329.
146
Dohler, Dora.
"The Role of Games, Puzzles, and Riddles
in Elementary ~·1athematics." The Arithmetic Teacher,
X (November, 1963), 450-452.
Edwards, K.J., O.L. DeVries, and J.P. Synder.
"Games and
Teams: A Winning Combination."
Simulation and ~ames,
III (September, 1972), 247-269.
Hoffman, Ruth I.
"l1athematics: Learning Through Games."
Instructor, LXXXIV (October, 1974), 69-70.
Hollingsworth, Caroline, and Eleanor Dean.
"Factoring
Puzzles." The Mathematics Teacher, LXVIII (May, 1975)
428-429.
Kerr, Donald R. Jr.
"Mathematics Games in the Classroom."
The Arithmetic Teacher, XXI (March, 1974), 172-175.
Metyner, Seymour, and Richard M Sharp.
"Cardematics lUsing Playing Cards as Reinforcers and Motivators in
Basic Operations." The Arithmetic Teachers, XXI
(May, 1974), 419-421.
Nies, Ruth H.
"Classroom Experiences with Recreational
Arithmetic." The Arithmetic Teacher, III (April, 1956),
90-93.
"Playing Problems." Nation's Schools and
Colleges, ed. Lonn C. Hickman.
II (May, 1975), 49.
Rutherford, Porter B.
"The Effects of Recreations in the
Teaching of Mathematics." School Review, XLVI (June,
1938), 423-427.
Simmons, Las G.
"The Place of the History of Mathematics
in Teaching Algebra and Geometry." The Mathematics
Teacher, XVI (January, 1923), 94-101.
Sobel, Max A.
"Junior High School Mathematics: Motivation v.s. r-tonotony." The Mathematics Teacher, LXVIII
(October, 1975), 479-485.
Steen, Lynn Arthur.
"What's in a Game?"
CXII (March, 1978), 204-206.
Science News,
147
other Sources
Devries, and K.J. Edwards.
"Student Terms and Instructional
Games: Their Effects on Cross-Race and Cross-Sex
Interaction." report no. 137, Learning Games and
Student Terms: Four Research Reports on Effects in
the Classroom, Baltimore: John Hopkins University
Center for Social Organization, 1973.
Taylor, Harold. A Think Twice Quiz for a Cold Night.
California Mathematics Council: pamphlet of mathematical puzzles, (date not given) .
APPENDIX I
DATA SHEET
148
149
COVER LETTER
Dear Colleague:
Please treat this sheet with care since it will be used
as evidence of my data. Notice that the puzzles are
named numerically followed by two columns:
(i) grade
level, and (ii) concept.
After reading both puzzle and solution, as best you can,
determine the grade level most appropriate. You may
specify simply the subject in place of grade level, for
example, trigonometry or algebra. However, you may feel
the puzzle is of a more general nature appropriate for the
junior high level, then please indicate 7th or 8th grade.
After reading both puzzle and solution, as best as you
can, determine, specifically, the concept the puzzle
illustrates. For example, the puzzle may illustrate
addition of fractions with unlike denominators or solving
simultaneous linear equations.
The puzzles are kept in a black binder in my bottom
file drawer in the mathematics office. The puzzles are
arranged two to a page with their corresponding solutions
on the next page.
I recommend that you read and analyze
three each day. You may want to vary this number
depending on your schedule.
See me if you have any further questions.
Thank you,
~~ r:?uu'~lekJ
Randy Baumback
150
DATA SHEET
NAME
PUZZLE #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
GRADE LEVEL
CONCEPT
151
DATA SHEET (Continued)
PUZZLE #
23
24
25
26
27
28
29
30
31
32
33
34
35
GRADE LEVEL
CONCEPT
APPENDIX II
RATING FORM AND PUZZLE EVALUATIONS
152
153
RATING FORM FOR EVALUATING MATHEMATICAL PUZZLES
Instructions:
I.
II.
III.
IV.
Rate each puzzle according to how well it
meets the following criteria.
Is this puzzle clear and understandable?
Does the puzzle possess a time factor practicality?
Does the puzzle possess motivational value?
Does the puzzle illustrate a mathematical concept?
Rating scale:
E--EXCELLENT
G--GOOD
F--FAIR
p--·POOR
PUZZLE 2
PUZZLE 1
Criterion
E G
Rating
F
P
I
II
III
IV
--- f - - -
Criterion
E G
I
I
II
I
III
I
IV
:
PUZZLE 3
Criterion
E G
I
II
III
IV
PUZZLE 4
Rating
F P
Criterion
E
III
IV
PUZZLE 5
I
II
III
IV
G
I
II
-
Criterion
E G
Rating
F P
PUZZLE 6
Rating
F P
I
I
!
I
Criterion
E G
I
II
III
IV
Rating
F
P
154
TABLE 1. Distribution of Rati~qs
of Criteria for Puzzles
PUZZLE 1
Criterion
E G
I
5
5
3
II
III
IV
4
PUZZLE 2
Rating
F P
--
2
1
Criterion
E G
I
3 1
II
III
IV
PUZZLE 3
Criterion
E G
I
4 1
II
4 1
III
3 2
IV
4
1
Rating
F P
I
2
3
3
II
III
IV
3
1
Rating
F P
2
1
3
1
1
1
1
1
Rating
F P
Criterion
E G
rI
II
3
2
2
2
III
IV
3
3
1
1
1
2
PUZZLE 6
Criterion
E G
I
II
2
2
1
2
PUZZLE 4
PUZZLE 5
Criterion
E G
3
3
Rating
F P
III
IV
3
3
3
2
Rating
F P
1
1
2_
1
1
1
1
1
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
155
PUZZLE 7
Criterion
E G
3 I 2
I
II
3 i 2
III
3 I 2
IV
4 I 1
PUZZLE 8
Rating
F P
Criterion
E G
I 5
I
II
5
III
5
IV
5
PUZZLE 9
Criterion
E G
4
4
4
2
I
II
III
IV
3
1
2
5
II
III
IV
I 1 I
I 1 i
I 1 !
1121
I
II
III
IV
III
IV
5
1
3
2
2 I
2 I 2
3 i
I
2
2
2
1
!
1
1
1
1
....
I
II
4-
III
IV
1
3
Rating
F P
I
3
3
3
1
2
1
1
I
!
I
PUZZLE 14
Rating
F P
1
5
4
4
3
Rating
F P
Criterion
E G
PUZZLE 13
I
II
I
!
PUZZLE 12
Rating
F P
Criterion
E G
I
Criterion
E G
PUZZLE 11
I
~
:
PUZZLE 10
Rating
F P
Criterion
E G
Rating
F P
1
Criterion
E G
I
II
III
IV
I 2
3
3
1
Rating
F P
21 1
11 1
11 1
31 1
I
I
!
I
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
156
PUZZLE 16
PUZZLE 15
criterion
E G
I
II
4
4
1
1
2
2
Rating
F P
Criterion
E G
1
II
III
IV
I
-r "t
III
IV
PUZZLE
Criterion
E G
I
II
III
IV
3
4
3
1
2
3
III
IV
2
PUZZLE 19
Criterion
E G
3
3
5
4
I
II
III
IV
Rating
F P
2
2
III
IV
5
5
5
5
I
II
III
IV
Rating
F P
1
18
Rating
F P
I
41 1
3 I 2
4 I 1
41
I
I
11
Criterion
Rating
E G -F P
3
4
3
3
1
1
1
2
1
1
PUZZLE 22
PUZZLE 21
Criterion
E G
2
1
PUZZLE 20
I
II
1
1
2
Criterion
E G
I
II
2
1
PUZZLE
17
Rating
F P
3
3
3
3
Rating
F P
Criterion
E G
I
II
III
IV
3
4
2
2
Rating
F P
2
1
3
3
KEY:
Rating
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
E
G
F
P
Excellent
Good
Fair
Poor
157
PUZZLE 24
PUZZLE 23
Ratin~
Criterion
----E--G
4 1
I
II
4 1
III '3- 1--2
IV ~- 1
F
P
..
_.-
Rating
F
III
IV
--
3
2
"4 --I
4
5
1
PUZZLE 26
PUZZLE 25
Criterion
E G
I
4 1
II
5
5
III
5
IV
I
II
P
Criterion
Rating
E G F P
4 Ii
I
I II
4 1
I
III
3
2
I
4 1
IV
I
"
PUZZLE 27
Rating
Criterion
I
II
III
IV
PUZZLE 28
E
G
1
5
4
5
4
F
P
1
PUZZLE 29
Criterion
E G
I
5
II
4
4 1
III
4 1
IV
Rating
F ..- P
1
Rating
Criterion
E G
4 1
I
4 1
II
III
3 2
4 1
IV
F
P
PUZZLE 30
Criterion
E G
I
5
II
4 1
III
4 1
4 1
IV
Rating
F
P
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
158
PUZZLE 32
PUZZLE 31
Criterion
Rating
E G F P
I
1 3 I 1
II
3 2 !
III
2 2 I 1
IV
4
1 I
Criterion
E G
I
4
1
II
4
III
IV
3
5
Rating
F P
I
I
I
Rating
F P
Ii 1 I
1
I
1 1 I
1
I
PUZZLE 37
Criterion
E G
I
3 ! 2
II
4 I 1
III
4 I 1
IV
4 I 1
2
Criterion
E G
4 1
I
2 1
II
2 3
III
IV
3
2
Rating
F P
Rating
F P
2
PUZZLE 36
PUZZLE 35
Criterion
E G
I
1121
II
1 I 3 I
III
2 I 1 I
IV
3 I 1 I
1
I
PUZZLE 34
PUZZLE 33
Criterion
E G
I
5 !
II
2 I 3
III
3 I 2
IV
5 I
Rating
F P
Criterion
E G
I
3
1
II
4
III
IV
4
Rating
F P
1
1
1
5
PUZZLE 38
Criterion
E G
I
I 4 1
II
! 4
1
III I 5
IV I 5
Rating
F P
KEY:
criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
159
PUZZLE 40
PUZZLE 39
Criterion
E G
I
II
2
4
III
IV
5
Rating
F P
3
1
I
I
II
III
IV
1
4
Criterion
E G
PUZZLE 41
criterion
E G
I
I 3 1
II
I 1
III
IV
I 2
2
2
2
i 2
E
Rating
F P
1
2
1
1
1
I
II
III
IV
3
F
I
II
III
IV
P
1
Ratina..
Criterion
E G
I
II
III
IV
1
3
2
1
1
Rating
F P
1
1 I
i
2
i 1 I
3
I 11
I
3 ill
I
PUZZLE 46
Rating
F P
Criterion
E G
I
II
I,
I
1
PUZZLE 44
1
1 I 2
121 2
4
III
IV
! 3 I :l I 1 I
3 I 1 I 1
i 2
5
Criterion
PUZZLE 45
Criterion
E G
I
i
I
II
Rating
G
3
4
4
2
1 i
1
PUZZLE 42
PUZZLE 43
Criterion
4
4
Rating
F , P
I
1
III
IV
I~I ~I
EfJ i I
Rating
F P
FI
: •
KEY:
Rating
criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
E
G
F
P
Excellent
Good
Fair
Poor
160
PUZZLE 47
Criterion
E G
I
2
2
II
III
IV
1
2
3
2
3
2
PUZZLE 48
Rating
F P
1
1
1
PUZZLE 49
Criterion
E G
3
2
3
2
I
II
III
IV
PUZZLE 50
Rating
F P
2
1
1
2
Criterion
E
II
3
2
III
IV
4
4
1
II
III
IV
2
1
1
Rating
F P
1
G
3
3
2
III
IV
4
4
1
1
1
F
1
Rating
F P
I
PUZZLE 54
Rating
E
P
lli=Er--+-I-
Criterion
E G
I
5
II
4
1
5
III
IV
5
PUZZLE 53
I
II
F
PUZZLE 52
1
Criterion
Rating
G
I
PUZZLE 51
Criterion
E G
3 1
I
Rating
F P
1 1 I
1 1
1 1
1
Criterion
E G
2
I
1
II
1 2
2
1
III
IV
2
2
P
Criterion
E G
I
I 2 3
II
III
IV
I
!
1
I 3
3
3
1
Rating
F
P
I
21
11
11
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
161
PUZZLE 55
PUZZLE 56
criterion
E G
Rating
F ~P4-'1"--.4 1
I
II
~._.
III
IV
Criterion
,..._.E G
3
1
I
II
~ 3
3
2
III
-3
1
4
f
IV
4
1
PUZZLE 57
Criterion
E G
I
I4 1
II ! 4 1
III I 4 i
IV I 5
PUZZLE 58
Rating
F
P
Criterion
E G
I
II
III
IV
PUZZLE 59
criterion
E G
3
I
II
III
IV
1
1
2
3
1
1
II
5
i
4 I 1
5 I
III
IV
i- -
' - -..
2
I
-4
II
3
III
4
IV
3
Rating
F P
1
1
1
1
1
1
PUZZLE 62
Rating
F
4
3
1
5
Criterion
E G
PUZZLE 61
Criterion
E G
I
5 I
4
f---
Rating
F P
PUZZLE 60
Rating
F P
2
2
4
."
1
1
1
P
Criterion
E G
I
2
1
II
2
III
IV
1
2
3
3
1
Rating
F P
2
1
1
1
--
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
162
PUZZLE 63
Criterion
I
II
III
IV
PUZZLE 64
~ating
E
G
4
4
5
5
1 e----
F
r-
P
-
I
II
3
III
IV
5
5
4
3
3
4
I
II
III
IV
Rating
F P
1
2
2
--
I
II
III
PUZZLE 67
Rating
F P
1
1
2
2
2
I
II
III
IV
1
2
1
2
2
Rating
F -P
1
1 1
1
2
4
2
2
2
Rating
F P
3
3
2
1
I
1
I
I
I
PUZZLE 68
Rating
F P
Criterion
E G
I
II
III
IV
PUZZLE 69
Criterion
E G
P
4" "':[
Criterion
E G
IV
1
Criterion
E G
I
3
1
II
3
1
III
4 1
IV
4 1
E'
2
PUZZLE 66
PUZZLE 65
Criterion
E G
~a.:ti.n~
Criterion
E G
5
3
3
5
2
2
PUZZLE 70
Criterion
E G
4 1
I
II
2
III
IV
1
4
Rating
F P
1
2
2
2
1
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
163
PUZZLE 71
criterion
E G
PUZZLE 72
-~----
3
1
I
II
III
IV
T3
,
2
....3
2
2
-1
Criterion
E G
I
II T
III T
2
IV
2
3
5
4
3
3
I
II
III
IV
Ratin9"
F P
Criterion
.....
E G
1
1
II
III
IV
2
--~
PUZZLE 75
Criterion
E G
4
3
4
5
I
II
III
IV
P
1
2
1
I
II
III
IV
Ratin9"
F P
2
1
3
3
4
4
Rating
F
P
2
1
1
1
1
PUZZLE 78
Criterion
E
I
1
3
5
4
5
Criterion
E G
PUZZLE 77
Criterion
E G
I
T
1
3 2
II
III
3 2
IV
3 1
Ratin9"
F P
~
PUZZLE 76
Rating
F
,.-
PUZZLE 74
I
1
P
"T=rc1
PUZZLE 73
Criterion
E G
~~!in9"
F
II
III
IV
Ratin9"
G
F
P
5
5
4
5
1
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
164
PUZZLE 79
PUZZLE 80
Criterion
E G
2
I
II
III
T
2
f-2
1- -
Criterion
--E G
1
-
2 3
IV
5
3
1
4"
IV
4
1
3
1
1
I
II
III
IV
i-'
1
3
5
Rating
F
P
I
3 i
1 I
II
III
IV
I
5
5
5
5
I
II
III
IV
P
I
II
-
Criterion
Rating
E G .. F P
3
1
3
4
III
IV
2
2
1
1
1
1
Criterion
E G
III
IV
PUZZLE 85
I
II
4
4
5
PUZZLE 84
Rating
F
3:-+-+---1
Criterion
Rating
E G F P
I
~ ,...._-
PUZZLE 83
Criterion
E G
P
PUZZLE 82
PUZZLE 81
Criterion
E G
F
'2
I
II
III
2
~at.in9
1 ~
1
1
2
4
5
Rating
F
P
4
3
1
PUZZLE 86
Criterion
E G
I
II
III
IV
4
4
4
5
Rating
F
P
1
1
1
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
165
PUZZLE 87
criterion
E G
4
3
2
4
I
II
III
IV
PUZZLE 88
Rating
F P
1
Criterion
-----_.E G
I
i 1
'X- l
II
III
IV
1
PUZZLE 89
Criterion
E G
4
4
4
3
I
II
III
IV
i-L
Rating
F -P
-+-
2
Criterion
E G
I
II
III
IV
PUZZLE 91
4
4
4
4
I
II
III
IV
Rating
F P
III
IV
PUZZLE 93
5
I
II
4
4
III
IV
5
3
1
"2
2
1
1
1
Rating
F P
5
4
4
4
1
1
1
PUZZLE 94
Ratin9:.
P
Criterion
E G
I
1
1
2
2
4
Criterion
E G
I
II
F
1
Rating
F P
PUZZLE 92
1
1
1
1
Criterion
E G
5
5
5
PUZZLE 90
1
Criterion
E G
--4-,-1,..·
II
III
IV
3
2
2
4
2
2
2
-r--
1
1
1
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
166
PUZZLE 95
criterion
E G
I
II
3
3
III
IV
4
1
1
1
3
2
PUZZLE 96
Rating
F
P
1
1
Criterion
E G
I
II
III
IV
PUZZLE 97
Criterion
E G
3
I
II
2
2
2
III
IV
1
2
1
III
IV
2
I
2
2
3
3
3
Rating
F P
1
1
1
1
1
I
II
III
IV
E
G
4
2
1
1
4
1
3
4 !
3 I 1
4 I 1
3
I 1
1
Criterion
E G
4 1
I
4 1
II
III
3
1
IV
4 1
PUZZLE 101
Criterion
Rating
F P
1
1
PUZZLE 100
PUZZLE 99
II
III
IV
1
Criterion
E G
I
II
3
Criterion
E G
3
2
1
1
PUZZLE 98
Rating
F P
1
3
4
2
4
Rating
F P
1
1
Rating
F P
I
2
1 [
1 I
I
Rating
F P
1
PUZZLE 102
Criterion
E G
I
II
III
IV
Rating
F P
3
2
2
2
2
1
1
2
1
4
KEY:
Rating
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
E
G
F
P
Excellent
Good
Fair
Poor
167
PUZZLE 103
Criterion
E G
4
3
3
3
I
II
III
IV
Rating
F P
1
2
2"
"2
PUZZLE 104
I
II
III
IV
PUZZLE 105
Criterion
Rating
E G F P
141
II
5
I I I ~5-t---+_-t---+
IV ' -5- - - ' - _.........- - - ' - - '
Rating
F P
5
5
5
5
I
II
III
IV
I
II
III
IV
I
II
III
IV
3
2
2
2
3
1
3
2
1
1
3
5
3
5
2
-_._. -
2
Criterion
E G
PUZZLE 109
Rating
F P
Rating
F P
PUZZLE 108
I
---~--,
EtHf3
Criterion
E G
II
III
IV
Criterion
E G
tIhl_~B
PUZZLE 106
PUZZLE 107
Criterion
E G
Rating
F P
Criterion
--'E -G
3
3
3
4
Rating
F P
2
2
2
1
PUZZLE 110
Criterion
E G
I
II
III
IV
4
3
3
1
2
2
5
KEY:
criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor
168
PUZZLE 111
Criterion
E G
I
4 1
II
4 1
III
3
2
IV
3 1
PUZZLE 112
Rating
F P
1
PUZZLE 113
Criterion
E G
I
3 2
II
1 3
III
2
2
IV
1 3
Ratin9:
F P
1
1
1
Criterion
E G
4
I
1
II
3 2
III
4
1
4
IV
1
Ratin9:
F P
I.
PUZZLE 116
Rating
F
I
I
PUZZLE 114
PUZZLE 115
Criterion
E G
3 2
I
II
3 2
III
3 2
IV
5
Rating
F P
Criterion
E G
I
5
5
II
III
5
IV
5
P
Criterion
E G
I
5
II
3
2
III
IV
4
1
Rating
F P
5
PUZZLE 117
Criterion
Ratin9:
E G F P
2
I
3 I
II
3 21
III
4
11
IV
3 21
KEY:
Criterion
I
II
III
IV
Clarity and understandability.
Time factor practicality.
Motivational value.
Illustration of a mathematical
concept.
Rating
E
G
F
P
Excellent
Good
Fair
Poor