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What do the following
people have in common?
Ralph Abernathy,
civil rights leader
Harry Blackmun, Associate Justice
of the U.S. Supreme Court
David Dinkins, Mayor of New York
city
Art Garfunkel, folk-rock singer
Alexander Solzhenitsyn, Nobel
prize winning novelist
J. P. Morgan, banking, steel,
and railroad magnate
Michael Jordan, basketball superstar
Find some puzzles,
tricks, or magic stunts that are based
on mathematics. Write a paper describing
the tricks and also indicate why they
work. References
William Schaaf, A Bibliography
of Recreational Mathematics (Washington,
D.C.: National Council of Teachers of
Mathematics, 1970).
See also the Journal of Recreational
Mathematics. Try this search engine:
YAHOO Search Results:
http://search.yahoo.com/bin/search?p=math+puzzles
(over 50 excellent sources)
A
4-by-4 magic square is shown in this
1514 engraving by Durer called Melancholia.
A detail of the magic square is shown:
 Notice
the date appears in the magic square.
Can you see additional properties in
addition to the usual magic square properties?
Hint: Add the corners, or the center
squares, or the slanting squares (2,
8, 9, 15, for example).
Write a short
paper about the construction of magic
squares.
You might include such facts as there
is 1 standard magic square of order 1,
0 of order 2, 8 of order 3, 440 of order
4, and 275,305,224 of order 5. According
to the Guinness Book of World Records,
Leon H. Nissimov of San Antonio, Texas,
has discovered the largest known magic
square with sum of 999,999,999,989. Show
that such a magic square is not possible.
You might also include the properties
of the magic square discovered by Benjamin
Franklin. References
William H. Benson and Oswald Jacoby,
New Recreations with Magic Squares
(New York: Dover Publications, 1976).
John Fults, Magic Squares
(La Salle, IL: Open Court, 1974).
Martin Gardner, "The Magic of
3 by 3; The $100 question: Can you Make
a Magic Square of Squares?" Quantum,
January/February, 1996, pp. 24-26.
Martin Gardner, "Mathematical
Games Department," Scientific American,
January 1976, pp. 118-122.
http://forum.swarthmore.edu/alejandre/magic.square/loshu.html
A process for
producing an artistic pattern using magic
squares is described in an article, "An
Art-Ful Application Using Magic Squares"
by Margaret J. Kenney (The Mathematics
Teacher, January 1982, pp. 83-89).
Read the article and design some magic
square art pieces.
An alphamagic square, invented by
Lee Sallows, is a magic square so that
not do when the numbers spelled out in
words form a magic square, but the numbers
of letters of the words also form a magic
square. For example,
gives rise to two magic squares:
The first magic square comes from the
numbers represented by the words in the
alphamagic square, and the second magic
square comes from the numbers of letters
in the words of the alphamagic square.
Find another alphamagic square.
Answer the question
posed in Problem 59 for your own state.
If you live in California, then use Florida.
References
Check an almanac to find the area of
your state. Also, most states have a
web site which provides this information.
Read the article,
"Mathematics at the Turn of the Millennium,"
by Phillip A. Griffiths, The American
Mathematical Monthly, January, 2000,
pp. 1-14. Briefly describe each of these
famous problems: a. Fermat's last theorem
b. Kepler's sphere packing conjecture
c. The four-color problem Which of these problems
are discussed later in this text, and
where?
The objective of this article was to
communicate something about mathematics
to a general audience. Discuss how well
did it succeed with you?
Top
What is the millionth positive integer
that is not a square or a cube?
What is the millionth positive integer
that is not a square, cube, or fifth
power?
Write a report discussing the creation
of colors using additive color mixing
and subtractive color mixing. 
Let S be the set of all real
numbers between 0 and 1. Prove that S
is not a countable set.
Top
What do the following people have
in common?
Ira Glasser, executive director
of the ACLU
Bram Stoker, author of Dracula
David Robinson, basketball star
Ed Thorpe, inventor of programmed-trading
on Wall Street
Clifford Brown, 1950s jazz trumpeter
Here is a problem by Bob Stanton
from the 1998 issue of Games magazine
which you can use to test your logical
reasoning.
"For her school's Science
Fair, your niece has pasted a deck of
cards onto a board to illustrate the
10 standard poker hands. You promised
to deliver her project to the school
auditorium. Unfortunately, most of the
cards fell off as you carried the board
across the parking lot. You recovered
the cards, but now you have to repair
the damage. Without moving any of the
remaining cards, can you construct the
10 hands on the board preferable before
your niece arrives?
Note: the ace of hearts is part of the
four-of-a-kind hand."
Do some research to explain the differences
between the words necessary and sufficient.
Sometimes statements p and q are
described as contradictory, contrary,
or inconsistent. Consult a logic text,
and then define these terms using truth
tables. Reference: The Internet Encyclopedia of Philosophy:
http://www.utm.edu/research/iep/u/universa.htm Introduction to Options
http://www.sjsu.edu/depts/itl/graphics/options/options.html
Between now and the end of the course,
look for logical arguments in newspapers,
periodicals, and books. Translate these
arguments into symbolic form. Turn in
as many of them as you find. Be sure
to indicate where you found each argument.
Suppose a prisoner must make a choice
between two doors. One door leads to
freedom and the other door is booby-trapped
so that it leads to death. The doors
are labeled as follows:
Door 1: This door leads to freedom
and the other door leads to death.
Door 2: One of these doors leads
to freedom and the one of these doors
leads to death.
If exactly one of the signs is true,
which door should the prisoner choose?
Give reasons for your answer.
Convention Problem. A mathematician
attended a convention of men and women
scientists. The mathematician observed
that if any two of them were picked at
random, at least one of the two would
be male. From this information, it is
possible to deduce what percentage of
the attendees were women?
Flower Problem. Three students
visited a very patriotic garden of red,
white, and blue flowers, but in addition
there were some yellow flowers. One student
observed that if any four flowers were
picked, one of them would be red. Another
observed that if any four were picked,
at least one of them would be blue. The
third shouted that if four were picked,
one would be yellow. Does this necessarily
mean that if any four were picked, one
would be white?
Baseball Problem. Nine men
play the positions on a baseball team.
Their names are Brown, White, Adams,
Miller, Green, Hunter, Knight, Smith,
and Jones. Determine from the following
information the position played by each
man.
a. Brown and Smith each won $10 playing
poker with the pitcher.
b. Hunter is taller than Knight and shorter
than White, but all three weigh more
than the first baseman.
c. The third baseman lives across the
corridor from Jones in the same apartment
house. Miller and the outfielders play
bridge in their spare time. White, Miller,
Brown, the right fielder, and the center
fielder are bachelors, and the rest are
married.
d. Of Adams and Knight, one plays an
outfield position.
e. The right fielder is shorter than
the center fielder.
f. The third baseman is a brother of
the pitcher's wife.
g. Green is taller than the infielders,
the pitcher, and the catcher except for
Jones, Smith, and Adams.
h. The second baseman beat Jones, Brown,
Hunter, and the catcher at cards.
i. The third baseman, the shortstop,
and Hunter each made $150 speculating
in General Motors stock.
j. The second baseman is engaged to Miller's
sister.
k. Adams lives in the same apartment
house as his own sister but dislikes
the catcher.
l. Adams, Brown, and the shortstop each
lost $200 speculating in grain.
m. The catcher has three daughters, the
third baseman has two sons, and Green
is being sued for divorce.
Build a simple device which will add single digit numbers.
Top
What do the following people have
in common?
Eamon de Valera, prime minister
and past president of the Republic of
Ireland
Tom Lehrer, songwriter-parodist
Edmund Husserl, the "Father of
Phenomenology"
Frank Ryan, past quarterback for
the Cleveland Browns
Write a paper discussing the Egyptian
method of multiplication. Reference: James Newman, The World of Mathematics,
Vol. I (New York: Simon and Schuster,
1956), pp. 170-178.
Howard Eves, Introduction to the History
of Mathematics, 3rd ed. (New York:
Holt, Rinehart, and Winston, 1969).
What are some of the significant
events in the development of mathematics?
Who are some of the famous people who
have contributed to mathematical knowledge?
Reference: Howard Eves, In Mathematical Circles,
Vols. 1 and 2 (Boston: Prindle, Weber,
& Schmidt, 1969), Mathematical Circles
Revisited (1971), Mathematical
Circles Adieu (1977).
Virginia Newell et al.,
Black Mathematicians and Their Works
(Ardmore, PA: Dorrence & Company,
1980). Mona Fabricant, Sylvia
Svitak, and Patricia Clark Kenschaft,
"Why Women Succeed in Mathematics," Mathematics
Teacher, February 1990, pp. 150-154
(with references).
Is it possible to have a numeration
system with a base that is negative?
Before you answer, see "Numeration Systems
with Unusual Bases," by David Ballew,
in The Mathematics Teacher, May
1974, pp. 413-414. Study the topic of
negative bases, and present a report
to the class.
"I became operational at the HAL Plant
in Urbana, Ill., on January 12, 1997,"
the computer HAL declares in Arthur C.
Clarke's 1968 novel, 2001: A Space
Odyssey. Now that time has passed
and many advances have been made in computer
technology between 1968 and today. Write
a paper showing the similarities and
differences between HAL and the computers
of today.
"Software bugs can have devastating
effects, for example the Y2K Millennium
Bug. During the Persian Gulf War, a bug
prevented a Patriot missile from firing
at an incoming Iraqi Scud missile, which
crashed into an Army barracks, killing
28 people. Another example involves Ashton-Tate
company that never recovered its reputation
after shipping bug-filled accounting
software to its customers. Do some research
to find recent (within the last 5 years)
examples of major problems caused by
software bugs.
Build a working model of Napier's
rods.
Write a paper and prepare a classroom
demonstration on the use of an abacus.
Build your own device as a project.
Write a paper regarding the inverntion
of the first electonic computer.
Visit a computer store, talk to a
salesperson about the available computers,
and then write a paper on your experiences.
Write a history of the held-held
calculator.
Find out what local, state, and federal
governments have stored in their computers
about you and your family. Find out what
you can see and what others can see.
This will provide you with an interesting
intellectual journey, if you wish to
take it.
Top
What do the following people have
in common?
Corazon Aquino, former President
of the Philippines
Leon Trotsky, revolutionary
Carole King, singer-songwriter
Heloise (Poncé Cruse Evans),
columnist, Hints from Heloise
Florence Nightingale, pioneer
in professional nursing
Investigate some of the properties
of primes not discussed in the text.
Why are primes important to mathematicians?
Why are primes important in mathematics?
What are some of the important theorems
concerning primes? Reference: Martin Gardner, "The Remarkable Lore
of Prime Numbers," Scientific American,
March 1964.
We mentioned that the Egyptians wrote
their fractions as sums of unit fractions.
Show that every positive fraction less
than 1 can be written as a sum of unit
fractions. References:
Bernhardt Wohlgemuth, "Egyptian Fractions,"
Journal of Recreational Math,
Vol. 5, No. 1 (1972), pp. 55-58.
The Egyptians had a very elaborate
and well developed system for working
with fractions. Write a paper on Egyptian
fractions. References:
George Berzsenyi, "Egyptian Fractions,"
Quantum, November/December 1994,
p. 45. Follow up comment in The College
Mathematics Journal, March 1995, p. 165.
Richard Gillings, Mathematics in the
Time of the Pharaohs (New York: Dover
Publications, 1982).
Spencer Hurd, "Egyptian Fractions: Ahmes
to Fibonacci to Today," Mathematics
Teacher, October 1991, pp. 561-568.
Prove that the positive square root
of 2 is not rational.
Write a paper or prepare an exhibit
illustrating the Pythagorean theorem.
Here are some questions you might consider:
What is the history of the Pythagorean
theorem?
What are some unusual proofs of the Pythagorean
theorem?
What are some of the unusual relationships
that exist among Pythagorean numbers?
What models can be made to visualize
the Pythagorean theorem?
Symmetries of a Cube
Consider a cube labeled as shown below:
List all the possible symmetries of this
cube. See Problem 60 in, Problem Set
5.6 to help you get started.
What is a Diophantine equation? Reference: Warren J. Himmelberger, "Puzzle Problems
and Diophantine Equations," The Mathematics
Teacher, February 1973, 136-138,
or a more complete reference, see any
number theory textbook.
Prepare an exhibit on cryptography.
Include devices or charts for writing
and deciphering codes, coded messages,
and illustrations of famous codes from
history. For example, codes are found
in literature in Before the Curtain
Falls, by J. Rives Childs; The
Gold Bug, by Edgar Allan Poe, Voyage
to the Center of Earth, by Jules
Verne. Reference: Andree, Richard V., "Cryptography
as a Branch of Mathematics," The Mathematics
Teacher, November 1952.
Gardner, Martin, "Mathematical Games
Department," Scientific American,
August 1972, pp. 114-118.
Shasta, Dennis, Codes, Puzzles, and
Conspiracy, Menlo Park, CA: Dale
Seymour Publications, 1993.
Write a paper on the importance of
cryptography for the internet. You might
begin with the Scientific American article
by Philip Zimmerman and conclude with
the RSA "Secret-Key Challenge." Zimmerman,
Philip, "Cryptography for the Internet,"
Scientific American, October,
1998, pp. 110-115.
RSA Data Security Secret-Key
Challenge:
http://www.rsasecurity.com/ Note: If you look at
the page Status and Prizes, you will
see the contest RC5-32/12/5 This $10,000
prize was won by deciphering the following
secret message: "Strong cryptography
makes the world a better place."
Top
What do the following people have
in common?
Carl T. Rowan, columnist for the
Washington Post
Lewis Carroll (Charles Dodgson)
author of Alice in Wonderland
Christopher Wren, architect of
St. Paul's Cathedral in London
Virginia Wade, tennis player,
Wimbledon champion
Lawrence Leighton Smith, conductor
and pianist
Write a paper on the relationship
between geometric areas and algebraic
expressions. Reference: Albert B. Bennett, Jr., "Visual Thinking
and Number Relationships," The Mathematics
Teacher, April, 1988.
Robert L. Kimball, "Sharing Teaching
Ideas: Using Pattern Analysis to Determine
the Squares of Three Consecutive Integers,"
The Mathematics Teacher, January
1986.
Write out a derivation of the quadratic
formula.
You can check almost
any high school algebra book.
A source I recommend
highly is the movie The Proof one of
PBS's shows on the NOVA series. The web
page for this move is:
http://www.pbs.org/wgbh/nova/proof/
Another general overview is at this site:
http://www-history.mcs.st-and.ac.uk/history/HistTopics/Fermat's_last_theorem.html
Top
What do the following people have
in common?
Eamon de Valera, Prime minister
and President of the Republic of Ireland
Tom Lehrer, songwriter-parodist
Edmund Husserl, the "Father of
Phenomenology"
Frank Ryan, quarterback for the
Cleveland Browns
What are optical illusions?
What is the following optical illusion:
(From Problem 1 of Section
7.1.)
Find some unusual optical illusions and
illustrate with charts, models, advertisements,
pictures, or illusions. Reference: Martin Gardner, "Mathematical Games,"
Scientific American, May 1970.
Richard Gregory, "Visual Illusions,"
Scientific American, November
1968.
Lionel Penrose, "Impossible Objects:
A Special Type of Visual Illusion," The
British Journal of Psychology, February
1958.
Jim Meador, "Pool Illusions," web site
found at:
http://www.billiardworld.com/puzzles.html
Many curves can be illustrated
by using only straight line segments.
The basic design is drawn by starting
with an angle, as shown below. Procedure for basic
angle design for aestheometry Step 1:
Draw an angle with two sides of equal
length
 Step 2: Mark off equally
distant units on both rays using a compass
Step 3: Connect #1 to #1; connect #2s,
#3s, ...
 The result is called
aestheometry and is depicted below. Make
up your own angle design.
 |
 |
 |
| a.
Angle design |
b.
Angle design |
c.
Circle design |
| Aestheometry
designs |
Many curves can be illustrated by
using only straight line segments. The
basic design is drawn by starting with
an angle, as shown below.
A second basic aestheometric
design (see Project 7.2) begins with
a circle as shown:
 a. Draw a circle and
mark off equally spaced points.
b. Choose any two points and connect
them.
c. Connect succeeding points around the
circle. Construct various designs
using circles or parts of a circle.
In 1928 Frank Ramsey, an English
mathematician, showed that there are
patterns implicit in any large structure.
The accompanying diagram (which resembles
the aestheometric design of Projects
7.3 and 7.4) typifies the problems that
Ramsey theory addresses.
"How many people does
it take to form a group that always contains
either four mutual acquaintances or four
mutual strangers?" In the diagram, points
represent people. A red edge connects
people who are mutual acquaintances,
and a blue edge joins people who are
mutual strangers. In the group of 17
points shown, there are not four points
whose network of edges are either completely
red or completely blue. Therefore, it
takes more than 17 people to guarantee
that there will always be four people
who are either acquaintances or strangers.
Write a report on Ramsey theory. What is Ramsey Tic-Tac-Toe? Reference: Ronald L. Graham and Joel H. Spencer,
"Ramsey Theory," Scientific American,
July 1990, pp. 112-117
Euclid clearly made a distinction
between the definition of a figure and
the proof that such a figure could be
constructed. Two very famous problems
in mathematics focus on this distinction:
1. Trisect an angle using only a straightedge
and compass.
2. Square a circle:
Using only a straightedge and compass,
construct a square with an area equal
to the area of a given circle. Reference:
http://mathforum.org/isaac/problems/pi3.html
This site has an interesting
interactive component to help you to
understand the problem. There are also
links to other sites. Other sites are:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html
These problems have
been proved to be impossible (as compared
with unsolved problems that might be
possible). Write a paper discussing the
nature of an unsolved problem as compared
with an impossible problem.
The Historical Note in Section 7.5
asks the question, Why did the Egyptians
build the pyramids using a slant height
angle of about 44 or 52 degrees? Write
a paper answering this question. Reference: Arthur F. Smith, "Angles of Elevation
to the Pyramids of Egypt," The Mathematics
Teacher (February 1982, pp. 124-127).
Kurt Mendelssohn, The Riddle of the
Pyramids, New York: Praeger Publications,
1974.
Do some research on the length-to-width
ratios of the packaging of common household
items. Form some conclusions. Find some
examples of the golden ratio in art.
Do some research on dynamic symmetry. Reference: Philip J. Davis and Reuben Hersh,
The Mathematical Experience (Boston:
Houghton Mifflin Company, 1981), pp.
168-171
H. E. Huntley, The Divine Proportion:
A Study in Mathematical Beauty (New
York: Dover Publications, 1970).
In Example 2 Section 7.6, we assumed
the width of the Parthenon to be 101
ft and found the height to be 62.4 (assuming
the golden ratio). If you worked Problem
8, you assumed the height to be 60 ft
and found the width using the golden
ratio to be 97 ft. Are the numbers from
Example 2 and from Problem 8 consistent?
Can you draw any conclusions? Reference: George Markowsky, "Misconceptions
About the Golden Ratio," The College
Mathematics Journal, January 1992.
The German artist Albrecht Durer
(1471-1528) is not only a Renaissance
artist, but also somewhat of a mathematician.
Do some research on the mathematics of
Durer. Reference: Charles Lenz "Modeling the Matrix
of Albrecht Durer," Modeling and Simulation,
1979, pp. 2149-2161.
Francis Russell, The World of Durer,
1471-1528 (New York: Time, 1967).
Karen Walton, "Albrecht Durer's Renaissance
Connections between Mathematics and Art,"
The Mathematics Teacher, April
1994, pp. 278-282.
Project 7.11
Write a paper on perspective. How
are three-dimensional objects represented
in two dimensions? Reference: Jan Garner, "Mathematics and Perspective
Drawing,"
http://forum.swarthmore.edu/sum95/math_and/perspective/perspect.html
Morris Kline, Mathematics, a Cultural
Approach (Reading, MA: Addison-Wesley,
1962), Chapters 10-11.
C. Stanley Ogilvy, Excursions in Geometry
(New York: Oxford University Press, 1969),
Chapter 7.
The discovery and acceptance of non-Euclidean
geometries had an impact on all of our
thinking about the nature of scientific
truth. Can we ever know truth in general?
Write a paper on the nature of scientific
laws, the nature of an axiomatic system,
and the implications of non-Euclidean
geometries. Reference: Philip J. Davis and Reuben Hersh,
The Mathematical Experience (Boston:
Houghton-Mifflin, 1981).
Top
In 1928 Frank Ramsey, an English
mathematician, showed that there are
patterns implicit in any large structure.
The accompanying diagram (which resembles
the aestheometric designs in the last
chapter) typifies the problems that Ramsey
theory addresses.
"How many people does
it take to form a group that always contains
either four mutual acquaintances or four
mutual strangers?" In the diagram, points
represent people. A red edge connects
people who are mutual acquaintances,
and a blue edge joins people who are
mutual strangers. In the group of 17
points shown, there are not four points
whose network of edges are either completely
red or completely blue. Therefore, it
takes more than 17 people to guarantee
that there will always be four people
who are either acquaintances or strangers.
Write a report on Ramsey theory. What is Ramsey Tic-Tac-Toe? Reference: Ronald L. Graham and Joel H. Spencer,
"Ramsey Theory," Scientific American,
July 1990, pp. 112-117
The Garden House of Ostia was constructed
in the 2nd century, in the city of Ostia,
whose population reached 50,000 at its
peak. (See the "What in the World" comment
at the beginning of Chapter 7.) This
city was a major port of Rome, which
was about 25 km away. The Garden Houses
are of interest because of the geometry
used in its construction. The key to
its construction, according to archeologists
Donald and Carol Watts, is a "sacred
cut." In searching the records of the
architect Vitruvius they found that the
basic pattern begins with a square (called
the reference square) and its diagonals.
Next quarter circles centered on the
corners of the square are drawn, each
with a radius equal to half of the diagonal.
The arcs pass through the center of the
square and intersect two adjoining sides;
together they cut the sides into three
segments, with the central segment being
larger than the other two. By connecting
the intersection points, you can divide
the reference square into nine parts,
as described in the article. At the center
of the grid is another square that can
serve as the foundation for the next
sacred cut. Experiment by drawing or
quilting some "sacred cut" designs. Reference: "A Roman Apartment Complex," by Donald
J. Watts and Carol Martin Watts. Scientific
American, December 1986, pp. 132-139.
An historical reference, along with a
very interesting site on the Garden Houses
of Ostia is found at this site:
http://www.ostia-antica.org/indexes.htm
The German artist Albrech Durer (1471-1528)
is not only a Renaissance artist, but
also somewhat of a mathematician. Do
some research on the mathematics of Durer.
Make drawings of geometric figures
on a piece of rubber inner tube. Demonstrate
to the class various ways in which these
figures can be distorted.
The problem shown in the News Clip was
first published by John Jackson in 1821.
Without the poetry, the puzzle can be
stated as follows: Arrange nine trees
so they occur in ten rows of three trees
each. Find a solution.
Top
Historical Quest
In the Boston Museum of Fine arts is
a display of carefully made stone cubes
found in the ruins of Mohenjo-Daro of
the Indus. The stones are a set of weights
that exhibit the binary pattern, 1, 2,
4, 8, 16, ... . The fundamental unit
displayed is just a bit lighter than
the ounce in the U.S. measurement system.
The old European standard of 16 oz for
1 pound may be a relic of the same idea.
Write a paper showing how a set of such
stones can successfully be used to measure
any reasonable given weight of more than
one unit.
Project 9.3
Construct models for the regular
polyhedra. References:
H.S.M. Coxeter, Introduction to
Geometry (New York: Wiley, 1961).
Jean Pederson, "Plaited Platonic Puzzles,"
Two-Year College Mathematics Journal,
Fall 1973, pp. 23-27.
Max Sobel and Evan Maletsky, Teaching
Mathematics: A Sourcebook (Englewood
Cliffs, N.J.: Prentice-Hall, 1975), pp.
173-184.
Charles W. Trigg, "Collapsible Models
of the Regular Octahedron," The Mathematics
Teacher, October 1972, pp. 530-533.
Project 9.4
What solids occur in nature? Find
examples of each of the five regular
solids. For example, the skeletons of
marine animals called radiolaria show
each of these forms.
 References:
David Bergamini, Mathematics
(New York: Time, Inc., Life Science Library,
1963), Chapter 4.
Juithlynee Carson, "Fibonacci
Numbers and Pineapple Phyllotaxy," Two-Year
College Mathematics Journal, June
1978, pp. 132-136.
James Newman, The World of
Mathematics (New York: Simon and
Schuster, 1956). "Crystals and the Future
of Physics," pp. 871-881, "On Being the
Right Size," pp. 952-957, and "The Soap
Bubble," pp. 891-900. Radiolaria:
http://www.ucmp.berkeley.edu/protista/radiolaria/rads.html
Historical Question Write an essay on John Napier. Include what he is famous
for today, and what he considered to be his crowning achievement.
Also Include a discussion of "Napier's bones."
Write an essay on earthquakes. In
particular, discuss the Richter scale
for measuring earthquakes. What is its
relationship to logarithms?
Use the following table, which shows
populations for eight cities to answer
the questions. More data are provided
than are required to answer these questions,
so you will need to make some assumptions
to arrive at your prediction. State the
assumptions you are making clearly.
 a. Which city seems
to have the greatest growth rate for
the period 1980-1990? Name the city and
predict its population in the year 2000.
b. Which city seems to have had the greatest
decline in population for the period
1980-1990? Name the city and predict
its population in the year 2000.
c. If the population of Denver, Colorado,
had continued to grow at its 1960-1980
rate, what would its 1990 population
have been? What statement can you make
about the rate of population growth in
Denver in the 1980s? In the 1970s?
From your local chamber of commerce,
obtain the population figures for your
city for the years 1980, 1990, and 2000.
Find the rate of growth for each period.
Forecast the population of your city
for the year 2010. Include charts and
graphs. List some factors, such as new
zoning laws, that could change the growth
rate of your city.
Write an essay on carbon-14 dating.
What is its relationship to logarithms?
Top
Project 11.1
Conduct a survey of banks, savings
and loan companies, and credit unions
in your area. Prepare a report on the
different types of savings accounts available
and the interest rates they pay. Include
methods of payment as well as interest
rates.
Do you expect to live long enough
to be a millionaire? Suppose that your
annual salary today is $39,000. If inflation
continues at 6%, how long will it be
before $39,000 increases to an annual
salary of a million dollars?
Consult an almanac or some government
source, and then write a report on the
current inflation rate. Project some
of these results to the year of your
own expected retirement.
Karen says that she has heard something
about APR rates but doesn't really know
what the term means. Wayne says he thinks
it has something to do with the prime
rate, but he isn't sure what. Write a
short paper explaining APR to Karen and
Wayne.
Some savings and loan companies advertise
that they pay interest continuously.
Do some research to explain what this
means.
Select a car of your choice, find
the list price, and calculate 5% and
10% price offers. Check out available
money sources in your community, and
prepare a report showing the different
costs for the same car. Back up your
figures with data.
Outline a program for your own retirement.
In the process of writing this paper
answer the following questions. You will
need to state your assumptions about
interest and inflation rates.
a. What monthly amount of money today
would provide you a comfortable living?
b. Using the answer to part a project
that amount to your retirement, calculating
the effects of inflation. Use your own
age and assume that you will retire at
age 65.
c. How much money would you need to have
accumulated to provide the amount you
found in part b if you decide to live
on the interest only?
d. If you set up a sinking fund to provide
the amount you found in part c, how much
would you need to deposit each month?
e. Offer some alternatives to a sinking
fund.
f. Draw some conclusions about your retirement. There are a multitude
of sites to help with retirement planning.
Here are a couple of samples:
http://www.troweprice.com/
http://money.cnn.com/retirement/
Top
Project 12.1
Write a paper on the famous Tower
of Hanoi problem. When my daughter was
2 years old, she had a toy that consisted
of colored rings of different sizes:
Suppose you wish to
move the "tower" from stand A to stand
C, and to make this interesting we agree
to the following rules:
1. move only one ring at a time;
2. at no time may a larger ring be placed
on a smaller ring. For three rings it will
take 7 moves (try it).
For four rings it will take 15 moves. The ancient Brahman
priests were to move a pile of 64 such
rings,and the story is that when they
complete this task the world will end.
How many moves would be required, and
if it takes one second per move, how
lond would this take? References
Frederick Schuh, The Masterbook of
Mathematical Recreations (New York:
Dover Publications, 1968).
Michael Schwager, "Another Look at the
Tower of Hanoi," The Mathematics Teacher,
Sepetember 1977, pp. 528-533.
How can all the constructions of
Euclidean be done by paper folding?
What assumptions are made when paper
is folded to construct geometric figures
from Euclidean geometry?
What is orgami?
What is a hexaflexagon?
What is a hexahexaflexagon?
This problem illustrates a numerical
solution for the Instant Insanity puzzle.
Let's associate numbers with the sides
of the cubes of the Instant Insanity
problem. Let White = 1; Blue =
2; Green = 3; Red = 5 
 Now, the product across
the top must be 30, and the product across
the bottom must also be 30 (why?). It
follows that a solution must have a product
of 900 for the faces on the top and bottom.
Consider the four cubes:
Top
What do the following people have
in common?
Paul Painleve, President of France
Omar Khayyam, author of The Rubaiyat
Emanual Lasker, world chess champion
James Moriarty, Sherlock Holmes's
nemesis, author of The Dynamics of an
Asteroid
Devise a fair scheme for eliminating
coins in this country. Here is an alternative
question:
Michael Rossides has come up with a scheme
for eliminating coins. This scheme involves
probability and the fact that most cash
registers today are computers. Suppose
that every cash register could be programmed
with a random number generator; that
is, suppose that it were possible to
pick a random number from 1 to 99. Rossides'
system works as follows. Suppose you
purchase items totaling $15.89. The computer
would choose a number from 1 to 99 and
then compare it with the cents portion
of the purchase. In this case, if the
random number is between 1 and 89 the
price would be rounded up to $16; if
it is between 90 and 99 the price would
be rounded down to $15. For example,
if you purchase a cup of coffee for $1.20,
and the random number generator produces
a random number from 1 to 20, the price
is $2, but if it produces a number from
21 to 99 the price is $1. Write a paper
commenting on this scheme.
Find the probability that the 13th
day of a randomly chosen month will be
a Friday. (Hint: It is not correct to
say that the probability is 1/7 because
there are seven possible days in the
week. In fact, it turns out that the
13th day of a month is more likely a
Friday than any other day of the week.)
Write a paper discussing this problem. Reference: William Bailey, "Friday-the-Thirteenth,"
The Mathematics Teacher, Vol.
LXII, No. 5 (1969), pp. 363-364.
C. V. Heuer, Solution to Problem E1541
in American Mathematical Monthly,
Vol. 70, No. 7 (1963), p. 759.
G. L. Ritter, et al, "An Aid to the Superstitious,"
The Mathematics Teacher, Vol.
70, No. 5 (1977), pp. 456-457.
The questions in this problem are
from a study by MacCrimmon, Stanbury,
and Wehrung, "Real Money Lotteries: A
Study of Ideal Risk, Context Effects,
and Simple Processes," in Cognitive
Processes in Choice and Decision Behavior,
edited by Thomas Wallsten, (Hillsdale,
N. J.: Lawrence Erlbaum Associates, 1980,
pp. 155-179). Question: You
have five alternatives from which to
choose. List your preferences for the
alternatives from best to worst. 1. sure win of $5 and
no chance of loss
2. 6.92% chance to win $20 and 93.08%
chance to win $3.98
3. 27.52% chance to win $20 and 72.48%
chance to lose 69 cents
4. 61.85% chance to win $20 and 38.15%
chance to lose $19.31
5. 90.46% chance to win $20 and 9.54%
chance to lose $137.20
a. Answer the question based on your
own feelings.
b. Answer the question using mathematical
expectation as a basis for selecting
your answer.
c. Conduct a survey of at least 10 people
and summarize your results.
d. What are the conclusions of the study.
Project 13.5
a. Answer each of the following questions*
based on your own feelings.
b. Answer questions (1) and (2) using
mathematical expectations as a basis
for selecting your answers.
c. Conduct a survey of at least 10 people
and summarize your results.
1. Choose between A and B:
A. A sure gain of $240
B. A 25% change to gain $1,000 and a
75% chance to gain $0
2. Choose between C and D:
C. A sure loss of $700
D. 75% chance to loose $1,000 and %25
change to lose nothing
3. Choose between E and F
E. Imagine that you have decided to see
a concert and have paid the admission
price of $10. As you enter the concert
hall, you discover that you have lost
the ticket. Do you pay $10 for another
ticket?
F. Imagine that you have decided to see
a concert where the admission is $10.
As you going to enter the concert ticket
line, you discover that you have lost
one of your $10 bills. Would you still
pay $10 for a ticket to the concert?
* These questions are from A Bird
in the Hand, by Carolyn Richbart
and Lynn Richbart in The Mathematics
Teacher, November 1996, pp. 674-676.
Do some research on Keno probabilities.
Write a paper on playing Keno. Reference: Karl J. Smith, "Keno Expectation,"
Two-Year College Mathematics Journal,
Vol. 3, No. 2, Fall 1972.
Top
"You can clearly see that Bufferin
is the most effective ... " "Penzoil
is better suited ... " "Sylvania was
preferred by ... ." "How can anyone analyze
the claims of the commercials we see
and hear on a daily basis?" asked Betty.
"I even subscribe to Consumer Reports,
but so many of the claims seem to be
unreasonable. I don't like to buy items
by trial and error, and I really don't
believe all the claims in advertisements."
Collect examples of good statistical
graphs and examples of misleading graphs.
Use some of the leading newspapers and
national magazines, or websites:
http://www.fedstats.gov
http://www.cdc.gov/nchswww/default.htm
http://www.lib.umich.edu/govdocs/stats.html
Carry out the following experiment:
A cat has two bowls of food, one bowl
contains Whiskas and the other some other
brand. The cat eats Whiskas and leaves
the other untouched. Make a list of possible
reasons why the cat ignored the second
bowl. Describe the circumstances under
which you think the advertiser could
claim: Eight out of ten owners said their
cat preferred Whiskas.
If you roll a pair of dice 36 times,
the expected number of times for rolling
each of the numbers is given in the accompanying
table. A graph of these data is shown. a. Find the mean, the
variance, and the standard deviation
for this model.
 b. Roll a pair of dice
36 times. Construct a table and a graph
similar to the ones shown in Figure 10.
Find the mean, the variance, and the
standard deviation for your experiment.
c. Compare the results
of parts a and b. If this is a class
problem, you might wish to pool from
the entire class data before making the
comparison.
Prepare a report or exhibit showing
how statistics are used in baseball.
Prepare a report or exhibit showing
how statistics are used in educational
testing.
Prepare a report or exhibit showing
how statistics are used in psychology.
Prepare a report or exhibit showing
how statistics are used in business.
Use a daily report of transactions on
the New York Stock Exchange. What inferences
can you make from the information reported?
Investigate the work of Adolph Quetelet,
Francis Galton, Karl Pearson, R. A. Fisher,
and Florence Nightingale. Prepare a report
or an exhibit of their work in statistics.
"We need privacy and a consistent
wind," said Wilbur. "Did you write to
the Weather Bureau to find a suitable
location?"
"Well," replied Orville, "I received
this list of possible locations and Kitty
Hawk, North Carolina, looks like just
what we want. Look at this . . ."
However, Orville and Wilbur spent many
days waiting in frustration after they
arrived in Kitty Hawk, because the winds
weren't suitable. The Weather Bureau's
information gave the averages, but the
Wright brothers didn't realize that an
acceptable average can be produced by
unacceptable extremes.
Write a paper explaining how it is possible
to have an acceptable average produced
by unacceptable extremes.
Select something that you think might
be normally distributed (for example,
the ring size of students at your college).
Next, select 100 people and make the
appropriate measurements (in this example,
ring size). Calculate the mean and standard
deviation. Illustrate your finding using
a bar graph. Do your data appear to be
normally distributed?
Top
This problem is a continuation of
Problem 30. Sectionn 15.2. A player's
batting average really isn't simply the
ratio h/a. It is the value of h/a rounded
to the nearest thousandth. It is possible
that a batting average could be raised
or lowered, but the reported batting
average might remain the same when rounded.
Write a paper on this topic. Reference: James M. Sconyers, "Serendipity:
Batting Averages to Greatest Integers,"
The Mathematics Teacher, April
1980, pp. 278-280.
The population in California was
31,910,000 in January 1995, and 32,231,000
in January 1996. Predict California's
population in the year 2005. Check the
Internet or an almanac to verify the
2005 population using the information
of this problem.
The population in Sebastopol, California
was 7,475 in January 1995, and 7,525
in January 1996. Predict Sebastopol's
population in the year 2005.
Predict the population of your city
or state for the year 2005.
Write a short paper exploring the
concept of the eccentricity of an ellipse.
The figure shown here shows some ellipses
with the same vertices, but different
eccentricities.
Top
Historical Question In the Boston
Museum of Fine arts is a display of carefully
made stone cubes found in the ruins of
Mohenjo-Daro of the Indus. The stones
are a set of weights that exhibit the
binary pattern, 1, 2, 4, 8, 16, ... .
The fundamental unit displayed is just
a bit lighter than the ounce in the U.S.
measurement system. The old European
standard of 16 oz for 1 pound may be
a relic of the same idea. Write a paper
showing how a set of such stones can
successfully be used to measure any reasonable
given weight of more than one unit.
Top
Research how voting is conducted
for the following events. Use the terminology
of this chapter, not the terminology
used in the original sources.
a. Heisman Trophy Award
b. Selecting an Olympic host city
c. The Academy Awards
d. The Nobel Prizes
e. The Pulitzer Prize
Compare and contrast the voting paradoxes.
Which one do you find the most disturbing,
and why? Which do you find the least
distrubing, and why?
Compare and contrast the different
apportionment plans. Which one do you
think is best? Support your position
with examples and facts.
Compare and contrast the apportionment
paradoxes. Which one of these do you
find the most disturbing,and why? Which
one of these do you find the least distrubing,
and why?
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