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 (over 50 excellent
sources):
http://search.yahoo.com/bin/search?p=math+puzzles
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://mathworld.wolfram.com/MagicSquare.html
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?
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.
References
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?
References
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.
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?
References
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 4.6 to help you get started.
What is a Diophantine equation?
References
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.
References
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."
References
Zimmerman, Philip, "Cryptography for the Internet," Scientific
American, October, 1998, pp. 110-115.
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.
References
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.
References
You can check almost any high school algebra book.
References
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
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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."
References
http://www.maxmon.com/1600ad.htm
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."
References
http://www.maxmon.com/1600ad.htm
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 1970, 1980, and 1990. Find the rate of growth for each period. Forecast the population
of your city for the year 2000. 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?
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.
References
There are a multitude of sites to help with retirement planning. Here are a couple of samples:
http://www.troweprice.com/common/index3/0,3011,lnp%253D10106%2526cg%253D920%2526pgid%253D7592,00.html?van=retirement
http://www.aoa.dhhs.gov/retirement/fpfr.html
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This problem is a continuation of Problem 30. 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 2000. Check the Internet or an almanac
to verify the 2000 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.
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.
Write a report on Ramsey theory.
The Garden House of Ostia was constructed in the 2nd century, in the city of Ostia,
whose population reached 50,000 at its peak. 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.
References:
"A Roman Apartment Complex," by Donald J. Watts
and Carol Martin Watts. Scientific American, December 1986, pp. 132-139.
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.
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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