The Nature of Mathematics, 12th Edition
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Chapter 1 Individual Research Projects

Project 1.1

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

Project 1.2

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)

Project 1.3

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).

Project 1.4

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

Project 1.5

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.

Project 1.6

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.

Project 1.7

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.

Project 1.8

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?

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Chapter 2 Individual Research Projects

Project 2.1

What is the millionth positive integer that is not a square or a cube?

Project 2.2

What is the millionth positive integer that is not a square, cube, or fifth power?

Project 2.3

Write a report discussing the creation of colors using additive color mixing and subtractive color mixing.

Project 2.4

Let S be the set of all real numbers between 0 and 1. Prove that S is not a countable set.

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Chapter 3 Individual Research Projects

Project 3.1

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

Project 3.2

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."

Project 3.3

Do some research to explain the differences between the words necessary and sufficient.

Project 3.4

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

Project 3.5

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.

Project 3.6

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.

Project 3.7

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?

Project 3.8

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?

Project 3.9

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.

Project 3.10

Build a simple device which will add single digit numbers.

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Chapter 4 Individual Research Projects

Project 4.1

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

Project 4.2

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).

Project 4.3

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).

Project 4.4

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.

Project 4.5

"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.

Project 4.6

"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.

Project 4.7

Build a working model of Napier's rods.

Project 4.8

Write a paper and prepare a classroom demonstration on the use of an abacus. Build your own device as a project.

Project 4.9

Write a paper regarding the inverntion of the first electonic computer.

Project 4.10

Visit a computer store, talk to a salesperson about the available computers, and then write a paper on your experiences.

Project 4.11

Write a history of the held-held calculator.

Project 4.12

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.

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Chapter 5 Individual Research Projects

Project 5.1

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

Project 5.2

Project 5.3

Project 5.4

Project 5.5

Project 5.6

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.

Project 5.7

Project 5.8

Project 5.9

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.

Project 5.10

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.

Project 5.11

Project 5.12

Prove that the positive square root of 2 is not rational.

Project 5.13

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?

Project 5.14

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.

Project 5.15

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.

Project 5.16

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.

Project 5.17

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."

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Chapter 6 Individual Research Projects

Project 6.1

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

Project 6.2

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.

Project 6.3

Write out a derivation of the quadratic formula.

References:

You can check almost any high school algebra book.

Project 6.4

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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Chapter 7 Individual Research Projects

Project 7.1

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

Project 7.2

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

Project 7.3

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

Project 7.4

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.

Project 7.5

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

Project 7.6

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.

Reference:

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.

Project 7.7

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.

Project 7.8

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).

Project 7.9

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.

Project 7.10

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.

Project 7.12

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).

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Chapter 8 Individual Research Projects

Project 8.1

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

Project 8.2

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

 

Project 8.3

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.

Project 8.4

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.

Project 8.5


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.

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Chapter 9 Individual Research Projects

Project 9.1

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 9.2

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

Chapter 10 Individual Research Projects

Project 10.1

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."

Project 10.2

Write an essay on earthquakes. In particular, discuss the Richter scale for measuring earthquakes. What is its relationship to logarithms?

Project 10.3

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?

Project 10.4

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.

Project 10.5

Write an essay on carbon-14 dating. What is its relationship to logarithms?

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Chapter 11 Individual Research Projects

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.

Project 11.2

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?

Project 11.3

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.

Project 11.4

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.

Project 11.5

Some savings and loan companies advertise that they pay interest continuously. Do some research to explain what this means.

Project 11.6

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.

Project 11.7

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/

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Chapter 12 Individual Research Projects

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.

Project 12.2

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?

Project 12.3

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:

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Chapter 13 Individual Research Projects

Project 13.1

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

Project 13.2

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.

Project 13.3

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.

Project 13.4

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.

Project 13.6

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.

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Chapter 14 Individual Research Projects

Project 14.1

"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

 

Project 14.2

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.

Project 14.3

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.

Project 14.4

Prepare a report or exhibit showing how statistics are used in baseball.

Project 14.5

Prepare a report or exhibit showing how statistics are used in educational testing.

Project 14.6

Prepare a report or exhibit showing how statistics are used in psychology.

Project 14.7

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?

Project 14.8

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.

Project 14.9

"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.

Project 14.10

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?

Project 14.11

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Chapter 15 Individual Research Projects

Project 15.1

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.

Project 15.2

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.

Project 15.3

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.

Project 15.4

Predict the population of your city or state for the year 2005.

Project 15.5

Project 15.6

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.

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Chapter 16 Individual Research Projects

Project 16.1

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.

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Chapter 17 Individual Research Projects

Project 17.1

    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

Project 17.2

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?

Project 17.3

Compare and contrast the different apportionment plans. Which one do you think is best? Support your position with examples and facts.

Project 17.4

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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