## Chapter 1 Individual Research Projects

### Project 1.1

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 1.2

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://mathforum.org/alejandre/magic.square/loshu2.html**

### Project 1.3

Design a piece of art based on a magic square.

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 1.4

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.

**a.** Verify that this is an alphamagic square.

**b.** Find another alphamagic square.

### Project 1.5

Answer the question posed in Problem 59, Section 1.3 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 1.6

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?

## Chapter 2 Individual Research Projects

### Project 2.1

The *KenKen* puzzle is an arithmetical puzzle that ties together sets, magic squares, and suduko-type puzzles.

Each of these puzzles uses the set consisting of the first *n* counting numbers. The rules are simple:

- Fill in the grid with the numbers 1 to without repeating a number in any row or column.
- The outlined portions are called
*cages*and these

portions must combine, in any order, to produce the given number using the given operation. - A number may be repeated in a cage as long as it is not in the same row or column.

Try solving the following KenKen puzzle.

Check out the KenKen website:

http://www-http://www.kenken.com/

### Project 2.2

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

### Project 2.3

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

### Project 2.4

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

### Project 2.5

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

## Chapter 3 Individual Research Projects

### Project 3.1 **Deck of Cards Puzzle**

Here is a problem by Bob Stanton from the 1998 issue of *Games* magazine which you can use to test your logical reasoning.

*Note*: the ace of hearts is part of the four-of-a-kind hand.”

### Project 3.2

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

### Project 3.3

Sometimes statements *p* and *q* are described as *contradictory*, *contrary*, or *inconsistent*. Consult a logic text, and then define these terms using truth tables.

**References:**

The Internet Encyclopedia of Philosophy:

**http://www.utm.edu/research/iep/u/universa.htm**

### Project 3.4

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 3.5

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 3.6 **Present Problem**

Happy Harry wrapped up a present for his wife, and he put a gift inside a small box. He then put this box inside a second box, and then placed the second box inside a third. He completed his task by butting the third box inside a larger fourth box. He tells his wife she can have the present, but only if she can identify the order of the boxes before opening them. Harry

gives the following clues:

*Clue 1*: The boxes (in no particular order are red, blue, gold, and black.

*Clue 2*: The gold box is not inside the blue box.

*Clue 3*: The blue box is inside two other boxes.

*Clue 4*: The black box is bigger than the red box, but smaller than the gold box.

Can you figure out the order of the boxes from smallest (first) to the largest (fourth)?

### 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

A man is about to be electrocuted but is given a chance to save his life. In the execution chamber are two chairs, labeled 1 and 2, and a jailer. One chair is electrified; the other is not. The prisoner must sit on one of the chairs, but before doing so, he may ask the jailer one question, to which the jailer must answer “yes” or “no.” The jailer is a consistent liar or else a consistent truth teller, but the prisoner does not know which. Knowing that the jailer either deliberately lies or

faithfully tells the truth, what question should the prisoner ask?

### Project 3.9

Build a simple device which will add single digit numbers.

**Reference:**

Circuit design

http://en.wikipedia.org/wiki/Adder_(electronics)

**http://cpuville.com/adder.htm **

## Chapter 4 Individual Research Projects

### Project 4.1

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).

### Project 4.2

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).

### Project 4.3

The people from the Long Lost Land had only the following four symbols as shown in the text. Write out the first 20 numbers.

Find

**a.** square plus star and

**b**. star times star.

(Use your imagination to invent a system to answer these questions.)

### 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. 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 invention of the first electronic computer.

### Project 4.10

In Chapter 1, we introduced Pascal’s triangle. The reproduction here is from a 14th century Chinese manuscript, and in this form is sometimes called Yang Hui’s triangle.

Even though we have not discussed these ancient Chinese numerals, see if you can reconstruct the basics of their numeration system. (Use your imagination.)

### Project 4.11 **Historical Quest**

The Ionic Greek numeration system approximately 3000 BC ) counts using the Greek letters. Do some research on this numeration system and report on the basics of this numeration system. If there are gaps in your research, use you imagination to complete the story you tell about this system.

## Chapter 5 Individual Research Projects

### Project 5.1

In the text we tried some formulas that might have generated only primes, but, alas, they failed. Below are some other formulas. Show that these, too, do not generate only primes.

a. *n ^{2} + n* + 41

b.

*n*79

^{2}−*n*+ 1,601

c. 2

*n*+ 29

^{2}d. 9

*n*498

^{2}−*n*+ 6,683

e.

*n*+ 1,

^{2}*n*an even integer

### Project 5.2

For what values of *n* is 11*14^{n} + 1 a prime?

*Hint*: Consider *n* even, and then consider *n *odd.

### Project 5.3

A formula that generates all prime numbers is given by David Dunlop and Thomas Sigmund in their book *Problem Solving with the Programmable Calculator *(Englewood Cliffs, N.J.: Prentice-Hall, 1983). The authors claim that the formula square root of (1 + 24*n*) produces every prime number except 2 and 3, but give no proof or reference to a proof. Create a table, and give an argument to support or find a counterexample to disprove their claim.

### Project 5.4

A large prime, 2^{30,402,457} , is a number that has 9,152,052 digits. A number this large is hard to comprehend. Write a paper making the size of this number meaningful to a nonmathematical reader.

### Project 5.5

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 5.6 ** Historical Quest**

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.

**Reference:**

**Bernhardt Wohlgemuth**, “Egyptian Fractions,” *Journal of Recreational Math,* Vol. 5, No. 1 (1972), pp. 55-58.

### Project 5.7 **Historical Quest**

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 5.8

You may have seen a geoboard, which is simply a board with pegs. Two simple geoboards with the pegs labeled A, B, C, ….

**a.** 2 x 2 geoboard **b.** 3 x 3 geoboard

**Distance on a Geoboard**

Suppose that the horizontal and vertical distance between the pegs is 1 inch. We can then calculate the distance from A to D to be the square root of two (by the Pythagorean theorem).m If we adopt the agreement that the distance from any point to itself is 0, we can find the distance from any peg to any other peg.

Use the 3 x 3 peg board shown here to fill in the entries in the following table. Note that some of the entries have been filled in for you.

### Project 5.9

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

### Project 5.10 **Historical Quest**

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 5.11 **Symmetries of a Cube**

Consider a cube labeled as shown below:

List all the possible symmetries of this cube. See Problem 60, Problem Set 5.6 to help you get started.

### Project 5.12

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 5.13

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:**

**Richard Andree**, “Cryptography as a Branch of Mathematics,” *The Mathematics Teacher,* November 1952.

**Martin Gardner**, “Mathematical Games Department,” *Scientific American,* August 1972, pp. 114-118.

**Dennis Shasta**,* Codes, Puzzles, and Conspiracy, *Menlo Park, CA: Dale Seymour Publications, 1993.

## Chapter 6 Individual Research Projects

### Project 6.1

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.

### Project 6.2

Write out a derivation of the quadratic formula.

*References:*

You can check almost any high school algebra book.

### Project 6.3

Find any replacements for *x*,* y,* and *z* such that*x ^{n}* +

*y*=

^{n}*z*, where

^{n}*n*is greater than 2 and where

*x, y,*and

*z*are counting numbers. Write a history of this problem, known as Fermat’s Last Theorem.

*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

### Project 6.4

Solve the following equation for *x*:

10−9(9−8(8−7(7−6(6−5(5−4(4−3(3−2(2−*x*))))))))= *x*

### Project 6.5

The idea of solving an equation sometimes uses the idea of a “balance” scale. If the balance point is not in the center, then the distance times the weight must be the same for the mobile to balance. For example, a weight of 2 units hanging three units from the pivot point will balance a 1 unit hanging 6 units from the pivot point. For the following puzzle, insert the weights through so that the mobile balances.

## Chapter 7 Individual Research Projects

### Project 7.1

What are optical illusions?

What are the following optical illusions:

(From Problems 1-4 of Section 7.1.) Find some unusual optical illusions and illustrate with charts, models, advertisements, pictures, or illusions.

*References: *

**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 7.2

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.

### Aestheometry Designs

### Project 7.3

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 7.4

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:*

*Trisecting:*

See if you can find the error in the given “proof.”

**http://www.cut-the-knot.org/pythagoras/archi.shtml**

**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 7.5 **Historical Quest**

The Historical Note in Problem Set 7.5 asks the question, Why did the Egyptians build the pyramids using a slant height angle of about 44° or 52°? Write a paper answering this question.

*References: *

**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 7.6

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.

*References: *

**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 7.7

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 7.8

What is the golden ratio?

What is the silver ratio?

Write a paper on the silver ratio.

### Project 7.9

Write a paper on perspective. How are three-dimensional objects represented in two dimensions?

*References: *

**http://mathforum.org/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 7.10

Is there a “best” rectangle. If you like to do origami, which size paper best suits your needs? What are “standard” sizes for paper?

### Project 7.11

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.

### Project 7.12

Find the one composite number in the following set:

31

331

3331

33331

333331

3333331

33333331

333333331

## Chapter 8 Individual Research Projects

### Project 8.1

Suppose you drive you car in a roundabout traffic circle that is 100 ft in diameter. Assume your tires are on the circumference of the circle, and also assume that your car is six feet wide.

- If you drive around the roundabout once, how much further (to the nearest foot) have your outer tires traveled than your inner tires?
- Now suppose the track was 1,000 miles in diameter. How much further would the outer tries travel?
- Finally, answer if the track were 1,000,000 miles in diameter.

### Project 8.2 **A 51-star Flag**

Recently there has been talk of Puerto Rico becoming the 51st state. If that happens, we will need to have a flag with 51 stars. Design a flag with 51 stars discussing the pros and cons of various possibilities.

### Project 8.3

Write a paper on Pythagorean triples (numbers *a*, *b*, and *c* that satisfy the Pythagorean theorem). For example, pick an odd positive integer; say 9.

Square it: 81.

Find consecutive integers that have the sum 81: 40 and 41.

You have generated a Pythagorean triple: 9, 40, 41

^{2}+ 40

^{2}= 41

^{2}

### Project 8.4

Look at Project 8.3, and extend the idea of a Pythagorean triplet. A Pythagorean quartet are four numbers *a*, *b*, *c* and *d* so that

*a*

^{2}+

*b*

^{2}+

*c*

^{2}=

*d*

^{2}

Outline a method for finding Pythagorean quartets.

What do you think is meant by a Pythagorean pentad?

### Project 8.5

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 8.6

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**

### Project 8.7

What is a Menger sponge, and what is interesting about this figure?

## Chapter 9 Individual Research Projects

### Project 9.1

In 1928 Frank Ramsey, an English mathematician, showed that there are patterns implicit in any large structure. The accompanying diagram (which resembles the aestheometric design of Projects 7.2 and 7.3) 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 9.2 **Historical Quest**

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 archaeologists 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.

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 9.3 **Historical Quest **

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 9.4

Prepare a classroom demonstration of topology by drawing geometric figures on a piece of rubber inner tube. Demonstrate to the class various ways in which these figures can be distorted.

### Project 9.5

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.

### Project 9.6 **Celebrate the Millennium**

**a.** Consider the product (1)(2)(3) … (1,998)(1,999)(2,000). What is the last digit?

**b.** Consider the product (1)(2)(3)… (1,998)(1,999)(2,000). From the product, cross

out all even factors, as well as all multiples of 5. Now, what is the last digit of the resulting product?

### Project 9.7 **Garment topology**

The garment industry uses mathematics and topology to investigate the draping behavior of a scanned garment model in irregular and regular mesh topology. Write an informational paper of at least one page regarding the topic of garment topology.

### Project 9.8 **What Good Are Fractals?**

Write an essay on the use and value of fractals.

## Chapter 10 Individual Research Projects

### Project 10.1 ** Historical Quest**

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.”

*Reference:*

**http://www.mathsisgoodforyou.com/artefacts/napiersbones.htm**

### 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

Write a paper using population analysis.

To get you started, select five U.S. cities of different sizes. Find the population of these cites in 1980, 1990, 2000, and 2010.

**a.** Which city seems to have the greatest growth rate for the period 1980-2010? Name the city and predict its population in 2020.

**b.** Which city seems to have had the greatest decline in population for the period 1980-2010? Name the city and predict its population in the year 2020.

### Project 10.4

From your local chamber of commerce, obtain the population figures for your city for the years 1980, 1990, 2000 and 2010. 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?

### Project 10.6

A store increases its prices 10% every odd month (January, March, May, …) and decreases them by 10% every even month (February, April, June, …). Do prices increase, decrease or stay the same over a long period of time?

## 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 ** Historical Quest**

Write a short paper about Fibonacci numbers.

You might check *The Fibonacci Quarterly*, particularly “A Primer on the Fibonacci Sequence,” Parts I and II, in the February and April 1963 issues. The articles, written by **Verner Hogatt** and **S. L. Basin**, are considered classic articles on the subject. You should investigate the relationship of the Fibonacci numbers to nature, as well as the algebraic properties of the sequence. You might also include the history of the sequence.

### Project 11.5

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 11.6>

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

### Project 11.7

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 11.8

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://money.cnn.com/retirement/**

### Project 11.9

Suppose you were hired for a job paying $21,000 per year and were given the following options:

*Option A:* Annual salary increase of $1,200.

*Option B:* Semiannual salary increase of $300.

*Option C:* Quarterly salary increase of $75.

*Option D:* Monthly salary increase of $10.

Write the arithmetic series for the total amount of money earned in 10 years under a different option. Which is the best option? Give reasons and show your calculations in the paper that you submit.

## 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:

- move only one ring at a time;
- 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 long 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,* September 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: Here are the products of top and bottom for the three possible arrangements of the first cube:

*Product*

Top: | Red = 5 | |

Bottom: | Red = 5 | 25 |

Top: | Blue = 2 | |

Bottom: | White = 1 | 2 |

Top: | Red = 5 | |

Bottom: | Green = 3 | 15 |

By a clever and systematic analysis of the products of top and bottom for cubes two, three, and four, you will find there are several ways of solving the top and bottom for a product of 900. Only one of these gives a product of 900 for front and back. This is the solution to the *Instant Insanity* puzzle. Find the solution using this method.

### Project 12.4

A puzzle sold under the name The Avenger, is pictured.

There are four problems posed in the article shown in the reference. Write a report on this article.

*References: *

“Group Theory, Rubik’s Cube and The Avenger,” *Games* , June/July 1987, pp. 44-45.

### Project 12.5

Consult one of the references and learn to solve Rubik’s cube.

Demonstrate your skill to the class. Nourse names the following categories:

20 minutes: WHIZ

10 minutes: SPEED DEMON

5 minutes: EXPERT

3 minutes: MASTER OF THE CUBE

- Stage a contest in front of the class to see which contestant can complete one face of a Rubik’s cube.
- Stage a contest to see who can solve the Rubik’s cube puzzle the fastest. Report the results to the class.

** References:**

**Ledbetter** and **Nering**,* The Solution to Rubik’s Cube *(Rohnert Park, CA, Noah’s Ark Enterprises, 1980).

**James G. Nourse**, *The Simple Solution to Rubik’s Cube *(New York: Bantam Books, 1981).

**David Singmaster**, *Notes on Rubik’s “Magic Cube,”* 5th ed. Hillside, N.J., Enslow Publishers, 1980).

## Chapter 13 Individual Research Projects

### Project 13.1

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 13.2

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.

**References:**

**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 13.3

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.

- sure win of $5 and no chance of loss
- 6.92% chance to win $20 and 93.08% chance to win $3.98
- 27.52% chance to win $20 and 72.48% chance to lose 69 cents
- 61.85% chance to win $20 and 38.15% chance to lose $19.31
- 90.46% chance to win $20 and 9.54% chance to lose $137.20
- Answer the question based on your own feelings.
- Answer the question using mathematical expectation as a basis for selecting

your answer. - Conduct a survey of at least 10 people and summarize your results.
- What are the conclusions of the study.

### Project 13.4

- Choose between A and B:
- A sure gain of $240
- 25% chance to gain $1,000 and 75% chance to gain $0

- Choose between C and D:
- A sure loss of $700
- 75% chance to lose $1,000 and 25% chance to lose nothing

- Choose between E and F:
- 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 your ticket. Would you pay $10 for another ticket?
- Imagine that you have decided to see a concert where the admission is $10. As you begin 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?

- Answer each of the questions based on your own feelings.
- Answer questions a and b using mathematical expectation as a basis for selecting your answers.
- Conduct a survey of at least 10 people and summarize your results. Answer each of the following questions* based on your own feelings.

* 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 13.5

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.

### Project 13.6

Consider the following classroom activity. Suppose the floor consists of square tiles 9 in. on each side. The players will toss a circular disk onto the floor. If the disk comes to rest on the edge of any tile, the player loses $1. Otherwise, the player wins $1. What is the probability of winning if the disk is:

**a.** a dime

**b.** a quarter

**c.** a disk with a diameter of 4 in.

**d.** Now, the real question: What size should the disk be so that the probability that the player wins is 0.45?

## 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.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 above. 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 **Historical Quest**

“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?

## Chapter 15 Individual Research Projects

### Project 15.1

This problem is a continuation of Problem 60, Section 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 33,871,648 in January 2000, and 37,353,956 in January 2010. Predict California’s population in the year 2020.

### Project 15.3

The population in Sebastopol, California was 7,475 in January 2000, and 7,598 in January 2013. Predict Sebastopol’s population in the year 2020.

### Project 15.4

Predict the population of your city or state for the year 2020.

### Project 15.5

Before GPS, ships at sea locate their positions using the LOng RAnge Navigation system known as LORAN. In this system, which ceased operation in 2010, a master station sends signals that can be received by ships at sea. To fix the position of a particular ship, a secondary sending station also emits signals that can be received by the ship. Since the ship monitoring the two signals will usually be nearer one of the two stations, there will be a difference in the distances that the two signals travel. Because *d = rt*, there will be a slight time difference between the signals. If the ship follows a path so that the time difference remains constant, what is the path the ship will follow? Suppose the difference in the arrival of the time signals is 300 microseconds. (*Note*: a microsecond is one millionth of a second.) Also, suppose that the foci are 100 miles apart. Finally, suppose that signals travel at 980 ft/sec.

### 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.

## Chapter 16 Individual Research Projects

### Project 16.1 **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 16.2

Five identical containers (shoe boxes, paper cups, etc.) must be prepared for this problem, with contents as follows: There are five boxes containing red and white items (such as marbles, poker chips, or colored slips of paper).

*Box Contents*

#1 15 red and 15 white

#2 30 red and 0 white

#3 23 red and 5 white

#4 20 red and 10 white

#5 10 red and 20 white

Select one of the boxes at random so that you don’t know its contents.

**Step 1** Shake the box.

**Step 2 **Select one marker, note the result, and return it to the box.

**Step 3 **Repeat the first two steps 20 times with the same box.

**a.** What do you think is inside the box you have sampled?

**b.** Could you have guessed the contents by repeating the experiment five times? Ten times? Do you think you should have more than 20 observations per experiment?

### Project 16.3

“I think I’m going crazy, Bill,” said George. “I can’t figure out a good rating system for the state league. I’ve got over a hundred teams and I need to come up with a statewide rating system to rank all of those teams. Any ideas?”

“As a matter of fact, yes!” said Bill enthusiastically. “Have you ever heard of the Harbin Football Team rating system? I think they use it in Ohio. They use it to determine the high school football teams that are eligible to compete in post-season playoffs. Each team earns points for games it wins and for games that a defeated opponent wins. Here’s how it works: Level 1 points are awarded for each game a team wins. A level-2 point is awarded for each game a defeated opponent wins.”

“I hear ya, but I still don’t get it. How do you put all that together?” asked George.

“Well,” said Bill, “it’s very easy. We simply use a matrix.”

Rank teams A, B, C, D, E, and F if you are given the following information.

Team A beats F, and ties C;

Team B beats A, C, and F;

Team C beats E and F;

Team D ties A and beats F;

Team E beats A and F;

Team F beats C and ties D.

If two teams tie, enter 0.5 in the zero-one matrix instead of 1.

### Project 16.4 **Knot Theory**

Get a piece of string with two free ends, and tie those ends together with a knot. Some knots that you can tie will hold the ends of the string together and other knots will not (no pun intended!) In mathematics, there is a branch of mathematics known as *knot theory*. Mathematically, a knot is defined as a *closed piecewise liner* curve in R^{3}. Two or more knots together is known as a* link.n *Knots can be cataloged according to the number of crossings (ignoring mirror reflections). There is only one knot with crossing number three (called the *cloverleaf knot*), one knot with crossing number four, two with crossing number five, and three with crossing number of six.

* This research project is adapted from **R. E. Kohn**, “A Mathematical Programming Model for

Air Pollution Control,”* Science and Mathematics*, June 1969, pp. 487-499.

- How many knots are possible with crossing number of seven?
- How many knots are possible with crossing number of eight?

Write a paper on knot theory.

## 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 disturbing, 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 disturbing, and why?