The Nature of Mathematics, 12th Edition
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Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
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individual projects
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Group Research Problems

    Working in small groups is typical of most work environments, and learning to work with others to communicate specific ideas is an important skill. Work with three or four other students to submit a single report based on each of the following questions.

Chapter 1

    G.1 It is stated in the text that "Mathematics is alive and constantly changing. As we complete the last decade of this century, we stand on the threshold of major changes in the mathematics curriculum in the United States." Report on some of these recent changes.


    Lynn Steen, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, D.C.: National Academy Press, 1989).
    See also, Curriculum and Evaluation Standards for School Mathematics from the National Council of Teachers of Mathematics (Reston, VA: NCTM, 1989).

    G.2 Do some research on Pascal's triangle, and see how many properties you can discover. You might begin by answering these questions:

    1. What are the successive powers of 11?
    2. Where are the natural numbers found in Pascal's triangle?
    3. What are triangular numbers and how are they found in Pascal's triangle?
    4. What are the tetrahedral numbers and how are they found in Pascal's triangle?
    5. What relationships do the patterns in Figure G.1 have to Pascal's triangle?

    Figure G.1 Patterns in Pascal's triangle


    James N. Boyd, "Pascal's Triangle," Mathematics Teacher, November 1983, pp. 559-560.
    Dale Seymour, Visual Patterns in Pascal's Triangle, (Palo Alto, CA: Dale Seymour Publications, 1986)
    Karl J. Smith, "Pascal's Triangle," Two-Year College Mathematics Journal, Volume 4 (Winter 1973).


Chapter 2

    G.3 A teacher assigned five problems, A, B, C, D, and E. Not all students turned in answers to all of the problems. Here is a tally of the percentage of students turning in each problem:

    A: 46%    A, B: 25%    A, B, C: 13%    A, B, C, D: 7%    A, B, C, D, E: 4%

    B: 40%    B, C: 26%    A, B, E: 19%    A, B, C, E: 8%

    C: 43%    C, D: 26%    A, D, E: 16%    A, B, D, E: 9%

    D: 38%    D, E: 22%    B, C, D: 12%    A, C, D, E: 11%

    E: 41%    A, E: 30%    C, D, E: 14%    B, C, D, E: 6%

    What percent of the students did not turn in any problems? Assume that no students turned in combinations not listed.

    G.4 Draw a Venn Diagram with five sets.
    For two sets, there are 4 regions.
    For three sets, there are 8 regions.
    For four sets, there are 16 regions.
    For five sets, there must be 32 regions.

    Symbolically name each of these 32 regions.

    G.5 A famous mathematician, Bertrand Russell, created a whole series of paradoxes by considering situations such as the following barber's rule: "Suppose in the small California town of Ferndale it is the practice of many of the men to be shaved by the barber. Now, the barber has a rule that has come to be known as the barber's rule: He shaves those men and only those men who do not shave themselves. The question is: Does the barber shave himself?" If he does shave himself, then according to the barber's rule, he does not shave himself. On the other hand, if he does not shave himself, then, according to the barber's rule, he shaves himself. We can only conclude that there can be no such barber's rule. But why not? Write a paper explaining what is meant by a paradox. Use the Historical Note below for some suggestions about mathematicians who have done work in this area. You might begin with this internet site:


Chapter 3

    G.6 Write a symbolic statement for each of the following verbal statements. (1) Either Donna does not like Elmer because Elmer is bald, or Donna likes Frank and George because they are handsome twins. (2) Either neither you nor I am honest, or Hank is a liar because Iggy did not have the money.

    G.7 Three schools have a track meet and enter one person in each of the events. The number of events is unknown, and so is the scoring system except that the winner of each event scores a certain number of points, second place scores fewer points, and third place scores fewer still. Georgia won with 22, and Alabama and Florida tied with 9 each. Florida won the high jump. Who won the mile run? *

    G.8 Consider the following question: "Of all possible collections of states that yield 270 or more electoral votesenough to win a presidential electionwhich collection has the smallest geographical area?" Hint: Let O be the set consisting of the optimal collection of states. Your group should choose a state and prove mathematically that the state is not in O or prove that the state is in O.


    This problem is found in the following article "Proof by Contradiction and the Electoral College," by Charles Redmond, Michael P. Federici, and Donald M. Platte in The Mathematical Teacher, November 1998.

    The U. S. Electoral College Calculator:

    National Archives and Records Administration Federal Register:

    G.9 Five cabbies have been called to pick up five fares at the Hilton Towers. On arrival, they find that their passengers are slightly intoxicated. Each man has a different first and last name, a different profession, and a different destination; in addition, each man's wife has a different first name. Unable to determine who's who and who's going where, the cabbies want to know: Who is the accountant? What is Winston's last name? Who is going to Elm Street? Use only the following facts to answer these questions:

    1. Sam is married to Donna.
    2. Barbara's husband gets into the third cab.
    3. Ulysses is a banker.
    4. The last cab goes to Camp St.
    5. Alice lives on Denver Street.
    6. The teacher gets into the fourth cab.
    7. Tom gets into the second cab.
    8. Eve is married to the stock broker.
    9. Mr. Brown lives on Denver St.
    10. Mr. Camp gets into the cab in front of Connie's husband.
    11. Mr. Adams gets into the first cab.
    12. Mr. Duncan lives on Bourbon St.
    13. The lawyer lives on Anchor St.
    14. Mr. Duncan gets into the cab in front of Mr. Evans.
    15. The lawyer is three cabs in front of Victor.
    16. Mr. Camp is in the cab in front of the teacher.

    * From "Ask Marilyn" by Marilyn Vos Savant, Parade Magazine, October 31, 1993.

    G.10 Consider the apparatus shown in Figure 3.9.

    Figure 3.9 Reward Game

    Note that there are 12 chutes (numbered 1 to 12), and if you drop a ball into the chute it will slide down the tube until it reaches an AND-GATE or an OR-GATE. If two balls reach an AND-GATE, then one ball will pass through, but if only one reaches an AND-gate, it will not pass through. If one or two balls reach an OR-GATE, then one ball will pass through. The object is to obtain a reward by having a ball reach the location called REWARD. What is the fewest number of balls that can be released in order to gain the reward?


Chapter 4

    G.11 Invent an original numeration system.

    G.12 Organize a debate. One side represents the algorists and the other side the abacists. The year is 1400. Debate the merits of the Roman numeration system and the Hindu-Arabic numeration system.


    Barbara E. Reynolds, "The Algorists vs. The Abacists: An Ancient Controversy on the Use of Calculators," The College Mathematics Journal, Vol. 24, No. 3, May 1993, pp. 218-223. Includes additional references.

    G.13 Organize a debate. The issue: "Resolved: Computers can think."

    G.14 In a now famous paper, Alan Turing asked, "What would we ask a computer to do before we would say that it could think?" In the 1950s Turing devised a test for "thinking" that is now known as the turing test. Dr. Hugh Loebner, a New York philanthropist, has offered $100,000 for the first machine that fools a judge into thinking it is a person. In 1991, the Computer Museum in Boston held a contest in which 10 judges at the museum held conversations on terminals with eight respondents around the world, including six computers and two humans. The conversations of about 15 minutes each were limited to particular subjects, such as wine, fishing, clothing, and Shakespeare, but in a true turing test, the questions could involve any topic. Work as a group to decide the questions you would ask. Do you think a computer will ever be able to pass the test?


    Betsy Carpenter, "Will Machines Ever Think?" U.S. News & World Report, October 17, 1988, pp. 64-65.
    Stanley Wellborn, "Machines That Think," U.S. News & World Report, December 5, 1983, pp. 59-62.

    G.15 Construct an exhibit on ancient computing methods. Some suggestions for your exhibit are charts of sample computations by ancient methods, pebbles, tally sticks, tally marks in sand, Roman number computations, abaci, Napier's bones, and old computing devices. You should consider answering the following questions as part of your exhibit: How do you multiply with Roman numerals? What is the scratch system? What is the lattice method of computation? What changes in our methods of long multiplication and long division have taken place over the years? How did the old computing machines work? Who invented the slide rule?


Chapter 5


    G.17 Four Fours Write the numbers from 1 to 100 (inclusive) using exactly four fours. See Problem 60, Problem Set 5.3, x-ref ok to help you get started.

    G.18 Pythagorean Theorem Write out three different proofs of the Pythagorean theorem.

    G.19 Modular Art Many interesting designs such as those shown here can be created using patterns based on modular arithmetic. Prepare a report for class presentation based on the article "Using Mathematical Structures to Generate Artistic Designs" by Sonia Forseth and Andrea Price Troutman, The Mathematics Teacher, May 1974, pp. 393-398. Another source is "Mod Art: The Art of Mathematics" by Susan Morris, Technology Review, March/April 1979.



Chapter 6




    G.24 Prove that there are infinitely many integers such that the sum of the digits in their square equals the sum of the digits in their cube.

    G.25 JOURNAL PROBLEM (From Journal of Recreational Mathematics, Vol. II, #2) Translate the following message: Wx utgtuz f pbkz tswx wlx xwozm pbkzr, f exbmwo cxlzm xm ts jzszmfi fsv cxlzm lofwzgzm tswx wlx cxlzmr xe woz rfnz uzsxntsfwtxs fkxgz woz rzpxsu tr tncxrrtkiz, fsu T ofgz frrbmzuiv exbsu fs funtmfkiz cmxxe xe wotr kbw woz nfmjts tr wxx sfmmxl wx pxswfts tw. Ctzmmz Uz Ezmnfw


Chapter 7

    G.26 In the above figure, there are eight square rooms making up a maze. Each square room has two walls that are mirrors and two walls that are open spaces. Identify the mirrored walls, and then solve the maze by showing how you can pass through all eight rooms consecutively without going through the same room twice. If that is not possible, tell why.

    G.27 In the text we considered different views of a cube. The figure shows a cube with a dot in the middle of each face.


    Draw a cube so that each dot is in the center of a face of the cube.

    G.28 Place a dollar bill across the top of two glasses that are at least 3.5 in. apart. Now, describe how you can place a $0.50 piece in the middle of the dollar bill without having it fall.


Chapter 8

    G.29 What exactly are fractals?

    To get you started on your paper, we ask the following question that relates the ideas of series and fractals using the snowflake curve. Cut an equilateral triangle of side a out of paper, as shown in part a of the figure. Next, three equilateral triangles, each of side a/3, are cut out and placed in the middle of each side of the first triangle, as shown in part b of the figure. Then 12 equilateral triangles, each of side a/9, are placed halfway along each of the sides of this figure, as shown in part c of the figure. Part d of the figure shows the result of adding 48 equilateral triangles, each of side a/27, to the previous figure. As part of the work on this paper, find the perimeter and the area of the snowflake curve formed if you continue this process indefinitely.

    Construction of a snowflake curve


    Anthony Barcellos, "The Fractal Geometry of Mandelbrot," The College Mathematics Journal, March 1984, pp. 98-114.
    "Interview, Benoit B. Mandelbrot," OMNI, February 1984, pp. 65-66.
    Benoit Mandelbrot, Fractals: Form, Chance, and Dimension (San Francisco: W. H. Freeman, 1977).
    Benoit Mandelbrot, The Fractal Geometry of Nature (San Francisco: W. H. Freeman, 1982).


Chapter 9

    G.30 Suppose a house has an 8-ft ceiling in all rooms except the living room, which has a 10-ft cathedral ceiling. Approximately how many marbles would fit into this house?

    G.31 Suppose you wish to build a spa on a wood deck. The deck is to be built 4 ft above level ground. It is to be 50 ft by 30 ft and is to contain a spa that is circular with a 14-ft diameter. The spa is 4 ft deep.

    1. How much water will the spa contain, and how much will it weigh? Assume that the spa itself weighs 550 lb.
    2. Draw plans for the wood deck.
    3. Draw up a materials list.
    4. Estimate the cost for this installation.



Chapter 10

    Before Hurricane Katrine in 2005, the entrance of the Aquarium of Americas in New Orleans has a gigantic building-size curve called a logarithmic spiral. Find out how to construct a logarithmic spiral, and write a paper about what you learned. Why do you suppose it would appear on the front of an aquarium?

    G.34 If we assume that the world population grows exponentially, then it is also reasonable to assume that the use of some nonrenewable resource (such as petroleum) will also grow exponentially. In calculus, it is shown that for some constant k, under these assumptions, the formula for the amount of the resource, A, consumed from time t = 0 to t = T is given by the formula

    where r is the relative growth rate of annual consumption.
    a. Solve this equation for T to find a formula for life expectancy of a particular resource.
    b. According to the Energy"Information Administration, the annual world production"(in billions of barrels" per day) of petroleum is shown in the following table: "
    Year: 1975 1980 1985 1990 1995 2000 2003
    Quantity: 52.42 62.39" 52.97 60.90 61.85 66.03 67.00
    Find an exponential"equation for these data.
    c. If in 1998,"the world petroleum reserves are 2.8 trillion barrels, estimate the life expectancy for petroleum.


Chapter 11

    G.35 Suppose you have just inherited $30,000 and need to decide what to do with the money. Write a paper discussing your options and the financial implications of those options. The paper you turn in should offer several alternatives and then the members of your group should reach a consensus of the best course of action.

    G.36 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. One member of your group should investigate the relationship of the Fibonacci numbers to nature, another the algebraic properties of the sequence, and another the history of the sequence.

    G.37 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

    Each person should write the arithmetic series for the total amountofmoneyearnedin10 years under a different option.

    Your group should reach a consensus as to which is the best option. Give reasons and show your calculations in the paper that your group submits.

    G.38 It is not uncommon for the owner of a home to receive a letter similar

    to the one shown below. Write a paper based on this letter. Different members of your group can work on different parts of the question, but you should submit one paper from your group.

    a. What is the letter about?

    b. A computer printout (above) was included with the letter. Assuming that these calculations are correct, discuss the advantages or disadvantages of accepting this offer.

    c. The plan as described in the letter costs $375 to sign up. I called the company and asked what their plan would do that I could not do myself by simply making 13 payments a year to my mortgage holder. The answer I received was that the plan would do nothing more, but the reason people do sign up is because they do not have the self-discipline to make the midmonthly payments to themselves. Why is a biweekly payment equivalent to 13 annual payments instead of equivalent to a monthly payment?

    d. The representative of the company told me that more than 250,000 people have signed up. How much income has the company received from this offer?

    e. You calculated the income the company has received from this offer in part d, but that is not all it receives. It acts as a bonded and secure "holding company" for your funds (because the mortgage company does not accept "two-week" payments). This means that the company receives the use (interest value) on your money for two weeks out of every month. This is equivalent to half the year. Let's assume that the average monthly payment is $1,000 and that the company has 250,000 payments that they hold for half the year. If the interest rate is 5% (a secure guaranteed rate), how much potential interest can be received by this company?


Chapter 12

    G.39 Prepare a strip of paper as shown. Turn it over and mark the other side as shown.

    Strips for constructing a hexahexaflexagon; Make sure that each of the numbered triangles is equilateral.

    Starting from the left, fold the 4 onto the 4, the 5 onto the 5, 6 onto 6, 4 onto 4, and so on until your paper looks like the one shown in Figures Gc and d.

    Hexahexaflexagon after the first fold

    Continue by folding 1 onto the 1 from the front, by folding the 1 onto the 1 from the back, and finally by bringing the 1 up from the bottom so that it rests on top of the 1 on the top. You paper should look like the one shown.

    Hexahexaflexagon after the second fold

    Paste the blank onto the blank, and the result is called a hexahexaflexagon, as shown. With a little practice you'll be able to "flex" your hexahexaflexagon so that you can obtain a side with all 1s, another with all 2s, ... and another with all 6s. After you have become fairly proficient at "flexing," count the number of flexes required to obtain all six "sides." What do you think is the fewest number of flexes necessary to obtain all six sides?

    To "flex" your hexahexaflexagon, pinch together two of the triangles (left two figures). The inner edge may then be opened with the other hand (rightmost picture). If the hexahexaflexagon cannot be opened, an adjacent pair of triangles is pinched. If it opens, turn it inside out, finding a side that was not visible before. Be careful not to tear the hexahexaflexagon by forcing the flex.

    G.40 A puzzle sold under the name The Avenger, is pictured.

    The Avenger Puzzle

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


    "Group Theory, Rubik's Cube and The Avenger,"
    Games, June/July 1987, pp. 44-45.

    G.41 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

    1. Stage a contest in front of the class to see which members of your group can complete one face of a Rubik's cube.
    2. Stage a contest to see which member of your group can solve the Rubik's cube puzzle the fastest. Report the results to the class.


      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

    1. In how many possible ways can you land on jail (just visiting) on your first turn when playing a Monopoly game?
    2. Is it possible to make it from GO to Park Place on your first roll of the dice in a Monopoly game? If so, what is the probability not only that would happen, but also that you would obtain a 2 on your next roll to complete a set (a monopoly).

    G.43 Birthday problem:

    1. Experiment: Consider the birthdates of some famous mathematicians:

      Abel    August 5, 1802

      Cardano    September 24, 1501

      Descartes    March 31, 1596

      Euler    April 15, 1707

      Fermat    August 18, 1602

      Galois    October 25, 1811

      Gauss    April 30, 1777

      Newton    December 25, 1642

      Pascal    June 19, 1623

      Riemann    September 17, 1826

      Add to this list the birthdates of the members of your class. But before you compile this list, guess the probability that at least two people in this group will have exactly the same birthday (not counting the year). Be sure to make your guess before finding out the birthdates of your classmates. The answer, of course, depends on the number of people on the list. Ten mathematicians are listed and you may have 20 people in your class, giving 30 names on the list.

    2. Find the probability of at least one birthday match among 3 randomly selected people. (See Example 4, Section 13.4.)
    3. Find the probability of at least one birthday match among 23 randomly selected people. Have each person in your group pick 23 names at random from a biographical dictionary or a Who's Who, and verify empirically the probability you calculated.
    4. Draw a graph showing the probability of a birthday match given a group of n people. How many people are necessary for the probability actually to reach 1?
    5. In the previous parts of this problem we interpreted two people having the same birthday as meaning at least 2 have the same birthday (see Example 4, Section .4). We now refine this idea. Find the following probabilities for a group of 5 randomly selected people:

      Exactly 2 of the 5 have the same birthday.
      Exactly 3 have the same birthday.
      Exactly 4 have the same birthday.
      All 5 have the same birthday.
      There are exactly two pairs sharing (a different) birthday.
      There is a full house of birthdays (that is, three share one birthday, and two share another).
      Show that the questions of this problem account for all the possibilities; that is, show that the sum of the probabilities for all of these possibilities is the same as for the original birthday problem involving 5 persons: What is the probability of a birthday match among 5 randomly selected people?

    G.44 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

    G.45 Toss a toothpick onto a hardwood floor 1,000 times or toss 1,000 toothpicks, one at a time, onto the floor. Let l be the length of the toothpick and d be the distance between the parallel lines determined by the floorboards.

    Buffon's needle problem

    Equipment needed: A box of toothpicks (of uniform length) and a large sheet of paper with equidistant parallel lines. A hardwood floor works very well instead of using a sheet of paper. The length of a toothpick should be less than the perpendicular distance between the parallel lines.

    1. Guess the probability p that a toothpick will cross a line. Do this before you begin the experiment. The members of your group should reach a consensus before continuing.
    2. Perform the experiment and find p empirically. That is, to find p, divide the number of toothpicks crossing a line by the number of toothpicks tossed (1,000 in this case). If you wish, you can use this interactive site to simulate the experiment.

    3. By direct measurement, find l and d.
    4. Calculate 2l and pd, and (2l)/(pd).
    5. Formulate a conclusion. This is an experiment known as Buffon's needle problem.

    G.46 You are interested in knowing the number and ages of children (0-18 years) in a part (or all) of your community. You will need to sample 50 families, finding the number of children in each family and the age of each child. It is important that you select the 50 families at random. How to do this is a subject of a course in statistics. For this problem, however, follow these steps:

    Step 1. Determine the geographic boundaries of the area with which you are concerned.

    Step 2. Consider various methods for selecting the families at random. For example, could you:

        (i)select the first 50 homes at which someone is at home when you call?

       (ii)select 50 numbers from a phone book that covers the same geographic boundaries as those described in step 1?

       (iii) Using (i) or (ii) could result in a biased sample. Can you guess why this might be true? In a statistics course, you might explore other ways of selecting the homes. For this problem, use one of these methods.

    Step 3. Consider different ways of asking the question. Can the way the family is approached affect the response?

    Step 4. Gather your data.

    Step 5 .Organize your data. Construct a frequency distribution for the children, with integral values from 0 to 18.

    Step 6. Find out the number of families who actually live in the area you've selected. If you can't do this, assume that the area has 1,000 families.

    1. What is the average number of children per family?
    2. What percent of the children are in the first grade (age 6)? If all the children aged 12-15 are in junior high, how many are in junior high for the geographic area you are considering?
    3. See if you can actually find out the answers to parts b and c, and compare these answers with your projections.
    4. What other inferences can you make from your data?

Chapter 15



    G.49 Investigate the topic of conic sections. Build models and/or find three-dimensional models for the conic sections. What did the Greeks know of the conic sections?

    G.50 Prepare a list of women mathematicians from the history of mathematics. Answer the question, "Why were so few mathematicians female?"


    Teri Perl, Math Equals: Biographies of Women Mathematicians plus Related Activities. (Reading, MA: Addison-Wesley Publishing Co., 1978).
    Loretta Kelley, "Why Were So Few Mathematicians Female?" The Mathematics Teacher, October 1996. cBarbara Sicherman and Carol H. Green, eds. Notable American Women: The Modern Period. A Biographical Dictionary. (Cambridge, MA: Belknap Press, Harvard University Press, 1980).
    Outstanding Women in Mathematics and Science (National Women's History Project, Windsor, CA 95492, 1991).

    G.51 Prepare a list of black mathematicians from the history of mathematics.


    Virginia Newell et al., eds. Black Mathematicians and Their Works (Ardmore, PA: Dorrance & Company, 1980).

    G.52 Prepare a list of mathematicians with the first name of Karl.

    G.53 Write a news article about a historical mathematician as if you were a contemporary of the person you are writing about. Put it in newspaper style and include other newsworthy items from the period.

Chapter 16

    G.54 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. Rank these teams. If two teams tie, enter 0.5 in the communication matrix instead of 1.

    G.55 Suppose your group conducts an experiment at a local department store. You walk up a rising escalator and you take one step per second to reach the top in 20 seconds. Next, you walk up the same rising escalator at the rate of two steps per second and this time it takes 32 steps. How many steps would be required to reach the top on a stopped escalator?

    G.56 Two ranchers sold a herd of cattle and received as many dollars for each animal as there were cattle in the herd. With the money they bought a flock of sheep at $10 a head and then a lamb with the rest of the money (less than $10). Finally, they divided the animals between them, with one rancher obtaining an extra sheep and the other the lamb. The rancher who got the lamb was given his friend's new watch as compensation. What is the value of the watch?

    G.57 Suppose your group has just been hired by a company called Alco. You are asked to analyze its operations and make some recommendations about how it can comply at a minimum cost with recent orders of the Environmental Protection Agency (EPA).

    To prepare your report, you study the operation and obtain the following information:

    • Alco Cement Company produces cement.
    • The EPA has ordered Alco to reduce the amount of emissions released into the atmosphere during production.
    • Alco wants to comply, but wants to do so at the least possible cost.
    • Present production is 2.5 million barrels of cement, and 2 pounds of dust are emitted for every barrel of cement produced.
    • The cement is produced in kilns that are presently equipped with mechanical collectors.
    • To reduce the emissions to the required level, the mechanical collectors must be replaced either by four-field electrostatic precipitators, which would reduce emission to 0.5 pound of dust per barrel of cement, or by five-field precipitators, which would reduce emission to 0.2 pound per barrel.
    • The capital and operating costs for the four-field precipitator are 14ó per barrel of cement produced; for the five-field precipitator, costs are 18ó per barrel.
    • To comply with the EPA, Alco must reduce particulate emission by at least 4.2 million pounds. *

    * 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.

    Use mathematical modeling to write your paper. Mathematical modeling involves creating equations and procedures to make predictions about the real world. Typical textbook problems focus on limited, specific skills, but in the real world you need to sift through the given information to decide what information you need and what information you do not need. You may need to do some research to gather data not provided.

    G.58 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 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 cubed. Two or more knots together is known as a link. 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.

    1. How many knots are possible with crossing number of seven?
    2. How many knots are possible with crossing number of eight?

    Write a paper on knot theory.


Chapter 17

    G.59 Write a history of apportionment in the United States House of Representatives. Pay particular attention to the paradoxes of apportionment.

    G.60 Your group should investigate some item of interest to your group.

    It might be to predict the outcome of an upcoming election, your favorite song or movie. Your group should make up a list of 5 or 6 choices; for example, you might be researching what is the best of the Star Wars movies. Make up a written ballot and ask at least 50 people to rank the items on your list. Summarize the outcome of your poll. Was there a majority winner; how about a plurality winner. Who wins the Borda count or the Hare methods? What about the pairwise comparison method. Present a summary of your results.