Extra Credit Possibilities for MTE 210

The deadline for handing in extra credit is the start of the final exam, but you may hand them in earlier, of course. Solutions of problems will be graded by the next class after they are submitted — ask to get them back, and if they are not correct, you can try again. All other items will not be graded until the end.

There have been several cases in which students have plagiarized film or book reports. My policy is to turn all such cases (as well as any other instances of cheating) over to the Academic Conduct Committee for investigation and sanctions. A student guilty of such conduct normally gets suspended from the university for at least one semester. Obviously, you don’t want to jeopardize your future that way!

Book report

I have three types of books in mind. One is a book about some aspect of mathematics, such as fractals, infinity, pi, etc. Another is a book about the history or culture of mathematics (e.g., biography of Emmy Noether or a look at mathematics among African peoples). The third is a book about mathematical learning, such as math anxiety or the sad state of mathematical literacy. Textbooks or books about teaching mathematics are not appropriate.

The list below will give you an idea of what you can read. Alternatively, browse in the mathematics section of a good bookstore (such as Borders) or library. If you see something good, let me know, so I can add it to my list (books not on the list need to be approved). Another alternative is to browse the shelves in my office; I can lend you a book for a few weeks.

The report should be about three pages (or more, if you have a lot to say). It should be well-written, make interesting reading, probably be somewhat personal (what did YOU get out of it?). It could be the sort of essay you’d find in the book review section of a newspaper. Please use a word processor (or typewriter), and make sure the spelling and punctuation are correct. You can earn up to 10 extra points, depending on the quality of what you do.

Books for book report

1. Edwin Abbott, Flatland (The entire book can be down-loaded from or read on the Web.)
2. David Acheson, 1089 and All That: A Journey into Mathematics
3. Amir Aczel, Chance: A Guide to Gambling, Love, the Stock Market and Just About Everything Else
4. Donald Albers (ed.), Mathematical People
5. David Auburn, Proof: A Play
6. W. W. Rouse Ball, Mathematical Recreations and Essays
7. Petr Beckmann, A History of Pi
8. Eric Temple Bell, Men of Mathematics
9. Deborah J. Bennett, Randomness
10. David Bergamini, Mathematics (Life Science Library). This is out of print but possibly available in libraries. I have a copy, too.
11. William Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics
12. Mark Buchanan, Nexus: Small Worlds and the Groundbreaking Science of Networks
13. Gregory Chaitin, Conversations with a Mathematician: Math, Art, Science, and the Limits of Reason
14. K. C. Cole, The Universe and the Teacup: The Mathematics of Truth and Beauty
15. Miriam P. Cooney, ed., Celebrating Women in Mathematics and Science
16. Richard Courant and Herbert Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods
17. Philip J. Davis and Reuben Hersh, The Mathematical Experience
18. Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics (see the NY Times review of this book)
19. Keith J. Devlin, The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip
20. A. K. Dewdney, 200% of Nothing
21. Apostolos K. Doxiadis, Uncle Petros and Goldbach’s Conjecture (this is a short novel)
22. A. W. F. Edwards, Cogwheels of the Mind: The Story of Venn Diagrams
23. Hans Magnus Enzensberger, The Number Devil: A Mathematical Adventure
24. Brian Everitt, Chance Rules
25. Clifton Fadiman, ed., Fantasia Mathematica: Being a Set of Stories, Together With a Group of Oddments and Diversions, All Drawn from the Universe of Mathematics
26. Sarah Flannery and David Flannery, In Code: A Mathematical Journey
27. Martin Gardner, (any of his collections of mathematical essays)
28. Timothy Gowers, Mathematics: A Very Short Introduction
29. Jan Gullberg, Mathematics: From the Birth of Numbers
30. Claudia Henrion, Women in Mathematics: The Addition of Difference
31. Peter Hilton and Jean Pedersen, Fear No More: An Adult Approach to Mathematics
32. Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth
33. Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid
34. Norton Juster, The Phantom Tollbooth
35. Robert Kaplan, The Nothing That Is: A Natural History of Zero
36. David Klarner, Mathematical Recreations (originally printed under the title The Mathematical Gardner)
37. Stanley Kogelman, Mind Over Math
38. Mario Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number
39. Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States
40. Richard Mankiewicz, The Story of Mathematics
41. Eli Maor, e
42. Eli Maor, To Infinity and Beyond
43. Joseph Mazur, Euclid in the Rainforest : Discovering Universal Truth in Logic and Math
44. Danica McKellar, Math Doesn't Suck: How to Survive Middle-School Math Without Losing Your Mind or Breaking a Nail
45. Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel lines to Hyperspace
46. Geoffrey Mott-Smith, Mathematical Puzzles
47. Derrick Neiderman and David Boyum, What the Numbers Say: A Field Guide to Mastering Our Numerical World
48. Christos H. Papadimitriou, Turing (A Novel about Computation)
49. Marla Parker, She Does Math!
50. John Allen Paulos, Beyond Numeracy
51. John Allen Paulos, Innumeracy
52. John Allen Paulos, A Mathematician Reads the Newspaper
53. John Allen Paulos, Once Upon a Number
54. Ivars Peterson, The Jungles of Randomness
55. Ivars Peterson, The Mathematical Tourist
56. Ivars Peterson, Mathematical Treks
57. George Polya, How To Solve It
58. George Polya, Mathematics and Plausible Reasoning
59. Alfred S. Posamentier and Ingmar Lehmann, Pi: A Biography of the World’s Most Mysterious Number
60. Constance Reid, From Zero to Infinity
61. Constance Reid, Hilbert
62. Constance Reid, Julia: A Life in Mathematics (This is a short book with lots of pictures, about the author’s sister, a very important modern mathematician who died in 1985. See this Web site for more details about Julia Robinson.)
63. Edward Rothstein, Emblems of the Mind: The Inner Life of Music and Mathematics
64. Rudy Rucker, Infinity and the Mind
65. Rudy Rucker, Spaceland: A Novel of the Fourth Dimension
66. Bruce Schechter, My Brain Is Open: The Mathematical Journeys of Paul Erdos
67. Charles Seife, Zero: The Biography of a Dangerous Idea
68. Simon Singh , Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem
69. Raymond Smullyan, The Lady or the Tiger
70. Sherman K. Stein, How the Other Half Thinks: Adventures in Mathematical Reasoning
71. Sherman K. Stein, Strength in Numbers: Discovering the Joy and Power of Mathematics in Everyday Life
72. Andrew Sterrett, editor, 101 Careers in Mathematics
73. Ian Stewart, Nature’s Numbers
74. Tom Stoppard, Arcadia [this is a play]
75. Sheila Tobias, Overcoming Math Anxiety [this is always a very popular choice]
76. Sheila Tobias, Succeed with Math
77. Andrew Vazsonyi, Which Door Has the Cadillac: Adventures of a Real-life Mathematician
78. Sue Woolfe, Leaning Toward Infinity: A Novel
79. Claudia Zaslavsky, Africa Counts
80. Claudia Zaslavsky, Fear of Math
81. Any biography or autobiography of a mathematician (e.g., Bertrand Russell or Gauss) or book about the history of mathematics (some of these are listed above, but there are dozens more)

Reports on math-related movie or TV program

You can do the same with math-related movies, such as the 1987 film Stand and Deliver (about a calculus teacher in Los Angeles), the 1996 film Infinity (about the Nobel-prize winning physicist/mathematician Richard Feynman), the 1997 film Good Will Hunting (about a math prodigy coming of age in Boston), the 1998 film Pi (strange! — see their Web site), the 2005 film Proof, the 2001 film Fermat’s Last Tango (a delightful musical about the proof of Fermat’s Last Theorem — see their Web site), and the 2001 film A Beautiful Mind (about a Nobel-prize winning economist/mathematician — see their Web site), or The Math Life (if you can find it — it’s a documentary about what being a mathematician is all about). I have quite a few math-related videos in my office, if you’d like to borrow one, such as one about an international math contest for high school students and another about a famous female mathematician. Again, ask about anything I haven’t mentioned.

The report should be about three pages (or more, if you have a lot to say). It should be well-written, make interesting reading, probably be somewhat personal (what did YOU get out of it?). A substantial portion should relate to the mathematical aspects of the work. Please use a word processor (or typewriter), and make sure the spelling and punctuation are correct. You can earn up to 10 extra points, depending on the quality of what you do.

Math education conference attendance

Journal report

Exploring the Web

Another fascinating page I’ve run across was devoted to a backlash against the NCTM’s approach to mathematics; the page is entitled Mathematically Correct. You could probably do an entire project just following the links from that page (let me know whether they’ve convinced you of how evil the NCTM is).

You can earn up to 10 points of extra credit by searching the Web for pages relevant to the subject matter of this course. Spend some hours “surfing” and keep track of what you find (the contents and the URLs, or addresses, of these pages). Print out some interesting pages. To get the extra credit, write up a brief report on what you did (e.g., what search engines you used, what key words you searched for) and what you found. Explain what you think of the quality of the sites you found. Include a list of the relevant URLs and the best of the printouts. Be selective; the number of points is not based on volume, but you need to find more than just a couple of sites. You should read everything you print out and hand in, and make sure to include some thoughtful comments on at least a few of the items in your report.

Mathnerds

Puzzles for extra credit

1. Jessica threw a party for 17 of her friends. Each guest was given a whole number to wear on his or her back; the numbers 1 to 18 were used, one per person, with Jessica keeping #1 for herself. At one point everyone was dancing (forming nine couples), and Jessica noticed that the sum of the numbers for each couple was a perfect square. What was the number on the back of the person with whom Jessica was dancing? Your solution has to show that your purported answer HAS to be the answer, not merely that it COULD BE the answer. You have to argue that the pairing you produce is FORCED to be the actual pairing — that no other is possible. You get no credit for simply producing a pairing that works, nor for saying something like “I tried all the possibilities and this is the only one that worked” — you need to CONVINCE the reader that this solution is the only one. Read the last three sentences very carefully before submitting your solution — I typically get lots of solutions to this problem that don’t do what is asked. (This problem comes from an eighth grade math book in use at a public school in Troy.)

2. In a large urn are 1999 maize balls and 2000 blue balls (sorry about that, MSU fans). Next to the urn is a large pile of blue balls. The following procedure is performed repeatedly: Two balls are drawn from the urn; if both are blue, then one is put back and the other thrown away; if both are maize, then they are both thrown away and a blue ball from the pile is put into the urn; and if they are of different colors, then the maize one is put back into the urn and the blue one is thrown away. What is the color of the last ball in the urn (and must the time be reached at which the urn has just one ball left)? Obviously you need justification for your claims.

3. Familiarize yourself with what a set of dominos is. A standard set has from 0 to 6 spots on each half, and the set contains 28 dominos in all — one for each combination possible. It is easy to generalize this to a formula for the number of dominos if the number of spots per side is 0 to n, rather than specifically 0 to 6. But one can also ask: how many spots are there in all? (It’s 168 for n=6.) Determine (with justification, of course) the total number of dots in this general situation (where each domino can have from 0 to n spots per side). You need to come up with an explicit formula involving n (without "dot dot dot" — for example, if the sum 1+2+3+...+n appears in your calculation, you need to use Gauss’s trick to rewrite this).

4. Compute (with justification) the last (ones) digit of the largest known prime number, 2^32582657 – 1. Finding the answer somewhere doesn’t count; you need to show how to compute it.

5. Suppose that there are 25 kids standing in the playground, each equipped with a water pistol. It happens that the distances between pairs of kids are all different. At a given signal, each kid shoots the kid closest to him or her. Show that there must be at least one person who stays dry. [HINT: just because A shoots B, it doesn’t follow necessarily that B shoots A, since while B may be the closest person to A, there might be someone closer to B than A is.]

6. You have two ropes, each of which will burn in an hour if you light one end (think of the rope as lying on a metal table). But the burning rate is not uniform (e.g., you can’t assume that half the rope will burn in half an hour), nor are the ropes necessarily the same. Find a way to measure out 3/4 hour. Explain why your method works.

7. You have a stack of coins of which 17 are turned heads up (the rest tails up). You don’t know which coins are heads up (i.e., you don’t know where in the stack the heads-up coins are located). Assume that you are blindfolded and cannot tell the heads/tails nature by feeling. Your job is to produce two stacks of coins with the same number of heads up in each stack. You don’t know how many coins there are in the original stack (but it’s more than 17).

8. You have a serious illness for which you need to take one of pill A and one of pill B every morning. Not taking your pills, or overdosing, has dire consequences, and these pills are expensive, so you don’t want to waste any. Unfortunately, the pills look exactly the same once you take them out of the labeled pill box — you can’t tell an A from a B. This morning you have a bottle of each kind. You open up the A bottle and shake a single pill into your palm. Then you open up the B bottle and, to your horror, two B pills pop into your palm. Being upset, you then drop the three pills on the table and now you can’t tell which is which. So you have three pills there, an A and two B’s, but they all look alike. Remember, you have to take exactly one of each pill each day, and you don’t want to waste any. What can you do?

9. Three suitors for the king’s daughter are engaged in a contest; the winner will get her hand. Each of them is pretty clever. Here’s how the contest works. First, each of them has either a red hat or a white hat placed on his head, but he can’t see the color of his own hat. Then they are all led into a room and sit in a circle. They are told that if they see a red hat, then they are to raise their hand, and once someone knows the color of the hat on his own head, then he is to stand up, and he wins the contest (if he is correct). All of them raise their hands. After a few minutes have gone by, one of the guys (the cleverest one) stands up. What color hat was he wearing, and how did he know?

10. Here is another suitor and hat problem (much harder than the previous one, which was already hard): The three suitors, A, B, and C, are wearing hats with positive integers painted on them. Each person sees the other two numbers, but not his own. Each person knows that the numbers are positive integers (natural numbers) and that one of them is the sum of the other two. A, B, then C take turns in a contest to see who can be the first to determine his number. In the first round, A, B, and C all pass, but in the second round, A correctly asserts that his number is 50. What are the other two numbers, and how did A determine his was 50?

11. You have four coins, each of which weighs either 9 grams or 10 grams (they could all be 9s, or three 9s and a 10, or ...). You also have an accurate scale, which will tell how many grams the items you put on it weigh. How can you determine the weight of each coin (i.e., all four of them) with just three weighings?

12. Three prisoners are in separate cells. Every once in a while the warden takes one of the prisoners into a special “switch room”, which contains a single switch (connected to nothing) with two settings, UP and DOWN. The prisoner can reverse the state of the switch or leave it alone. The warden is committed to taking the prisoners to the room infinitely often: in other words, whenever they leave the room, they know they will eventually be taken back. At any time, a prisoner can announce “all three of us have visited the switch room”, in which case they will all be released if the statement is true, or executed if the statement is false. The prisoners may consult with each other before the room visits begin in order to formulate a strategy, but after that they will never see each other in the prison. And the warden can put the switch in any position he or she likes at the start (though after that the warden may not touch the switch). Formulate a strategy that will guarantee that the prisoners will eventually be released. Note: No probability is involved...the warden can arrange the visits any way he or she wants, subject to his or her commitments.

13. Emma and Ben are playing cards. The loser pays $1 to the winner in the first game, $2 in the second game, $4 in the third game and so on. (The prize is always doubled.) Emma starts with $601 but loses all her money in 10 games (in other words, in the tenth game she loses all she has at that point when she pays Ben the $512 she owes for losing that game). Exactly which games did Emma win? You need to show how to figure this out — just coming up with the answer and showing that it works is not sufficient.

14. This problem came in a call from a random adult who called the OU math department for help. He was referred to me. Can you give him the right answer? He has material that comes in sheets 1.8 mm thick. The specific gravity of the material is 0.96. The cost of the stuff is 23 cents per square foot. How much does it cost per pound? (You’ll need to look up — the Web is a good place to do so — the conversions between American and metric units, as well as the definition of specific gravity and the mass of water.)

15. A basketball player early in the season had a free-throw percentage (the fraction she made out of the shots attempted) of less than 80%. Later in the season her percentage had improved to more than 80%. Must there have been a time during the season at which her percentage was exactly 80%? You have to justify your answer, of course.

16. Alice and Bob are being tested for their logical reasoning ability by a diabolical math teacher. They will be blindfolded and the teacher will place a hat on each one’s head; then the blindfolds will be removed. The hats can be red or blue (so there are four possibilities as to how the hats can be placed: red on Alice and red on Bob; red on Alice and blue on Bob; and so on). Each of the people can see the other’s hat, but not his/her own. They are not allowed to communicate once the hats are placed. Then each of them has to write down the color of his/her own hat and give it to the teacher. If at least one of them is correct, then they pass the test; if both are wrong, they fail. Alice and Bob can devise a strategy ahead of time for how they will respond, but, again, they cannot communicate in any way once the hats are placed. How can they guarantee passing the test?

17. Five pirates have to divide 100 gold coins among them. There is a pecking order among the pirates, so pirate #1 makes the first proposal for how the coins should be divided. The pirates take a vote. If a majority of the pirates vote to accept the proposal, the deal is done. All votes get counted, including the lead pirate who made the proposal. If the proposal fails to win a majority (a tie is not good enough), then the lead pirate is killed, and 20 gold coins are paid to the executioner. The second pirate in the pecking order takes over and makes a new proposal on how to divide up the remaining 80 gold coins. The process continues until either there is a majority in a vote or there is one pirate left. Assume that each pirate votes in a way that maximizes his final outcome (if a vote does not affect the outcome, he votes “no” — after all, they're pirates!). What proposal should the first pirate make to maximize his profit?

18. Ron walks a one-hour round trip from his house and back each day. One day, he decides to run part of the way. Ron runs three times as fast as he walks. If he cut 10 minutes off his usual trip, how many minutes did Rob run?

19. Say your two visiting cousins want to meet a couple of women who play chess. You know all of the weird Fischer sisters, some of whom fit this description. You tell your cousins that if they hang around the sisters’ favorite coffee house long enough to run into two of them, there is a 50-50 chance that both women will be chess-players. How many such sisters are in the family?

20. There are six different ways to arrange four points in the plane so that there are only two different distances between the points. One of these is to put the points at the corners of a square, say with side length 1 inch. Then the each of the points is 1 inch from the two neighboring vertices of the square, but is the square root of 2 inches from the opposite vertex. There are six distances in all (four around the square and two diagonals) but only two different numbers that these distances can be. Find the other five ways to arrange the points to have the property that only two values arise among the six point-to-point distances.

21. Alice's husband picks her up from work every day at the same time. Today she decides to leave early, so she starts walking home 20 minutes before her husband would have arrived at work. He picks her up along the way, and they arrive home 6 minutes earlier than normal. How many minutes did Alice walk? (You can make the obvious assumptions about constant speeds, no time lost in pickup, etc.) Obviously you need to explain the reasoning leading to the answer.

22. (This comes from the president of Harvey Mudd College, who started her career as a mathematics professor here at OU.) Maria is terrified of spiders and wants to create a structure to prevent the infinitesimally small spider in her bedroom from crawling on her while she is asleep. The spider is somewhere on the walls or ceiling of her room. The spider can crawl on any surface except water, and can drop down on a vertical thread from any point. Maria puts each of the legs of her bed into a bowl of water to stop the spider from crawling up the legs. She then constructs another structure and happily sleeps through the night. The only constraint on the structure is that Maria and the spider remain in the same connected component of the air space. The correct answer can't involve unrealistically complex things like a continually running waterfall system in the bedroom.

Errors in the textbook