Mathematics 1512             Calculus II             Fall, 2000


Current reading and homework assignments

Due for the final:


Current contests

NOTE: Submit your entries either in printed form or by electronic mail.

  1. Why a cat? What does Arnold the cat have to do with the mystical golden matrix
    
           [1    1 ]
           [       ]     ?
           [1    2 ]
    
    
    The prize will go to the first correct and informed answer about this math-cultural question by Friday, 8 December.
  2. In this problem, A,B,C, and I (the identity) are all 3 by 3 matrices.
    1. Can you find an example where AB - BA = I? Please either give the example or prove that there cannot be one.
    2. Can you find an example where ABC - CBA = I (with B not the identity)? Please either give the example or prove that there cannot be one.
    Deadline extended indefinitely. This could be a good term paper!
  3. The blooper contest. This one is a standing contest. It has no deadline, and may be won more than once. Find and document an egregious mathematical mistake in All entries must be recent, and the standards for receiving points will be demanding.
If your appetite for puzzles is insatiable, try out for the Georgia Tech Putnam Competition team, or just go to the practices for fun. The William Lowell Putnam Mathematical Competition is the premiere intercollegiate mathematical competition. Fame and fortune may come your way!

Past homework assignments

Due Thursday, 24 August:
  • Reading:
  • Exercises Due Thursday, 31 August:
  • Reading:
  • Exercises There was a test on Friday, 8 September. It covered the following sections of the text SHE: 9.3,9.4,10.5, 10.6, 11.1-11.6, as well as section 0.6 of H2.

    Due Thursday, 7 September:

    Due Thursday, 14 September: Due Thursday, 21 September: There was a test on Friday, 29 September.
    Due Thursday, 28 September: Due Thursday, 5 October: Due Thursday, 12 October:

    Due Thursday, 19 October:

    There was a test on Friday, 27 October.
    Due Thursday, 26 October: Due Thursday, 2 November: Due Friday, 10 November: Due Thursday, 16 November: Due Thursday, 30 November:

    Past contests

    1. The first contest was purely aesthetic: Use Maple or another graphing utility to graph the most creative or beautiful curve.
        Congratulations to Christos Tsonis for a graph which looked something like a hairy carneval mask:
        > plot[sin(260*t)^2 + cos(2*t),t,t=-Pi..Pi];
    2. A professor had ten different pairs of socks, but a malicious dryer ate six (individual) socks. An optimist would say that with luck, there will still be 7 matching pairs, while a pessimist would say that there will more probably be only 4 matching pairs. Who is more likely to be right, and by how much? Give a quantitative comparison of the probabilities.
        Congratulations to Karthik Balakrishnan and Jay Underwood for correctly showing that the pessimist is 112 times more likely to be right than the optimist.
    3. Over in Athens, Georgia, many people - far, far, too many - believe that (f g)' = f' g'.
      This is known as the Ugamath formula. Once in a while it is even right, for example when f and g are constants, or f(x) = g(x) = exp(2 x). How many examples can you find where (f g)' = f' g' but NOT = 0?
        Congratulations to Karthik Balakrishnan for the largest number of solutions.
    4. As you know from a class in Euclidean geometry, two triangles are congruent if they share two angles and a side (ASA) or two sides and the angle between them (SAS), but not necessarily if, for example they share two sides and a different angle (SSA). The relative positions of the 6 possible elements (3 S's and 3 A's) need to be given for these theorems. This does not exclude the possibility that two triangles might share 5 of the 6 elements but fail to be congruent, if we don't say which angles are adjacent to which sides. How many examples can you find of such pairs of triangles with integer sides?
        Congratulations to Karthik, again, for many solutions.!
    5. Define a sequence of polynomials by P0 := 1, P1 := x+1, and Pn+1 := Pn + x Pn-1. Show that all roots of Pn are real, for all n.
        Sadly for the polynomials, this one was not solved by the extended deadline. Keep trying for an exciting prize, though no points!
    6. Arithmetic codes. There is a category of arithmetic puzzles, in which each digit is encoded as a letter, and the arithmetic message spells something meaningful. The classic one is a letter sent home from college, which reads
      Dear Father,
      
                       S E N D
                     + M O R E
                     _________
                     M O N E Y
      
      
      This is a good puzzle because there is exactly one solution.
      Our contest has two parts. The first is to solve for the arithmetic encoded by
                      AB*CD=EEE
                      E*CD-AB=CC
      
      in order to find A*B*D . This much may get you an exciting prize.
      The second part is more creative. Make up a puzzle, preferably something like the SEND + MORE = MONEY code by using concepts from this class.
      The winning puzzle will have one and only one solution, and will be judged on the basis of beauty.

      P.S., the father's response to the letter above was:

                         U S E
                     + L E S S
                     _________
                     S O N N Y
      
        Congratulations to Tim Lee for two sums, including
                               I
                         + AGREE
                         +   ITS
                       _________
                           TOUGH
        
    7. Three-way duel, if "duel" is the proper word. Three duelists stand at the corners of an equilateral triangle. The rules of the duel are that they shoot in order, and they draw straws to see who begins. At each turn a given duelist gets one shot, and the shooting continues until there is only one left standing.

      Now, they all know that Dead-Eye Dana is a great sharpshooter who will hit the target 100% of the time, while Average Alex hits the target 80% of the target 80% of the time and So-So Sal hits the target 50% of the time. Although the duel began over some passionate irrationality, now their minds are focused, and they each act rationally to maximize the chances of their survival. The contest is to figure out the probability each of the three has to survive, and to explain your answer clearly.

        Congratulations to Steven Paul for showing (quantitatively) that the worst shot is the most likely to win!
    8. The infidelity contest.

      Note: This logic puzzle is the creation of John Allen Paulos, and was recounted in his book Once upon a Number, which was chosen by the Los Angeles Times as one of the best nonfiction books of 1998. The innovation of alien genders is, however, my own.


      A nominally monogamous species on the planet Thrae consists of two genders, "wen" and "momen," the singulars being "wan" and "moman." Now, wen are very intelligent and logical, and each wan knows that all the other wen are equally so. They have the following well-known habits. 1. If a wan has proof that the moman to whom the wan is married has been unfaithful, the wan will kill the moman within one day. 2. Wen's intuition always allows them to know when another wan's spouse has been unfaithful. 3. Wen never tell any wan else about infidelity. 4. For some reason, wen's intuition fails them as to their own spouses. 5. Every wan takes as proved whatever is printed in the Equinchet, a local oracle.

      In the village of Chetag there are forty married couples, and all the momen have been unfaithful. Of course, every wan lives in the belief of being the one wan with a faithful spouse. Well, maybe. One day the Equinchet prints the statement that it has evidence that at least one moman in Chetag has been unfaithful.

      Does any wan learn anything not already known? Do any killings ensue? If so, how many and when? To win this one, you have to have the correct answer and the correct logical reasons for it. (Hint: One wins when one wonders well when one wan wonders whether one wan's one wanders.)

        Ah, well, no correct solutions to this one by the deadline
    9. The literature contest. Who can find the best examples of realmath in real literature? I am aware of some examples in work by Poe, Calvino, Borges, Stephenson, and some others. If you find a good example our panel of judges was unaware of, and back it up with the evidence, you may win a few points.
        A split win this time, between Karthik Balakrishnan and Eric Rahm! Karthik provided a really nice website on math and fiction, while Eric came up with the movie pi; check out the movie's web site. Honorable mention to Steven Paul for finding Flatland on the Web.
    10. Three politicians are having a debate around a round table on the planet Thrae. To the left of A sits B, to the left of B sits C, and to the left of C sits A. Now, there are three political parties, the Truthtellers, who always tell the truth, the Liars, who always lie, and the Wafflers, who lie and tell the truth in strict alternation. In the course of the debate they report the following facts (in order) about their opponents:
      
      speaker   The person to my left is   The person to my right is     I am
      
      A            a Liar                    a Waffler                 a Truthteller
      
      B           a Waffler                  a Waffler                 a Truthteller
      
      C            a Liar                    a Truthteller             a Truthteller
      
      
        Congratulations to Tim Lee and, minutes later, Casey, Zach, Steven, and Karthik, for showing that two of the politicians are Wafflers and one a Truthteller.
    11. Eigendoggerel. Write some rhymed couplets about eigenvectors and eigenvalues. Your couplet must contain at least one true mathematical statement (and no false mathematical statements), and must be original by you. I.e., not downloaded from the Web, where there is no doubt a vast literature of wonderful poetry about linear algebra. The winner will be judged on the basis of wit. Your entries are due on Monday, 27 November.
        Congratulations to Rebecca Fink, whose poem was voted the best, though there were also awe-inspiring eigenpoems by Tim Lee and Zach Eaton.

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