The William Lowell Putnam Mathematical Competition 2001

It is a challenging opportunity for you to test your mathematical mettle. In the 2000 competition, 2818 individuals participated, representing 434 colleges and universities across Canada and the U.S.

Stanford placed seventh, and many individuals did very well.

The competition emphasizes ingenuity rather than knowledge, so freshmen are not at much of a disadvantage compared to seniors. Interest in or experience with problem solving is a plus. Not just math majors have done well; some recent winners have come from nearby disciplines, including physics, computer science, and engineering.

Completely solving even one of the twelve problems is a significant achievement, and in most years would place you well above the median. (Keep in mind that the particpants are self-selected from among the best in the continent.)

If you might like to try the Putnam, please let me know as soon as possible, definitely by November 7. The preliminary list of competitors has already been submitted, but I can send in changes as late as early November. If you need to take the exam at a different time, please let me know as soon as possible as well.

Links:

• The introductory poster.
• Meetings: We will have dinner-time problem solving practice sessions this quarter (Oct. 9 - Nov. 27): Tuesdays, 380-383N, 6-7:30. Practice problems will include some on the topics discussed, and some others.
  • Here is the handout for the introductory meeting.
  • October 9:
    • topics: pigeonhole principle and induction
    • notes and practice problems.
    • Problem of the week: Find positive integers n and a_1, a_2, ..., a_n such that a_1 + a_2 + ... + a_n = 1979 and the product a_1 a_2 ... a_n is as large as possible.
      (Follow-up: what year did this problem appear on the Putnam?)
  • October 16:
    • topic: number theory.
    • notes and problems.
    • link to one of the "million-dollar problems", the Birch-Swinnerton-Dyer Conjecture.
    • Problem of the week: Write down the first few powers of 2 in a column, and the first few powers of 5:

      2 5
      4 25
      8 125
      16 625
      32 3125

      Notice that the total number of digits in each row are 2, 3, 4, 5, 6! Why is the number of digits in the nth row, n+1?

  • October 23:
    • topics: linear recursions (recurrence relations) and linear recursive sequences; fancy linear algebra.
    • problems.
    • Instead of a "Problem of the Week", here's a "Website of the Week": Rich Schwartz's "Lucy and Lily" game. This may not work on some browsers. On my machine at home, it works on internet explorer, but not on netscape. There's actually a huge amount of cool math buried in this game, including links to Penrose tiles. Try playing the "single game" to see how hard it is. Then try the double game, which looks even harder. Finally, read the simple winning strategy for the double game. (It's amazing that the double game ends up being easier than the single game.)
  • October 30:
    • topics: generating functions; analysis on the real line.
    • notes and problems.
    • Problem of the week (found on the web; this one is an old chestnut). Follow-up once you've figured it out: Find Fibonacci numbers in the puzzle. They are somehow central to what's going on. Can you make a similar problem with "bigger" Fibonacci numbers?
  • November 6
    • topics: the theory of games; how to approach a Putnam problem
    • problems.
    • The game of Nim.
    • The 1993 Putnam, which I'll discuss.
    • Problem of the week: Figure out how to win the game of Nimskulls.
  • November 13
    • topics: invariants (presented by Vin da Silva), and how to tackle a general Putnam problem.
    • problems (including the 1988 and 1993 Putnams)
    • Website of the week: Why Mathematicians Now Care About their Hat Color. (Problems: (i) Figure out the 3-person strategy before reading the entire article. (ii) Explain why there's an n person strategy that's at least as successful as the n-1 person strategy. (iii) Find a seven-person strategy (very hard, unless you've happened to have seen "Hamming Codes")
  • November 20
    • topics: the "hat problem" (see the website of last week), and more practice.
    • problems (including the 1988 and 1992 Putnams)
  • November 27: No seminar. But here's a handout with sixteen analysis problems and solutions.
  • December 1: The Putnam. Part A: 8-11 am, Part B: 1-4 pm. The solutions are available here.
• The official Putnam website.
• Recommended reading: I really like Loren Larson's "Problem Solving Through Problems" --- definitely worth owning. It's great preparation for the Putnam. It's now on reserve at the math library, in Building 380. It is also great help trying many old problems; a compilation of old Putnam problems and solutions, collected by Kiran Kedlaya. Older Putnam problems have appeared in two books; I'll add the precise reference here soon.
• Some more links are available through this good UCSD site, by Patrick Fitzsimmons.

Want more? Additional suggestions for others? Please let me know.