The William Lowell Putnam Mathematical Competition 2006

The photo on the right was taken before the afternoon session of the 2006 Putnam, thanks to Alex Flury! Most of the Stanford competitors are shown.

The William Lowell Putnam Mathematical Competition will take place on Saturday, December 2, 2006. The Putnam will take place in the Vidalakis room at the Schwab Residential Center, on Serra Street right by the corner with Campus Drive East. Here is a map (it's actually the building just below the one in red). We'll meet at 7:40 am. The first session will be 8-11 am, and the second session will be 1-4 pm. We'll meet again for the second session at 12:50.

The Putnam is a challenging opportunity for you to test your mathematical mettle. In the 2005 competition, 3545 individuals participated, representing 500 colleges and universities across Canada and the U.S. The Stanford team placed seventh, and many individuals did very well. (I have a text file of results that I can forward to anyone interested.) The measure I look to is the number of competitors on the list of top performers sent out with the results; we had 37, making us one of the top three.

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; many 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 almost all years would place you well above the median. (Keep in mind that the particpants are self-selected from among the best in the continent.)

Links:

• The introductory poster.
• The introductory meeting will take place Monday Oct. 2, 5:15-5:45 pm, in 380-383N (in the corner of the third floor of the math department). Here is the introductory handout.
• Meetings: We will have informal dinner-time problem solving practice sessions this quarter, on Mondays 5:15-6:45. (Rough schedule: 5-5:15 beforehand, people can drop by to discuss ``left-overs'' from the previous week. 5:15-5:30: short discussion of new technique. 5:30-6:15: work on problems. 6:15-6:45: eat pizza, watch ``volunteers'' present solutions and explain how they thought about the problems.) Some faculty and graduate students always turn up with wise words, including hopefully Mark Lucianovic, Daniel Mathews, Dragos Oprea, Andy Schultz, Henry Segerman, Kannan Soundararajan, and others. Practice problems will include some on the topics discussed, and some others. Handouts will appear here.
  • Monday, October 9: general problem-solving strategy, induction, and the pigeonhole principle. (We start with this every year.)
    Problems: ps pdf.
    Problem of the week (from Sound): Does there exist a circle with radius 100 in the plane containing 31415 lattice points (points of the form (a,b) where a and b are integers)?
  • Monday, October 16: generating functions.
    Cihan Baran pointed out that the wonderful book generatingfunctionology by Herbert Wilf is available on-line here.
    Problems: ps pdf.
    Problem of the week (from Sound): In a hotly contested election year, each Senator has slapped the face of one other Senator. Can you form a committee of 34 Senators none of whom has slapped another committee member?
  • Monday, October 23: analysis.
    Problems: ps pdf.
    Problem of the week (from Sound): Two players A and B play the following game. A thinks of a polynomial with non-negative integer coefficients. B must guess the polynomial. B has two shots: she can pick a number and ask A to return the polynomial value there, and then she has another such try. Can B win the game?
  • Monday, October 30: Mark Lucianovic explained number theory.
    Problems: ps pdf.
    Problem of the week from Sound: The Ritz-Carlton has 80 rooms, but a hundred guests. Fortunately for the hotel at any point of time no more than eighty guests show up. At check in, the hotel manager craftily gives each guest a number of keys such that no matter which eighty guests arrive the manager can assign each guest a room to which he already has a key. (E.g. the manager can do this by giving each guest all eighty keys.) What is the minimum number of keys the manager can get away with?
  • Monday, November 6: invariants.
    Problems: ps pdf. (With typo corrected)
    Problem of the week from Sound (with very clever political satire added by Ravi): An indicted senator gives you an m by n chocolate bar. What is the minimum number of times you must break it in order to get 1 by 1 squares?
  • Monday, November 13: algebraic tools and techniques.
    Problems: ps pdf.
    Problem of the week from Sound: Look at the leading decimal digit of 2^n. Does 7 appear more frequently or 8?
  • Monday, November 20: no meeting (Thanksgiving week!).
  • Monday, November 27: last regular meeting: Putnam problems and strategy.
    Problems: ps pdf.
• There will also be a Masterclass for a few experts (everyone is welcome), on Mondays (immediately after the regular seminar, roughly 7-8 pm).
  • Monday, October 9. Problems (Natth's problem now corrected): ps, pdf.
  • Monday, October 16. Problems: ps, pdf.
  • Monday, October 23. No meeting due to problem shortage.
  • Monday, October 30. Problems: ps, pdf.
  • Monday, November 6. Problems: ps, pdf.
  • Monday, November 13. Problems: ps, pdf.
  • Monday, November 20: no meeting (Thanksgiving!).
  • Monday, November 27. Problems: ps, pdf.
• 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 is also great help (more than you would think) trying many old problems. I'd especially recommend the collection of recent Putnams The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, by Kiran Kedlaya, Bjorn Poonen, and myself.
  • There are more good problems in the previous Putnam book, The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965-1984, by Alexanderson, Klosinski, and Larson (look especially at those in the 1980's).
  All of these books are in the math library (2-day reserve, under the course ``VAKIL-SEM''), in Building 380 (fourth floor). Also, on the web you can find
  • a compilation of old Putnam problems and solutions on the web, collected by Kiran Kedlaya;
  • to find solutions to older Putnams, you can read the official solutions in the American Mathematical Monthly, which you can see electronically (from any Stanford computer) here. At that page, 'Search this journal' for ``William Lowell Putnam'' in the title.
  • Some more links are available through this good UCSD site, by Patrick Fitzsimmons.
• Last year's Stanford Putnam website.

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