The Pólya Problem-Solving Seminar 2015

The Pólya Problem-Solving Seminar is a series of informal dinner-time problem solving practice sessions this fall quarter, on Mondays 6-8 pm.  Some faculty and graduate students always turn up with wise words.   The seminar can be taken for credit (in the guise of Math 193).

There will also be a Masterclass for a few experts (everyone is welcome), on Mondays (an hour, immediately after the regular seminar, starting roughly at 7:45 or 8).

The Pólya Seminar can be useful preparation for the William Lowell Putnam Competition, but is logically independent.

(For information about the Putnam competition, click here.  For some resources, click here.  For the Math 193 Coursework site, click here — those not in the class but students at Stanford can still sign up here.)

If you are (even potentially) interested in hearing about the seminar, please let us know (by emailing either of us).  We maintain an email list (not the coursework list associated to Math 193) for those interested, where we send announcements.

Practice problems will include some on the topics discussed, and some others. Handouts will be posted on this page.

• Monday, September 21: introductory meeting. Here is the introductory handout.
• Monday, September 28: warming up: induction, and the pigeonhole principle.  Here is the handout.
• Monday, October 5:  number theory.  Here is the handout.  We also had the first meeting of the Masterclass.  The Masterclass problems are here.  At risk of making you hungry, here is a picture of a cake Lisa Sauermann’s mom baked in summer 2011 (relevant to one of the masterclass problems): 
• Monday, October 12:  combinatorics.  Here is the handout.  Here are the Masterclass problems.
• Monday, October 19:  invariants.  Here is the handout.  Here are the Masterclass problems.
• Monday, October 26:  inequalities.  Here is the handout. Here are the masterclass problems.
  • Here is Zeb Brady’s one-line (but evil) proof of the Arithmetic-Geometric Mean Inequality:  For any , we have
    which is manifestly nonnegative.
• Monday, November 2:  probability and expected values.  Here is the handout.  Here are the Masterclass problems.
• Monday, November 9:  recursions and generating functions.  Here is the handout.  (No Masterclass this week.)
  
• Monday, November 16:  analysis.  Here is the handout. Here are the masterclass problems.
• Monday, November 23:  no meeting (Thanksgiving break).
• Monday, November 30:  general problem-solving practice.  Here is the handout.  Here are the masterclass problems.  Also, Gyujin Oh has collected some nonstandard techniques that have appeared on the Putnam that he is willing to share.  Here it is!
• Saturday, December 5:  The Putnam Competition!

Want more? Additional suggestions for others? Please let us know!

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Resources

(This has not been thought through particularly thoroughly, but I wanted to get this page posted.)

Recommended reading:

• I (=Ravi) really like Loren Larson’s Problem Solving Through Problems — definitely worth owning. It’s great preparation for the Putnam, but also great in general.
• Ditto for Paul Zeitz’ The Art and Craft of Problem Solving.  It is ostensibly aimed at high school students, but there is no sell-by date on this stuff.
• 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 Ravi Vakil.
• 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).

Larson and Putnam ’85-’00 are in the math library.  Also, on the web you can find

• a compilation of old Putnam problems and solutions on the web, collected
  by Kiran Kedlaya (link to be added — but you can find them by google);
• 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.
• The Art of Problem Solving is terrific.
• Some more links are available through this good UCSD site, by Patrick Fitzsimmons.

Please pass on any suggestions you like!

The William Lowell Putnam Mathematical Competition (Saturday, December 5, 2015)

The Putnam is a challenging opportunity for you to test your mathematical mettle. In the 2014 competition, about 4300 individuals participated, representing over 550 colleges and universities across Canada and the U.S.  The Stanford team placed seventh, and many individuals did very well.  The measure we look to is the number of competitors on the list of top performers sent out with the results.

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 throughout the mathematical sciences, 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.)

You can just take the Putnam without thinking about it beforehand.  But you are also welcome to come to the Pólya Problem-Solving Seminar held on Monday evenings over dinner.  And you can also check out these resouces.  (Here is the official Putnam website.)

If you are (even potentially) interested in trying the Putnam, please let us know (by emailing either of us).  We maintain an email list (not the coursework list associated to Math 193) for those interested, and we will announce the location, etc. there.

Want more? Additional suggestions for others? Please let us know!