Math 294A: Problem-Solving Seminar

The goal of this seminar is to improve students' problem solving skills by presenting several important techniques and providing numerous examples and problems for each technique. The seminar meets informally once a week, and anyone is welcome to attend. It is also possible to register for the seminar as a one-credit pass-fail course, Math 294A. In Fall 2009, the seminar meets on Mondays from 4:15 to 5:30 in Math 501.

The seminar will cover six or seven major problem-solving techniques or concepts each semester. We will spend two weeks on each topic: an introductory session and a presentation session. During the introductory session, the technique will be presented with examples. Several problems that make use of the technique will be distributed, and students are asked to work on those problems during the week, writing up careful solutions to those that they have solved. During the presentation session, students will present their solutions to one another.

The techniques and problems we will consider are interesting and important in their own right, but are also designed to prepare students for the Putnam Exam, a competitive nationwide exam for mathematics undergraduates. The William Lowell Putnam Mathematics Competition is expected to take place on Saturday, December 5, 2009. See below for additional resources that may be useful in preparing for this difficult exam.

Students who are officially enrolled in the course will receive an alternative grade, which may be S (Superior Pass), P (Pass), C, D, or E. The grades S and P are NOT included in the calculation of the GPA, nor do they count toward meeting the criteria for dean's list, honorable mention, or academic distinctions. A grade of Superior Pass will be assigned to students with exemplary attendance (no more than two absences) who present one or more problems during the seminar. Students who attend at least half of the seminar meetings and present at least one problem during the semester will receive a grade of Pass. Otherwise, a regular grade of C, D, or E will be assigned at the discretion of the instructor. All students who attend the seminar, but particularly those who are enrolled in Math 294A, are strongly urged to take the Putnam Exam. Taking the Putnam exam may substitute for the attendance requirement for a superior pass.

Problems for Fall 2009

The Pigeonhole Principle: (8/24, 8/31) handout

Parity and Invariants: (9/14, 9/21) handout

Number theory: (9/28, 10/5) handout

Combinatorics: (10/12, 10/19) handout

Calculus: (10/26, 11/2) handout

Probability: (11/9, 11/16) handout

Games: (11/23, 11/30) handout

Note: Few if any of the above problems are original to us, but for the most part we have not included attributions.

Putnam Competition Resources

  • Past Putnam Problems
    Here you can find problems, solutions and scores from 1985 to 2005.

  • More Past Putnam Problems
    Here you can find problems and solutions from 1938 to 2003.

  • Proposed Putnam Problems by Richard Stanley
    Richard Stanley is one of the most respected people in the problem-solving community. These are some problems proposed for Putnam that didn't make the cut. The problems were intended to be A1 or B1 so they are are solvable and make very good practice.

  • Problem Solving Seminar at Dalhousie University by Richard Hoshino
    This resource has both problems and explanations of the most popular topics. Good place to start, since more explanations are given than on the following sites.

  • Problem Solving Seminar at MIT
    MIT Putnam prep site. Most problems are doable, but difficult.

  • Problem Solving Seminar at Stanford by Ravi Vakil
    Problems of various difficulty and a master class for experts.

  • Putnam Preparation Problems at University of Waterloo
    Here you can find small problem sets geared specifically for Putnam. Some of the more advanced calculus topics are covered here.

Books about Problem-Solving

  • Problem Solving Through Problems by Loren C. Larson
    This is a practical anthology of some of the best elementary problems in different branches of mathematics. They are selected for their aesthetic appeal as well as their instructional value, and are organized to highlight the most common problem-solving techniques encountered in undergraduate mathematics. Readers learn important principles and broad strategies for coping with the experience of solving problems, while tackling specific cases on their own. The material is classroom tested and has been found particularly helpful for students preparing for the Putnam exam. For easy reference, the problems are arranged by subject.

  • The Art and Craft of Problem Solving by Paul Zeitz
    Zeitz covers the essentials of Algebra, Combinatorics, Number Theory and Calculus. Excellent book for beginners, since it teaches how to approach a problem, how to think about a problem rather than just presenting numerous examples.

  • The William Lowell Putnam Mathematical Competition 1985-2000 by K. S. Kedlaya, B. Poonen, and R. Vakil
    This book includes problems, solutions and results from 1985 through 2000. Nice exposition, and includes hints to solutions in the middle of the book in case you get stuck, and you don't want to read the solution. The authors are all Putnam fellows.

  • Problem-Solving Strategies by Arthur Engel
    The author is a coach for the German IMO (International Mathematical Olympiad) team. The book gives strategy-by-example exposition of most essential ideas. Topics include the invariance principle, coloring proofs, the extremal principle, the box principle (aka the pigeonhole principle), enumerative combinatorics, number theory, inequalities, mathematical induction, sequences, polynomials, functional equations, geometry, and games.

  • Concrete Mathematics: A Foundation for Computer Science by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik This is a good book to study the basics of discrete mathematics. Topics include induction, sums, recurrences, floors and ceilings, elementary number theory, binomial coefficients, generating functions, and discrete probability. This book was intended to develop the mathematical apparatus for computer scientists to analyze algorithms, but it's very useful for problem solvers, because the theory is very meticulous, and most of the examples from the book are olympiad-type problems.