Math 488A (and 575P): Problem Solving

Information Page, Fall 2005

(The following course description borrows heavily from the introduction to the textbook.)

Problem solving is a part of mathematics organized around methods rather than theorems. For this reason it is somewhat amorphous, and difficult to study, but the ideas are applicable to almost any part of mathematics. For those not familiar with the area, there are some obvious questions about this field. For example, don't we do problem-solving in every area of mathematics, and every course we take? The answer is no, and to understand why, one needs to know what is meant in this context by a "problem". For purposes of this class, there is a distinction between problems and exercises. An exercise is a question that tests mastery of a narrowly focused technique. For example, if you have studied the chain rule for differentiation, finding the derivative of sin(cos x) is an exercise. In a problem, by contrast, the technique to be used is not immediately apparent. Problems are sometimes open-ended, poorly-defined, or even unsolvable. To give you an idea of what will be studied in this class, here are some sample problems.

This course will be run partly in a seminar format; students will be expected to regularly present their own work to the class. However, grades will be based mostly on a combination of homework and in-class, closed-book tests. In particular, students will be required to take the Putnam exam in December to obtain credit for the course.

The course is worth 4 credits and will meet MWF 3:30-4:30.

Text: The Art and Craft of Problem-Solving, by Paul Zeitz.

Prerequisite: Math 330 or consent of instructor. From the preface to the textbook: "This is a book about mathematical problem-solving, for college-level novices. By this I mean bright people who know some mathematics (ideally, at least some calculus), who have at least a vague notion of proof, but who have spent most of their time doing exercises rather than problems."

Grading and other technical issues: see course information

Homework

Here is a solution to problem 3.2.7.

Putnam Problems Handouts: Level 1 Level 2

I am too lazy to get the diagram from 1999 B-1 to work in LaTeX. However, a correct version of the problem, together with all problems from recent years, can be found here.

Tests

Final Exam Requirements You must be able to solve the following questions from past Putnams: 2000-2005 A-1 and B-1. We are well aware that solutions for all these problems can be found on the web. You should be aware that these solutions are not sufficiently detailed to get full credit on your exam. You should also be aware that we reserve the right to put minor variations on these problems on your exam. Here are some examples of solutions that are sufficiently detailed to get full credit.

Sample Test 2 is now available.

Solutions to Test 2 are also available.