MATH 190 "Topics in Problem Solving"

Description

This seminar is designed for undergraduates with an interest in the problem solving approach to mathematics, including students who are considering participating in college-level mathematics competition such as The William Lowell Putnam Mathematical Competition. Students should have completed Calculus I-II, multivariable calculus, and linear algebra (with differential equations).

The purpose of the seminar is not to provide a comprehensive survey of Putnam mathematics, but to develop a small, durable repertoire of problem-solving techniques and, more importantly, the ability to recognize when those techniques might apply. The skills developed in the seminar are intended to be useful well beyond the Putnam competition: students should become more comfortable approaching unfamiliar mathematical problems, testing ideas, learning from unsuccessful approaches, and constructing complete proofs.

Format

There are seven meetings of 75 minutes each. The seminar emphasizes active problem solving, discussion of failed approaches, and the presentation of complete proofs.

Students will work on problems independently during the seminar. An important goal of the course is to give students sustained practice in thinking through a difficult problem on their own: deciding where to begin, testing ideas, recognizing when an approach is unproductive, and developing a solution without relying on someone else to supply the key idea.

The problems will draw on a range of mathematical ideas, but the emphasis will be on general problem-solving strategies rather than on mastering a particular collection of topics. Depending on the problems selected, we may encounter techniques involving, for example, induction, invariants, extremal arguments, counting, number theory, algebraic manipulation, inequalities, and geometry.

Students should leave the course better able to attack unfamiliar mathematical problems independently, whether or not they ultimately become Putnam contestants. The ability to work independently through a difficult problem is a valuable mathematical skill and one that can serve students well in their future studies and careers.

Approaching an unfamiliar problem

A central goal of the seminar is to develop a productive response to an unfamiliar problem. Rather than waiting for a familiar theorem or technique to announce itself, students will practice a deliberate process of exploration and selection. When faced with an unfamiliar problem, try to:

1. Spend ten productive minutes experimenting. Test small cases, make calculations, draw pictures, look for patterns, and try simple examples and counterexamples.
2. Identify several plausible structures. Ask what the problem might be really about: an invariant, an extremal object, a symmetry, an induction, a counting argument, an algebraic identity, or some other structure.
3. Choose a promising line of attack. Commit to an approach and investigate it rather than waiting for a familiar theorem or standard solution to appear.

An unsuccessful approach is not necessarily wasted effort. Understanding why an idea fails is often an important part of discovering what the problem requires.

Grading

The grade for the course is based entirely on in-class participation (40%) and written assignments (60%).

• Participation: 40%. Participation includes actively working on problems, contributing to discussions, asking questions, explaining approaches, and presenting or discussing solutions.
• Written assignments: 60%. There will be seven assignments, of which only the best five will count toward the course grade.

There will be no make-up homework. The homework should be submitted by email to the instructor as a single PDF file before the start of the following lecture.

Independent work and academic honesty

All problem solving for this seminar is intended to be done independently. Students should work on their own during class and should complete the homework without assistance from other students or from outside sources. This policy is intentional: learning to make progress on a difficult, unfamiliar problem without having someone else provide the key idea is an important part of mathematical development.

Students are encouraged to ask questions of the instructor when they are stuck and to discuss ideas and unsuccessful approaches during the class discussion. However, students should develop their own solutions before looking at or discussing another person's solution.

In particular, submitted homework should represent the student's own mathematical thinking and writing. Students should not obtain solutions, hints, or substantial assistance from other students, solution manuals, online forums, or other external sources.

You are responsible for knowing and abiding by the University of Rochester's academic honesty policy . Any violation of academic honesty will be pursued according to the specified procedures.

Extra resources

The following books on problem solving can be found as e-books through the University of Rochester Libraries:

• Putnam and Beyond, by Razvan Gelca and Titu Andreescu. This book provides a broad collection of Putnam-style problems and techniques.
• Problem-Solving Strategies, by Arthur Engel. This book emphasizes general-purpose strategies for attacking challenging mathematical problems.

You may also find the Art of Problem Solving community useful as a source of additional problems and mathematical discussions. For purposes of this seminar, however, students should not use online solutions or discussions to obtain assistance with assigned homework.

Tentative weekly schedule