Advanced Problem Solving and Proof

Explorations in Number Theory, Combinatorics, and Graph Theory

Advanced Problem Solving & Proof is a rigorous, proof-based course for students prepared to engage seriously with abstract reasoning and mathematical argument.

This course draws on the traditions of math circles and advanced problem-solving seminars to develop students’ mathematical maturity through sustained engagement with challenging problems and carefully structured explorations. Emphasis is placed on constructing clear, rigorous proofs; articulating mathematical ideas with precision; and understanding the underlying structures that unify diverse areas of mathematics. Close mentorship, discussion, and feedback play a central role in supporting students’ growth as independent mathematical thinkers.

Topics include combinatorics, number theory, and graph theory, with problems selected to emphasize proof techniques, logical structure, and mathematical exposition rather than routine computation.

The problems are intentionally crafted by Dr. Paquin not only to build deep problem-solving skills, but also to guide students toward ideas that culminate in open research questions pursued in the Research Seminar in Mathematics (Spring/Summer 2026).

Enrollment for all courses is limited to 12 students to ensure high-contact mentorship, seminar-style discussion, and detailed feedback on problem solving and proof writing.

  • Advanced Problem Solving & Proof is a rigorous, proof-based mathematics course designed to help students learn how mathematicians think, reason, and communicate. The course emphasizes deep problem solving, logical reasoning, and clear mathematical writing rather than speed or rote techniques.

  • Students will work through a carefully designed sequence of challenging problems and write complete, rigorous solutions to each one. Over the term, they will build a cumulative mathematical portfolio that reflects sustained effort, revision, and growth. The focus is on explaining why results are true and presenting arguments clearly and confidently.

  • Students receive ongoing, individualized feedback on both their mathematical reasoning and written exposition. Revision is an expected and important part of the course, and students are supported as they refine their work over time.

  • By the end of the term, each student will have a polished portfolio of high-level mathematical proofs and exposition. This portfolio demonstrates depth of understanding, persistence, and clarity of thought and can be used for undergraduate and graduate program applications, research programs, math enrichment opportunities, and letters of recommendation.

  • Yes. While the course is challenging, it is designed to support motivated students who are new to proof-based mathematics. Clear guidance, examples, and feedback are provided throughout, and students are encouraged to grow into mathematical maturity over time.

  • Students should expect to spend time each week engaging thoughtfully with the material — thinking about problems, writing solutions, and revising their work — but Dr. Paquin is very flexible and understands that students have other academic, extracurricular, and personal commitments. There is no fixed expectation for the number of hours students must spend outside of class. Students are encouraged to engage as deeply as they are able, and may spend as little or as much time as they would like working on the course material outside of scheduled meetings. The emphasis is always on depth, clarity, and quality of thinking, not speed or volume, and students are supported in finding a pace that works well for them.

  • The skills developed in this course — reasoning carefully, writing clearly, persevering through challenging problems, and revising work thoughtfully — are foundational for success in advanced mathematics, research, and related fields. Students leave the course better prepared, more confident, and more independent as mathematical thinkers.

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