Math 235: Mathematical Problem Solving, Fall 2023 and Fall 2026
Professor: Darij Grinberg
Organization
Course description
An introduction to mathematical problem solving.
We will learn techniques and tools for solving problems of the kind that appear in mathematical competitions and journals.
These techniques (like induction, the Pigeonhole Principle, modular arithmetic or the Cauchy-Schwarz inequality) have uses all over mathematics; we will explore these uses through hands-on problem solving.
A typical week will include some preparatory reading, a few problems solved together in class, and a homework set.
Prerequisites: Math 200.
Course materials
- Recommended:
- Darij Grinberg, Notes on mathematical problem solving: The notes for my Math 235 course in Fall 2020. We will not follow them, but there will be some overlap, and we will occasionally refer to them for background and additional examples.
- Virginia Tech Regional Mathematics Contest. A discontinued competition similar to the Putnam.
- David Galvin, Math 43900 - Problem Solving in Math (Fall 2020). Lecture notes from a course similar to ours (follow the link to Overleaf, allow javascript and note the "Download PDF" button).
- Simon Foucart, Problem Solving. Brief Math 235 notes from a few years ago.
- Răzvan Gelca, Titu Andreescu, Putnam and Beyond, 2nd edition. Standard text for university competition training. At 857 pages, it goes far beyond what we can do in a quarter.
- Kiran Kedlaya et al., Putnam archive. Contains solutions to several years of Putnam exams.
- John Scholes aka kalva, Maths problems. Classical (early 2000s) collection of contest problems with (terse) solutions.
- Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, 2nd edition. Collection of all problems ever suggested for the International Mathematical Olympiads, often with solutions. You can also get official solutions for recent shortlist problems from imo-official.org. We aren't training for the IMO, but there is a lot of overlap between IMO and Putnam topics, and IMO shortlists are among the best sources for high-quality problems.
- Evan Chen, OTIS Excerpts. Olympiad training lecture notes (with focus on high school contests, but lots of relevance to undergraduate ones too).
- Arthur Engel, Problem-Solving Strategies. One of the first books written explicitly for math contest training. Somewhat dated and not always well-written, but a venerable collection of problems sorted by theme, and an inspiration for newer texts.
- Alexander Bogomolny, Cut the Knot. Famous collection of maths puzzles, including various contest problems (most hidden behind the "and more..." links). Unfortunately, the Java applets no longer run on most browsers. Use the Wayback Machine for some links that have disappeared.
- Padraic Bartlett, various courses. Lots of problem solving materials, very enjoyably written and split into bite-sized (mostly self-contained) pieces.
- Berkeley Math Circle archives. Olympiad training worksheets on various levels.
- The Fall 2021 version of this course: We will (mostly) reuse Worksheets 1--5 from that version. Worksheet 6 has extra material on graph theory, which is another topic worth exploring but usually not that important for Putnam.
- Contests:
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Course calendar
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- No Copyright:
- The above lecture notes and assignments have been released under the CC0 license, i.e., are dedicated to the public domain. They can be copied, modified and distributed without permission. See the license for details.
Grading and policies
- Grading matrix:
- 100%: homework sets. The lowest score will be dropped if all graded homework sets are submitted.
- Grade scale:
- These numbers are tentative and subject to change:
- A: (50%, 100%].
- B: (40%, 50%].
- C: (20%, 40%].
- D: [0%, 20%].
- Homework policy:
- Collaboration and reading are allowed, but you have to write solutions in your own words and acknowledge all sources that you used.
- If you have already seen a homework problem before, you are free to reuse the solution that you know. Anything you come across in the literature is fine, but you must not deliberately seek solutions to homework problems in the literature or contact outsiders (anyone apart from Math 235 students and Drexel employees) for help with the problems (online or offline). (In particular, you cannot post homework as questions on math.stackexchange before the due date!) Likewise, consulting artificial intelligence (e.g., GPT or Claude) for help with the problems is not allowed.
- Late homework will not be accepted.
- Solutions must be typeset; handwriting will not be accepted! To typeset text with formulas, you can use any of LaTeX (a free online editor is Overleaf), Markdown (a free online editor is Stackedit), LibreOffice (which has a built-in equation editor), Google Docs (which, too, has an equation editor inside), or many other tools. You can even type in a text editor if you use standard conventions to write your formulas in ASCII. In either case, convert to PDF (e.g., by printing to PDF) and submit to Gradescope.
- Solutions have to be submitted electronically via the Gradescope that is included in Canvas. If there are problems with submission, send your work to me by email for good measure (I can read PDF and TeX, but not doc/docx).
- Expected outcomes:
- Students should have obtained some hands-on experience solving competition-type mathematical problems. They should be aware of standard problem solving techniques in mathematics (such as the pigeonhole and extremal principles) and be familiar with examples of their application.
Other resources
- University policies:
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- Disability resources:
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Back to Darij Grinberg's teaching page.