MAT475H1F, Problem Solving Seminar
Fall 2026
- Instructor
- Prof. Kasra Rafi, rafi@math.toronto.edu
- Office
- BA 6236
- Meetings
- Tuesdays, 3–5 p.m.; Thursdays, 2–3 p.m.
- Office hours
- Tuesdays, 2–3 p.m.; Thursdays, 1–2 p.m.; or by appointment
- Teaching assistant
- Amalrose Vayalinkal, amalrose.vayalinkal@mail.utoronto.ca
What this course is about
This course addresses the question: How do you attack a problem the likes of which you have never seen before? Students will apply Pólya’s principles of mathematical problem solving, draw upon their previous mathematical knowledge, and explore the creative side of mathematics in solving a variety of interesting problems and explaining those solutions to others.
In practice, we will make examples, find patterns, choose useful notation, change the problem, compare approaches, and explain ideas clearly. The aim is not to collect tricks, but to become more resourceful and more precise when the route to a solution is not obvious. By the end of the course, you should be better able to choose and adapt strategies for unfamiliar problems.
TRY THIS When you are stuck: try some examples, draw a picture, work backwards, replace the question with a simpler one, or explain the obstacle to someone else.
How the course works
Most weeks will focus on one broad strategy or family of ideas. Class time will mix short explanations with substantial time for problem solving and discussion. There will be weekly readings and problem sets, but homework will not be collected. The problems are preparation for the weekly quizzes, the class discussions, and the final examination.
Course textbooks
Weekly readings and problems are drawn from the following books. The pink abbreviations are used in the course calendar.
- PSTP: Loren C. Larson, Problem-Solving Through Problems, 1st edition (Springer, 1983);
- PSS: Arthur Engel, Problem-Solving Strategies, 1st edition (Springer, 1998);
- ACPS: Paul Zeitz, The Art and Craft of Problem Solving, 2nd edition (Wiley, 2007).
Editions and numbering. All assigned section, page, example, and problem numbers refer to the editions listed above. In particular, we use the second edition of Zeitz, not the newer third edition. Page numbers are those printed in the books, not the PDF page counter. If you use another edition, please check that the problem statements match.
We will not follow any one book from beginning to end.
Course calendar: readings and homework
All readings are optional suggestions to help with the homework. Homework prepares you for class and quizzes and is not collected.
| Week | Dates | Textbook | Sections / pages | Homework / practice problems |
|---|---|---|---|---|
| 1 | Sept. 8–10 | ACPS | Ch. 2 | 2.1.22, 2.1.23, 2.2.13, 2.2.19, 2.3.11, 2.3.12, 2.3.18, 2.3.21, 2.4.8, 2.4.12. |
| 2 | Sept. 15–17 | ACPS | §3.4, pp. 92–102 §3.2, pp. 73–77 |
3.4.19, 3.4.24, 3.4.26, 3.4.27; 3.2.7, 3.2.8, 3.2.12. |
| PSS | Ch. 1, pp. 1–8 Ch. 3, pp. 39–47 |
Ch. 1: 1, 3(a,b), 5 (generalization optional). Ch. 3: 13. |
||
| PSTP | — | 1.11.8. | ||
| 3 | Sept. 22–24 | PSS | Ch. 2, pp. 25–28 |
Ch. 2: 1, 3, 5, 7, 8, 9, 11, 12, 19, 29. Suggested order: 5 → 8 → 7. For Problem 9, prove both necessity and sufficiency. For Problem 12, include a configuration attaining the minimum. Optional challenges: 26, 28, 34. |
| ACPS | pp. 54–55, 101–102 | — | ||
| PSTP | §1.10, pp. 47–49 | — | ||
| 4 | Sept. 29–Oct. 1 | PSS | Ch. 4, pp. 59–63 | Pigeonhole principle. Ch. 4: 16, 19, 24, 27, 31, 35. |
| ACPS | §3.3, pp. 84–89 | 3.3.14, 3.3.15, 3.3.24. | ||
| PSTP | §2.6, pp. 79–83 | 2.6.6, 2.6.9, 2.6.11(a). | ||
| 5 | Oct. 6–8 | TBD | TBD | To be posted |
| 6 | Oct. 13–15 | TBD | TBD | To be posted |
| 7 | Oct. 20–22 | TBD | TBD | To be posted |
| — | Oct. 26–30 | Fall Reading Week — no classes or quizzes | ||
| 8 | Nov. 3–5 | TBD | TBD | To be posted |
| 9 | Nov. 10–12 | TBD | TBD | To be posted |
| 10 | Nov. 17–19 | TBD | TBD | To be posted |
| 11 | Nov. 24–26 | TBD | TBD | To be posted |
| 12 | Dec. 1–3 | TBD | TBD | To be posted |
Quizzes and solutions
Quizzes will be administered in class. In place of a separate midterm, they will provide regular opportunities to practise solving problems and give students regular feedback. Quizzes and their solutions will be posted here after the corresponding quiz has been administered.
- Quiz 0
- Quiz 1
- Quiz 2
- Quiz 3 — to be posted
- Quiz 4 — to be posted
- Quiz 5 — to be posted
- Quiz 6 — to be posted
- Quiz 7 — to be posted
- Quiz 8 — to be posted
- Quiz 9 — to be posted
- Quiz 10 — to be posted
- Quiz 11 — to be posted
Assessment
| Component | Weight |
|---|---|
| Weekly in-class quizzes | 50% |
| Class participation | 15% |
| Final examination | 35% |
The lowest two quiz grades will be dropped. The final examination will be scheduled later.
Prerequisites
MAT224H1/MAT247H1, (MAT235H1, MAT236H1)/MAT235Y1/MAT237Y1/MAT257Y1, and at least 1.0 credit at the 300+ level in APM/MAT.
Recommended preparation. You should be comfortable reading and writing proofs. Familiarity with linear algebra, elementary number theory, modular arithmetic, and plane geometry will be useful.
Communication, participation, and regrading
Course announcements and materials will appear on Quercus. Before emailing, please check whether your question is answered there. Email must come from your official U of T address and include MAT475 in the subject line.
This seminar depends on discussion. The participation grade reflects consistent preparation and constructive engagement, not mere attendance. Contributions may include trying problems, explaining partial ideas, asking useful questions, and responding thoughtfully to classmates. Students who need an alternative way to participate because of an absence or an accommodation should contact the instructor.
Regrade requests must identify a specific issue in writing and be submitted within two weeks after the work is returned. A submission may be reviewed in full, so its mark may go up, stay the same, or go down.
Missed work
Students seeking consideration for missed work must document the absence as appropriate through the Absence Declaration (available once per term), a Verification of Illness or Injury form, a College Registrar letter, or an Accessibility Services accommodation letter. See the Arts & Science absence policy.
The two dropped quiz grades are intended to cover ordinary short absences. If illness, injury, an accommodation, or another significant circumstance causes you to miss additional quizzes, contact the instructor as soon as possible to discuss an appropriate accommodation.
Academic integrity and use of AI
Work submitted for credit must represent your own honest mathematical thinking and writing. No outside resources, computational tools, or AI tools may be used during quizzes or examinations unless explicitly authorized. Such tools may be used for ungraded practice, but their output can be unreliable and is no substitute for working through the argument yourself.
Suspected academic dishonesty will be handled under the Code of Behaviour on Academic Matters. Further guidance is available from the Academic Integrity website.
Accessibility, inclusion, and support
Students who need disability-related accommodations should contact Accessibility Services as early as possible and are welcome to discuss how accommodations apply in this course. We will work to maintain a respectful learning environment in which students can participate and express mathematical ideas without discrimination or harassment.
For current policies and support links, consult the Arts & Science Academic Handbook and Academic Dates & Deadlines.