MAT 475: Problem Solving Seminar, Fall 2020

• Kasra Rafi (2020).
• Jacob Tsimerman (2019).
• Dror Bar-Natan (2016).

• Problem solving is an important aspect of mathematics, but in many courses you focus more on absorbing new material. The goal of this class is to introduce you to various methods of problem solving, so that you will become better at solving math problems and also at writing out solutions.
• Usually, each week we will focus on a new topic. We'll introduce new material on Thursday and then there will be a roughly 20 to 25-minute long quiz at the beginning of class on the next Thursday.
• This seminar class is meant to be very interactive (hopefully we'll manage this well in the online format!). We'll be discussing lots and lots of problems, and you will split into groups to work on them. Participation will count in this class! Discussing and presenting your ideas and solutions is a great way to improve your problem solving abilities!

• Work in groups. Try small cases. Plug in small numbers. Do examples. Look for patterns. Draw pictures. Use LOTS of paper. Talk it over. Choose effective notation. Look for symmetry. Divide into cases. Work backwards. Argue by contradiction. Consider extreme cases. Modify the problem. Generalize. Don’t give up after five minutes. Don’t be afraid of a little algebra. Sleep on it if need be. Ask.

• Week 1 (due Wed Sep 16): Read Chapter 1 (Invariance Principle). Think about some of the following problems: any on Jacob's sheet, as well as Engel 2, 3, 4, 7, 8, 14, 15, 26, 27, 31, 49, 51, 55.
• Week 2 (due Wed Sep 23): Read Chapter 2 (Coloring Proofs). Think about some of the following problems: any on Jacob's sheet, as well as Engel 1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 26.
• Week 3 (due Wed Sep 30): Read Chapter 3 (Extremal principle). Think about some of the following problems: any on Jacob's sheet, as well as Engel 4, 6, 7abc, 11a, 13 (hint: consider the person who won the most games), 14 (harder), 22 (hard), 27, 28, 32, 33.
• Week 4 (due Wed Oct 7): Read Chapter 4 (Box/pigeonhole principle). Think about some of the following problems: any on Jacob's sheet, as well as Engel 13, 15, 16, 17, 18, 19, 20, 24, 25, 27, 28, 32, 33, 35, 36, 52, 74.
• Week 5 (due Wed Oct 14): Read Chapter 6 (Number Theory). Think about some of the following problems: any on Jacob's sheet, as well as Engel 8, 10, 11, 12, 14, 18, 21, 22, 29, 31, 32, 34 (irreducible here means fraction in lowest term), 36, 37, 39, 43, 46, 53, 57, 66, 68, 72, 74, 82, 98 ('pairwise prime' here means coprime, i.e. gcd = 1), 137, 167.
• Week 6 (due Wed Oct 21): Read Chapter 7 (Inequalities). Think about some of the following problems: Engel 2, 4, 10, 11 (one way is to use the rearrangement inequality), 14, 15, 16, 17, 29 (e.g. induct), 45, 49, 52, 53, 54, 61, 67, 80 (use CS).
• Week 7 (due Wed Nov 4): Read Chapter 8 (Induction). Think about some of the following problems: any on Jacob's sheet, as well as Engel 4, 7, 10(ab), 15, 16, 17, 18, 19, 20, 22, 25, 26, 28.
• Week 8 (optional): Read Chapter 9 (Sequences). Think about some of the following problems: 1, 2, 3, 5, 17, 18, 20, 22, 27, 52, 58, 59, 60, 61, 62, 63, 64. [Class on Oct 29 cancelled.]
• Week 9 (due Wed Nov 18): Read Chapter 10 (Polynomials). Think about some of the following problems: any on Jacob's sheet, as well as Engel 2, 3, 6, 16, 17, 19, 22, 23, 24, 25, 26, 29, 33, 34, 39, 40, 44, 45, 46, 53, 55.
• Week 10 (due Wed Nov 25): Read Chapter 13 (Games). Think about some of the following problems: any on Jacob's sheet, as well as Engel 2, 3, 5, 6, 7, 8, 10, 11, 12, 17, 20, 21, 22, 23, 24, 25, 26, 28.
• Week 11 (due Wed Dec 2): Read Chapter 14.1 (Graph Theory) as well as the introduction of Jacob's sheet on this topic. Think about some of the following problems: any on Jacob's sheet, as well as Engel 1, 2, 3.
• Week 12: we will discuss a mix of problems in class, e.g. these.

• Quizzes (about 11): 55%
• Participation: 10%
• Final assessment: 35%

• Please familiarise yourself with the University of Toronto Code of Behaviour on Academic Matters. See also a simplified version.
• The University of Toronto treats cases of academic misconduct very seriously. All suspected cases of academic dishonesty will be investigated following the procedures outlined in the Code. The consequences for academic misconduct can be severe, including a failure in the course and a notation on your transcript. Every year, students get expelled permanently for academic offences.

• Please review the Minimum Technical Requirements.
• A good internet connection is essential. Headphones, microphone and webcam are highly recommended for in-class participation.
• In addition, you will need a scanner or phone camera to submit assignments and tests. If you are using your phone, I recommend you download a free scanning app (such as scanbot or camscanner), as the quality will be much better.
• Contact Student Tech Support in case of technology problems.

• A Verification of Illness (also known as a "doctor's note") is temporarily not required. Students who are absent from class for any reason (e.g., COVID, cold, flu and other illness or injury, family situation) and who require consideration for missed academic work should report their absence through the online absence declaration. The declaration is available on ACORN under the Profile and Settings menu. Students should also advise their instructor of their absence. Visit COVID-19 Information for University of Toronto Students page on the Vice-Provost, Students website for information on this and other frequently asked questions.

• University of Toronto is committed to accessibility. If you require accommodations, or have any accessibility concerns about the course, please contact Accessibility Services as soon as possible.