MAT 475: Problem Solving Seminar, Fall 2021

• my page from last year (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. 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 15): 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 22): 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 29): 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 6): 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 13): 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 20): 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 Oct 27): 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 (due Wed Nov 3): 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.
• Week 9 (due Wed Nov 17): 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 24): 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 1): 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, see quercus.

• Quizzes (about 11): 50%
• Participation: 10%
• Final assessment: 40%

• Quizzes (about 11): 83%
• Participation: 17%

• 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.

• Masks are an inexpensive and effective measure that limits the spread of COVID and will facilitate the return to normal life as quickly as possible. Failure to wear a mask properly entails unnecessary risks to public health and may disrupt learning by creating unwelcome distractions. It is the policy of the Math Department that in-person instruction cannot take place unless all students are wearing a mask that covers both mouth and nose, with exceptions only for students who have received documented exemptions.
• As with other accommodations, any student who has an official exemption from wearing a mask is expected inform the instructor BEFORE classes begin, providing documentation.

• 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.

• 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.