Florian Herzig - Math 475

MAT 475: Problem Solving Seminar, Fall 2022

Instructor: Florian Herzig; my last name at math dot toronto dot edu
Office Hours (online/zoom): Wed 3:45-4:45pm or by appointment
TA: Daniel Spivak
TA Office Hours (online/zoom): Tue 7-8pm
Official syllabus (preliminary)
Lectures: Tuesdays 1-2pm, Thursdays 1-3pm
The class location will be BA2155.

Textbook: Problem-solving strategies by Arthur Engel (click link for online access!)
Another helpful book: Larson's Problem Solving Through Problems.

Final assessment: Tue Dec 13, 9am-12pm

Some previous course homepages for MAT475:

Course description:

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!

General hints for this course:

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.

Homework: Weekly homework will be assigned but not collected. Quiz problems will be related to the assignment. Practice is essential in this course! You are encouraged to work together with other students on homework!

Week 1 (due Wed Sep 14): 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 21): 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 28): Read Chapter 3 (Extremal principle). Think about some of the following problems: any on Jacob's sheet, as well as Engel 4, 6, 7abcd, 11a, 13 (hint: consider the person who won the most games), 14 (harder), 22 (hard), 27, 28, 32, 33.
Week 4 (due Wed Oct 5): 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 12): 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 19): 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 26): Read Chapter 8 (Induction). Think about some of the following problems: any on Jacob's sheet, as well as Engel 7, 10(ab), 15, 16, 17 [note: 2n+1 should be 2n-1!], 18, 19, 20, 22, 25, 26, 28 [tricky], 37 [note typo: last number should be 100117], 39.
Week 8 (due Wed Nov 2): Read Chapter 9 (Sequences). Think about some of the following problems: 1, 2, 3, 5, 17, 18 [solve #3 first!], 20, 22, 27, 49 [typo: show a_k <= 0!], 50, 52, 58, 59, 60, 61, 62, 63, 64.
Week 9 (due Wed Nov 16): 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 23): 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 Nov 30): 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.

Marking scheme:

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

There are no make-up quizzes, but the lowest three quiz scores will be dropped.

Academic integrity:

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.

Accessibility:

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.