Art and Craft of Mathematical Problem Solving

Course Description

Let an award-winning professor and former champion “mathlete” demonstrate how solving math problems can be fun by teaching techniques you can use in many aspects of life.

Video Lectures

01: Problems versus Exercises (30 min)

Begin your problem-solving journey with three entertaining problems that teach “wishful thinking” and “get your hands dirty” strategies, comparing math problems to hiking adventures.

02: Strategies and Tactics (29 min)

Learn the difference between strategies, tactics, and tools in problem solving through puzzles like a cryptic census reply and jumping frogs on a number line.

03: The Problem Solver’s Mind-Set (33 min)

Explore psychological aspects of problem solving – concentration, creativity, and confidence – while avoiding overreliance on narrow mathematical tricks through “think outside the box” challenges.

04: Searching for Patterns (29 min)

Develop pattern recognition skills by investigating trapezoidal numbers and Pascal’s triangle, experiencing both successful conjectures and cautionary failures.

05: Closing the Deal – Proofs and Tools (33 min)

Master different proof techniques including deductive proof, proof by contradiction, and algorithmic proof to validate earlier conjectures.

06: Pictures, Recasting, and Points of View (29 min)

Solve word problems through three breakthrough strategies: drawing pictures, changing perspectives, and recasting problems in new forms.

07: The Great Simplifier – Parity (29 min)

Apply parity concepts to wizard puzzles and locker problems, then see how this leads naturally into graph theory applications.

08: The Great Unifier – Symmetry (29 min)

Discover how to impose symmetry where none is obvious, demonstrated through problems like finding the shortest detour to fetch water.

09: Symmetry Wins Games! (31 min)

Develop winning strategies for combinatorial games like “puppies and kittens” by uncovering hidden symmetrical patterns.

10: Contemplate Extreme Values (30 min)

Solve challenging puzzles by focusing on minimal or maximal values in problems – a simple idea with powerful applications.

11: The Culture of Problem Solving (30 min)

Explore the global community of problem solvers, particularly strong in Russia and Eastern Europe, who specialize in non-traditional mathematics.

12: Recasting Integers Geometrically (27 min)

Tackle the famous “chicken nuggets” problem (largest unobtainable combination of 7 and 10) using visual geometric approaches.

13: Recasting Integers with Counting and Series (32 min)

Apply recasting and rule-breaking to classical number theory including Fermat’s Little Theorem and Euler’s proof of infinite primes.

14: The Pigeonhole Tactic (30 min)

Use the pigeonhole principle (n+1 items in n containers) to solve diverse problems and explore Ramsey theory’s patterns in randomness.

15: The Greatest Unifier of All – Invariants (31 min)

Discover how invariants – quantities that remain unchanged – form the most powerful problem-solving tactic, encompassing symmetry and parity.

16: Optimizing 3s and 2s (32 min)

Solve an International Mathematical Olympiad problem about maximizing products with fixed sums using algorithmic proof techniques.

17: Using Physical Intuition (31 min)

Apply developed skills to challenging problems including marble collisions on tracks, Gardner’s airplane puzzle, and laser beam reflections.

18: Geometry and Transformations (31 min)

Solve baffling geometry problems by applying rotations, reflections and other transformations pioneered by Felix Klein.

19: Mathematical Induction (30 min)

Learn when proof by mathematical induction is essential, typically in recursive situations where complex structures build from simple ones.

20: Induction on a Grand Scale (32 min)

Calculate the probability of even numbers in Pascal’s triangle using sophisticated inductive proof techniques.

21: Weird Dice (32 min)

Explore whether differently numbered dice can have the same probability distribution as standard dice through polynomial recasting.

22: Solving Very Hard Problems (30 min)

Apply the pigeonhole principle to find patterns in seemingly random, enormous structures – demonstrating limitless problem-solving potential.

23: Conway’s Infinite Checkers (32 min)

Examine John Conway’s famous checkers problem and learn about mathematical icons Paul Erdős and Évariste Galois.

24: How versus Why (34 min)

Review problem-solving tactics, introduce complex numbers, and advocate for understanding why solutions work, not just how they’re found.