Art and Craft of Mathematical Problem Solving

Art and Craft of Mathematical Problem Solving
Course Trailer
Problems versus Exercises
1: Problems versus Exercises

Solving a math problem is like taking a hike, or even climbing a mountain. It's exciting, challenging, and unpredictable. Get started with three entertaining problems that plunge you into thinking like a problem solver and illustrate two useful strategies: "wishful thinking" and "get your hands dirty."...

30 min
Strategies and Tactics
2: Strategies and Tactics

Learn the difference between strategies, tactics, and tools when applied to problem solving. Try to decipher a puzzling reply to a census question, and determine whether three jumping frogs will ever land on a given point....

29 min
The Problem Solver's Mind-Set
3: The Problem Solver's Mind-Set

Delve deeper into the psychological aspects of problem solving-especially concentration, creativity, and confidence-and ways to enhance them. Learn to avoid overreliance on very narrowly focused mathematical tricks, and investigate a number of "think outside the box" problems, including the original that gave the name to this strategy....

33 min
Searching for Patterns
4: Searching for Patterns

Brainstorm an array of problems with the goal of building your receptiveness to discovery. See how far you can go by just letting yourself look for interesting patterns, experiencing both conjectures that work as well as cautionary examples of those that don't. The core of the lecture is an investigation into trapezoidal numbers and a search for patterns in Pascal's triangle....

29 min
Closing the Deal-Proofs and Tools
5: Closing the Deal-Proofs and Tools

Learn how to "close the deal" on some of the outstanding conjectures from the previous lecture by using airtight arguments, or proofs. These include deductive proof, proof by contradiction, and algorithmic proof-along with the narrow (and often overestimated) power of specific tools or "tricks," such as the "massage" tool, to make a mathematical expression simpler....

33 min
Pictures, Recasting, and Points of View
6: Pictures, Recasting, and Points of View

Explore three strategies for achieving a problem-solving breakthrough: draw a picture, change your point of view, and recast the problem. Try these strategies on a selection of intriguing word problems that almost magically yield an answer, once you find a creative way of analyzing the situation....

29 min
The Great Simplifier-Parity
7: The Great Simplifier-Parity

Applying the problem-solving tactic of parity, test your wits against an evil wizard, an open-and-shut row of lockers, and other colorful conundrums. Then see how parity leads naturally into graph theory, a playground for investigation that has nothing to do with conventional graphs....

29 min
The Great Unifier-Symmetry
8: The Great Unifier-Symmetry

Having used symmetrical principles to tackle problems in earlier lectures, take a closer look at this powerful tactic. Discover that when symmetry isn't evident, impose it! This approach lets you compute the shortest distance to grandma's when you first have to detour to a river to fetch water....

29 min
Symmetry Wins Games!
9: Symmetry Wins Games!

Devise winning strategies for several fun but baffling combinatorial games. One is the "puppies and kittens" exercise, a series of moves and countermoves that can be taught to children but that is amazingly hard to play well; that is, until you uncover its secrets with symmetry and a few other ideas....

31 min
Contemplate Extreme Values
10: Contemplate Extreme Values

Take your problem-solving skills to extremes on a variety of mathematical puzzles by learning how to contemplate the minimal or maximal values in a problem. This "extreme" principle is a simple idea, but it has the nearly magical ability to solve hard problems almost instantly....

30 min
The Culture of Problem Solving
11: The Culture of Problem Solving

Detour into the hidden world of problem solvers-young people and their mentors who live and breathe nontraditional, nontextbook mathematics such as what you have been studying in this course. The movement is especially strong in Russia and eastern Europe but is catching on in the United States....

30 min
Recasting Integers Geometrically
12: Recasting Integers Geometrically

Delve deeply into the famous "chicken nuggets" problem. In brief, what's the largest number of nuggets that you can't order by combining boxes of 7 and 10 nuggets? There are many roads to a solution, but you focus on a visual approach by counting points in a geometric plane....

27 min
Recasting Integers with Counting and Series
13: Recasting Integers with Counting and Series

Apply the powerful strategies of recasting and rule-breaking to two classical theorems in number theory: Fermat's "little" theorem and Euler's proof of the infinitude of primes....

32 min
Things in Categories-The Pigeonhole Tactic
14: Things in Categories-The Pigeonhole Tactic

According to the pigeonhole principle, if you try to put n + 1 pigeons into n pigeonholes, at least one hole will contain at least two pigeons. See how this simple idea can solve an amazing variety of problems. Also, delve into Ramsey theory, a systematic way of finding patterns in seemingly random structures....

30 min
The Greatest Unifier of All-Invariants
15: The Greatest Unifier of All-Invariants

To Professor Zeitz, the single most important word in all of mathematics is "invariants." Discover how this granddaddy of all problem-solving tactics-which involves quantities and qualities that stay unchanged-can be used almost anywhere and encompasses such ideas as symmetry and parity....

31 min
Squarer Is Better-Optimizing 3s and 2s
16: Squarer Is Better-Optimizing 3s and 2s

What is the largest number that is the product of positive integers whose sum is 1,976? Tackle this question from the 1976 International Mathematical Olympiad with the method of algorithmic proof, in which you devise a sequence of steps-an algorithm-that is guaranteed to solve the problem....

32 min
Using Physical Intuition-and Imagination
17: Using Physical Intuition-and Imagination

Draw on your skills developed so far to solve a tricky problem about marbles colliding on a circular track. Martin Gardner's airplane problem and a question about how many times a laser beam reflects between two intersecting mirrors help you warm up to a solution....

31 min
Geometry and the Transformation Tactic
18: Geometry and the Transformation Tactic

Focusing on geometry, consider some baffling problems that become almost trivial once you know how to apply rotations, reflections, and other geometric transformations of your normal point of view. This clever tactic was pioneered by the 19th-century mathematician Felix Klein....

31 min
Building from Simple to Complex with Induction
19: Building from Simple to Complex with Induction

Sometimes a problem demands a different type of proof from the ones you learned in Lecture 5. Study cases in which proof by mathematical induction is the only feasible approach. These typically occur in recursive situations, where a complicated structure emerges from a simpler one....

30 min
Induction on a Grand Scale
20: Induction on a Grand Scale

Continuing your use of inductive proof, calculate the probability that a randomly chosen number in Pascal's triangle is even. This problem is surprisingly easy to investigate, but it requires sophistication to resolve. But by now you have a good grasp of the methods you need....

32 min
Recasting Numbers as Polynomials-Weird Dice
21: Recasting Numbers as Polynomials-Weird Dice

Is it possible to find weird dice that "play fairly"? These are two dice that are numbered differently from standard dice but that have the same probability of rolling 2, 3, 4, and so on through 12. Learn that, amazingly, the answer is yes....

32 min
A Relentless Tactic Solves a Very Hard Problem
22: A Relentless Tactic Solves a Very Hard Problem

In a lecture that Professor Zeitz compares to walking along a mathematical cliff edge, use the pigeonhole principle to find patterns within apparently random and mind-bogglingly large structures. You'll discover there is no limit to what the intrepid problem solver can do....

30 min
Genius and Conway's Infinite Checkers Problem
23: Genius and Conway's Infinite Checkers Problem

No course on problem solving is complete without a look at the checkers problem, formulated by contemporary mathematician and puzzle-master John Conway. Also learn about two other icons in the field: Paul Erdos, who died in 1996, and evariste Galois, who lived in the early 1800s....

32 min
How versus Why-The Final Frontier
24: How versus Why-The Final Frontier

Professor Zeitz reviews problem-solving tactics and introduces one final topic, complex numbers, before recommending a mission to last a lifetime: the quest for why a solution to any given problem is true, not just how it was obtained. He closes by sharing some of his favorite examples of this elusive intellectual quest....

34 min
Paul Zeitz

Practicing a scale is an exercise; learning jazz improvisation is a problem. The math that you learned in school was mostly exercises. My course is all about problems.


University of California, Berkeley


University of San Francisco

About Paul Zeitz

Dr. Paul Zeitz is Professor of Mathematics at the University of San Francisco. He majored in history at Harvard and received a Ph.D. in Mathematics from the University of California, Berkeley, in 1992, specializing in ergodic theory. One of his greatest interests is mathematical problem solving. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO. He has helped train several American IMO teams, most notably the 1994 Dream Team, which, for the first time in history achieved a perfect score. He founded the San Francisco Bay Area Math Meet in 1994 and cofounded the Bay Area Mathematical Olympiad in 1999. These and other experiences led him to write The Art and Craft of Problem Solving (1999; second edition, 2007). He was honored in March 2002 with the Award for Distinguished College or University Teaching of Mathematics by the Northern California Section of the Mathematical Association of America (MAA), and in January 2003, he received the MAA's national teaching award, the Deborah and Franklin Tepper Haimo Award.

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