Art and Craft of Mathematical Problem Solving

Course No. 1483
Professor Paul Zeitz, Ph.D.
University of San Francisco
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Course Overview

One of life's most exhilarating experiences is the "aha!" moment that comes from pondering a mathematical problem and then seeing the way to an elegant solution. And many problems can be solved relatively quickly with the right strategy. For example, how fast can you find the sum of the numbers 1 + 2 + 3 up to 100? This was famously answered in the late 1700s by the 10-year-old Carl Friedrich Gauss, later to become one of history's greatest mathematicians. Young Gauss noticed that by starting at opposite ends of the string of numbers from 1 to 100, each successive pair adds up to 101:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

and so on through the 50th pair,

50 + 51 = 101

Gauss was already thinking like a good problem solver: The sum of the numbers from 1 to 100 is 50 × 101, or 5,050—obtained in seconds and without a calculator!

In 24 mind-enriching lectures, The Art and Craft of Mathematical Problem Solving conducts you through scores of problems—at all levels of difficulty—under the inspiring guidance of award-winning Professor Paul Zeitz of the University of San Francisco, a former champion "mathlete" in national and international math competitions and a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.

These are not mathematical exercises, which Professor Zeitz defines as questions that you know how to answer by applying a specific procedure. Instead, problems are questions that you initially have no idea how to answer. A problem by its very nature requires exploration, resourcefulness, and adventure—and a rigorous proof is less important than no-holds-barred investigation.

Think More Lucidly, Logically, Creatively

Not only is solving such problems fun, but the techniques you learn come in handy whenever you are presented with an unfamiliar problem in mathematics, giving you the confidence to try different approaches until you make a breakthrough. Also, by learning a range of different problem-solving approaches in algebra, geometry, combinatorics, number theory, and other fields, you see how all of mathematics is tied together, and how techniques in one area can be used to solve problems in another.

Furthermore, entertaining math problems sharpen the mind, stimulating you to think more lucidly, logically, and creatively and allowing you to tackle intellectual challenges you might never have imagined.

And for those in high school or college, this course serves as an enriching mathematical experience, equal to anything available in the top schools. Professor Zeitz is a masterful coach of math teams at every level of competition, from beginners through international champions, and he knows how to inspire, encourage, and instruct.

Strategies, Tactics, and Tools of Math Masters

The Art and Craft of Mathematical Problem Solving is more than a bag of math tricks. Instead, Professor Zeitz has designed a series of lessons that take you through increasingly more challenging problems, illustrating a variety of strategies, tactics, and tools that you can use to overcome difficult math obstacles. His goal is to give you the persistence and creativity to turn over a problem in your mind for however long it takes to reach a solution.

The first step is to come up with a strategy—an overall plan of attack. Among the many strategies that Professor Zeitz discusses are these:

  • Get your hands dirty: Dive in! Plug in numbers and see what happens. This is a superb starting strategy because it almost always shows a way to keep on investigating. You'll be surprised at how often a pattern emerges that takes you to the next step.
  • Think outside the box: Break the bounds of conventional thinking. Professor Zeitz shows you the original think-outside-the-box problem, in which the key idea is to disregard the boundaries of an implied box. He also explains why he prefers to call this strategy "chainsaw the giraffe."
  • Wishful thinking: Turn a hard problem into an easy one by removing the hard part. For example, substitute small numbers for big ones. This is a confidence-builder that often gives you a partial solution that shows you how to solve the original problem.
  • Change your point of view: Every problem has a natural point of view, such as a time or place where something is happening. Step back and try a different point of view. This could mean recasting an algebra problem as one in geometry, or vice versa.

The next step after choosing a strategy is to find a suitable tactic. For example, suppose you live in a cabin that is two miles north of a river that runs east and west, and your grandma's cabin is 12 miles west and 1 mile north of your cabin. Every day you go to visit grandma, but first you stop by the river to get fresh water for her. What is the length of the route that has the minimum distance?

You start with the "draw a picture" strategy. Once you have something to look at, you realize that the "symmetry" tactic will give you the shortest distance. Here's how it works: Imagine an alternate you on the same errand but on the south side of the river, in a mirror image of the situation on the north side. By drawing a line connecting your real cabin with the alternate grandma's cabin, and another line connecting the real grandma's cabin and the one belonging to the alternate you, you find an intersecting point at the river that is the perfect place to stop.

On some problems you also need a special-purpose technique—a tool. For example, the 10-year-old Gauss's trick of pairing numbers in the earlier example is a tool whose underlying idea—symmetry—can be applied to a wide range of problems. You learn the strengths, as well as possible pitfalls, of such tools.

Prepare for an Exhilarating Experience

Professor Zeitz compares this systematic approach to problem solving—in which you deploy strategies, tactics, and tools—to the mountaineer's quest to reach the top of a high peak. The mountain may seem insurmountable, but there is always a way to conquer it by proceeding one step at a time.

Looking at an impressive mountain, you can't but feel a sense of awe at the prospect of climbing it. Math problems, too, can produce this same reaction. But don't be daunted: You are more ready than you think. So sharpen your pencil, get some paper, and prepare for the exhilarating experience of The Art and Craft of Mathematical Problem Solving.

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24 lectures
 |  Average 30 minutes each
  • 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." x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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 Évariste Galois, who lived in the early 1800s. x
  • 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. x

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Paul Zeitz

About Your Professor

Paul Zeitz, Ph.D.
University of San Francisco
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...
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