ne 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. Try your hand at these:
- Can you cut a sheet of typing paper with scissors into only one piece so that an elephant can step through it?
- What is the first time after noon at which the hour and minute hands meet?
- Seventeen pennies are on a table. Two players take turns removing 1, 2, 3, or 4 pennies. The last to make a legal move wins. Is there a winning strategy?
Each of these problems can be solved relatively quickly with the right strategy, as you discover in The Art and Craft of Mathematical Problem Solving, 24 mind-enriching lectures taught by award-winning Professor Paul Zeitz of the University of San Francisco. A former champion "mathlete" in national and international math competitions, Professor Zeitz is a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.
Under his inspiring guidance, you investigate scores of problems at all levels of difficulty. 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.
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, 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.
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.
- Think outside the box: Break the bounds of conventional thinking.
- Wishful thinking: Turn a hard problem into an easy one by removing the hard part.
The next step after choosing a strategy is to find a suitable tactic, and sometimes you need a special-purpose mathematical tool. Professor Zeitz compares this systematic approach 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.
Math problems sometimes seem as formidable as high mountains. 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.