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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:
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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 x 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.