Customer Reviews

The Art and Craft of Problem Solving

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Sometimes a piece of music or a painting or a film just leaves me speechless. Sometimes it is a book, and this is such a book. When I first saw this book, I looked at no more than a handful of pages and bought it instantly. This book is truly thrilling, certainly for young and beginning mathematicians but even for mature ones. Every new page I read is full of thought and insight and elegance (both in the mathematical sense and otherwise). I don't know of any other book in its class. I honed my problem solving skills on the classics by Polya, but Polya did not cover this turf in even nearly the comprehensive way and the full and extensive detail of this book by Paul Zeitz. I wish I had had this extraordinary book when I was in high school -- I think it would have changed my life -- but I am so pleased that I at least have it now. Polya's books are of course classics in this area, but this book takes its place clearly beside them. It is destined to become a classic itself. In my eyes, it already has.

When I was in high school, I placed second in the Alabama State Mathematics Contest and won many others. However, I might could have been competitive with the IMO style problems had I had this book and would be much better off today had I seen this book earlier.

This book is for the exceptionally brilliant and the mentally tough. It is absolutely necessary to approach this book in a different way from a standard math textbook. You MUST attempt the examples BEFORE looking at the example solutions, NO MATTER HOW DIFFICULT OR FRUSTRATING. You may be bamboozled by the problems, but even trying to understand the problems before looking at the solutions and thinking about how a solution might proceed will pay huge dividends in the long run.

For example, in the first chapter Zeitz presents an example asking the reader to prove that the product of four consecutive integers cannot be a perfect square. The solution involves some clever algebraic trickery not visible to the inexperienced, but persistence and getting your hands dirty is key.

If you persist in spite of the considerable difficulty, you will find that you get better very, very quickly. You will also notice that it isn't just contest problems it helps you solve. I have found that I have solved my homework sets in the Berkeley graduate engineering program much more easily since working these problems. You will start to see creative and clever solutions where they exist in everything problem oriented.

PATIENCE PATIENCE PATIENCE!

This book is indeed one of the best problem-solving textbook so far. As a frequent lecturer of Taiwan IMO team, I have many many MO books. Most of the books available are well-written by professionals and excellent mathematicians. However, since IMO does really prevail in recent years, these authors could not be the participants themselves (^^). Furthermore, usually these books (except those are merely problems collections) contains a good proportion of "harder" and beautiful problems, and the easier and basic training problems are relatively few. It often get the beginners frustrate.

Now this maybe is the first book written by a member of former MO team, and now a training lecturer. (The author himself won the USAMO and IMO in 1974, and helped train several USA IMO teams, including the 1994 "perfect score team"). So here is the precious experience! Besides, the ratio between the harder problems and the easier problems is really good. In my opinion this is an excellent textbook for ambitious beginners (both teachers and students), for self-studys and problem-solving fans. Highly recommended.

The Art and Craft of Problem Solving is an excellent book that covers the essentials of Algebra, Combinatorics, Number Theory, and even Calculus from a problem-solving point of view. However, there are very few solved problems. I strongly recommend this book, but also suggest that the student study other books as well, books with many more solved examples, because this book alone does not provide enough experience in putting the concepts into practice.

I join the ranks of previous reviewers here who honestly feel that having read this book in high school would have almost certainly changed my life. I, too, did very well in high school math competitions, but the maturity I am gleaning from this gem may have vaulted me into a different league.

It contains hundreds of problems from various levels of competition, from AIME problems all the way through some of the toughest Putnam problems (which, if you know anything about the Putnam, are about as hard as competition problems come). But the biggest help are the vital insights and exciting ways of looking at these problems. Don't take my word for it--many past IMO contestants have suggested this book too.

Particularly helpful is the way the author divides the book into sections based on often-used concepts and techniques. For example, you will see applications of the pigeonhole principle from the most basic (e.g. "In a drawer with socks of 2 colors, show that after picking any 3 socks, we must have a pair of same-colored socks.") through some rather difficult ones (1994 Putnam A4, an Erdos problem, and more).

The same goes for a multitude of others--the invariants section includes both the classic chocolate bar-cutting problem and Conway's rather difficult checker problem. Then, not only does he solve the latter beautifully, but incorporates nontrivial questions that ensure the reader has completely understood the solution (e.g., "Could we have replaced lambda with an arbitrary integer? Why not?").

You don't have to be a math competition buff to gain from this book, however. If you're simply interested in mathematical puzzles and problems, and are looking to expand your repertoire, this book will help you. Anyone with a good dose of intelligence and motivation will benefit.

For an additional problem book, check out Mathematical Olympiad Challenges by Andreescu and Gelca. For purely Putnam treatment, there are several volumes written by Kedlaya. And if you're a CS student, looking for honing those CS math skills to be razor sharp, you should definitely look into Concrete Mathematics by Graham, Knuth, and Patashnik.

Happy solving.

Paul Zeitz has himself a masterpiece of a book in THE ART AND CRAFT OF PROBLEM SOLVING. As a student who was bored by the conventional curriculum in high school, I was interested in a more theoretical and challenging approach to mathematics. I was not disappointed. While this book does not go into considerable depth, it covers almost all major areas needed for an introduction to problem-solving. The examples he chooses are excellent and sometimes awe-inspiring-- everything from John Conway's amazing solution of the checkers problem to some fascinating proofs of common theorems.

Everything from algebraic combinatorics, probability, methods of proof, overarching mathematical ideas, problem-solving strategies, and more specific techniques are introduced. Zeitz explains everything in an understandable yet informed manner. If you ever wonder how some mathematicians manage to do what they do, look no further than this book.

A warning to the impatient: this book is not for you. If you can't stand thinking for longer than 5 minutes about a problem, DO NOT BUY THIS BOOK. You will be frustrated by many of the problems (not exercises, as Zeitz poignantly points out) presented at the end of each chapter. There are hints in the back, but no full solutions.

This book is also pretty good for those wanting to do well in math competitions. A lot of the problems come from national high school/college exams, and the all the ideas he presents are very relevant to solving those kinds of problems.

In summary, I would definitely recommend purchasing this book if you are an aspiring mathematician or just someone who likes problem-solving.

I happened to come across this book in my universities' used book store last summer. As a read and worked through the problems in this book I realized that this type of mathematics was infinitely different from the math I had learned in my first 2 years of engineering. If you are someone who is bored with routine college calculus and wants an intellectual challenge buy this book! The many problems in this book are excellent, the more time you spend on them the more you'll get out of the book. One note to people in love with solutions manuals, this book does not have one. You will have to work the problems yourself and think independently. This has many advantages in that you are forced to really analyze the validity of your solutions. Also, you'll realize that sometimes the solutions to problems will instantly come to you sometimes months after you first saw the problem. This book is the reason that I have decided to study for a double major in math, and I plan to attend graduate school in mathematics as well. This book is highly recommended, it will change the way you look at mathematics. Happy problem solving!

Charles Dickens wrote that some things are "like being touched in the marrow with some pungent and searching acid." The allusion is potent, but not unfavorable; like Dickens, Zeitz goes straight to the heart and crux of important mathematical problems. He communicates his ideas clearly and lucidly, without compromising mathematical rigor for style. Indeed, his style is humorous and informal, and always concerned with *teaching* and *communicating* important strategies and tools, not just showcasing a litany of the usual tricky problems and solutions. His examples are realistic, interesting, and transcedent--they'll keep you thinking and find ways to seep into your other intellectual queries. I'm lucky enough to have Paul as a teacher, but for those who aren't, his book is certainly the next best thing.

As a high school student that is essentially bored with the regular, ho-hum classes that my school offers, this book is perfect. It gives a problem-solving foundation for math enthusiasts desiring to compete nationally in contests like the AMC, AIME, and USAMO. The problems are excellent and cover a wide range of difficulty (past ASHMEs, USAMOs, and, finally, IMOs); and the solutions are well-written, logical, and intelligible. In short, if you are looking to "get better" at problem solving, this is the book for you.

Note: I also bought Problem-Solving Strategies by Arthur Engle. Those, perhaps more advanced, problem-solvers that want even more of a challenge should purchase this book as well (as both books give very challenging problems, but Engel's is undoubtedly more advanced).

Perfect match for all math problem solvers.
Wonderful Book with around 660 problems.
Level National Math competition, IMO, Putnam.
If I have to pick the best two problem solving books so far publish in the English Language
Problem-Solving Strategies (Problem Books in Mathematics) by Arthur Engel and this Book by Paul Zeitz are the clear winners.

This particular book has very clear explanations of the main problem solving strategies illustrated with carefully sample problems. Reading this book brings to my memory the works of Polya. One of the only things I think the book is lacking is on strategies to solve Geometry problems in particular or to use the same strategies in the book to solve more Geometrically flavor problems. Nevertheless is a Joy to read.
Please Paul keep writing this beautiful problem solving books.