Customer Reviews

Challenging Problems in Algebra (Dover Books on Mathematics)

5 star:		(2)
4 star:		(3)
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While this book is obviously a good resource for math teams (and apparently the audience towards which the book is geared), the book is also useful for teachers of regular algebra classes. While some of the problems are considerably beyond the grasp of all but the most gifted students, others lie more reasonably on the more challenging side of what "average" students are capable of. Solving these problems really encourages students to mentally stretch more than the average story problems in a textbook, and the level of achievement students feel after solving one is truly gratifying. I've found these problems to be especially good for small group work or "team" competitions.

I do math for fun, but I only like to do problems that are, well, PROBLEMS, and not just exercises. That's what this book is, a book of unique, nonroutine problems in algebra.

While I've taken courses in AP Calculus and Statistics and such, I find that the problems contained in this little book re-open the avenues of precalculus mathematics and show the reader/solver what the math teachers and AP and all that jazz never could: problems that require insight, ingenuity, and diligence. Solving each problem in this book is a joy! There's always something new to be learned with each problem, even if you don't succeed.

Anyway, yeah, the problems are great, the solutions lucid, and the price dirt cheap. If you want to challenge your math ability, then buy this book and spend a good amount of time doing the problems. If you want to just do well in algebra class, please redirect yourself to the Schaum's "Algebra" guide for more routine, classroom work.

This text contains non-routine algebra problems, their answers, some of their solutions, and a set of appendices covering results that are useful in solving the problems. By non-routine, I mean that the problems require insight and, in some cases, ingenuity to solve. The authors presume that you have acquired strong algebra skills and ask you to use those skills to solve unfamiliar problems. The level of difficulty of the problems is similar to those in the American Mathematics Competition (AMC), which is not surprising since Charles T. Salkind was the editor of the American High School Mathematics Examination (AHSME), now the AMC, from its inception in 1950 until his death in 1968. Unlike the problems in those competitions, these are not multiple-choice questions. While many of the problems in this text are similar to mathematics competition problems in which a numerical answer or an algebraic expression is sought, others require you to write a proof or investigate an algebraic relationship. Working through these stimulating problems and reading the solutions that the authors provide will expand your algebra skills and extend your knowledge of the subject.

The problems draw upon topics covered in elementary, intermediate, and advanced algebra courses. They are grouped, somewhat loosely, by topic. The topics include equations and inequalities; systems of linear equations; arithmetic, geometric, and harmonic means; relations and functions; maxima and minima; the relationship between algebra and geometry; sequences and series; logarithms; combinatorics and probability; non-decimal bases; number theory; and Diophantine equations. While rational expressions crop up occasionally, they are not explored in any depth. Some of the problems include extensions or variations labeled "Challenges." Despite the name, the Challenges are not usually more difficult than the problem with which they are associated. Some problems explore topics of interest in higher mathematics. A knowledge of calculus enables you to find simpler solutions to some of the problems than those provided by the authors in the solution key, but it is not strictly necessary.

The problems are generally well-posed. However, it was not always clear from context when the authors meant integer or natural number (positive integer) when they used the word integer. When I found a problem unclear (which was not often), it was usually attributable to an implicit assumption. However, there was one problem in which the authors used the phrase "divided by" rather than "divided into," making the problem impossible to solve.

Answers are provided to the problems for which a numerical solution or algebraic expression is sought in a section at the back of the book. This gives you an opportunity to check your work before reading the solutions the authors provide. Most, but not all, of the answers are correct. In some cases, there is a discrepancy between the answer provided in the answer key and that provided in the solution key. In all but two such cases, which were easy to discover in the context of the solution, the answer in the solution key was correct.

You can learn a lot by reading the solutions the authors provide, even if you have solved the problem correctly. In many cases, the authors not only provide a solution to the problem, they also show you how to solve an entire class of problems related to the problem you solved or attempted to solve. In some cases, the techniques they use to solve a problem can be applied to subsequent problems in the text. Solutions are provided for all the principal problems. However, solutions are provided for only some of the problems labeled Challenges.

The appendices cover terminating digits; the remainder and factor theorems; maximum product, minimum sum problems; arithmetic, geometric, and harmonic means; divisibility tests; the binomial theorem; some useful algebraic relationships; and how to write a proof by mathematical induction. Since the relationships and techniques examined in the appendices are used in solving the problems and are not necessarily covered in the high school curriculum, it is useful to read the appendices before beginning to solve the problems. Proofs are given for some of the results in the appendices, although not the Binomial Theorem. However, the motivation and strategies for the proofs are not discussed. While the strategies become clear if you understand the proof, the motivations for those strategies do not.

If you enjoy the challenge of working through this text, you may also wish to explore its companion volume, Challenging Problems in Geometry. Both texts were completed by Alfred S. Postamentier, who had collaborated with Salkind on them, after Salkind died.

Working through this book is a great way to enhance your problem-solving skills. In that sense, it is useful preparation for mathematics contexts. However, a better resource for that purpose would be the books in the Contest Problem Book Series. The Contest Problem, Book VIII (MAA Problem Book Series) prepares you for the AMC 10 examination; The Contest Problem Book IX (Maa Problem Books) (Bk. 9) prepares you for the AMC 12 examination. Older books in the series, including those edited by Salkind, were written when the competitions had a different format and somewhat different contents. They are still useful as a means to develop your problem-solving skills, particularly The Contest Problem Book V: American High School Mathematics Examinations (AHSME) / American Invitational Mathematics Examinations (AIME) 1983-1988 (Anneli Lax New Mathematical Library), which is the only volume in the series that contains problems from the American Invitational Mathematics Examination (AIME). The books in that series are better edited than this text.

This is the opinion of a math teacher, who bought this book for self-improvement. It's a good book and I sure reccomend it.

If you are a teacher you will find out that the questions vary, from very easy, to medium and difficult. Mathematics Olympiad-type questions are a little scarce. So if you're looking for an upper level you could always buy it together with "Mathematical Olympiad Challenges" by Titu Andreescu.

Still, the easiest questions fit the stronger students very well, and the harder ones can be very helpful for teachers.

Also, the solutions are very well written.

The book was a bit harder than i thought, however a little extra challenge is never too bad. Has some proofs, some logic problem solving. The reason i rated this 4 is because of where everything is placed in the book. After you solve the problem, u have to go to another page to see the answer, or if you dont enderstand the problem, u have to dig in the book to see where its solution is.