Mathematics as Problem Solving

Various elementary techniques for solving problems in algebra, geometry, and combinatorics are explored in this second edition of Mathematics as Problem Solving. Each new chapter builds on the previous one, allowing the reader to uncover new methods for using logic to solve problems.  Topics are presented in self-contained chapters, with classical solutions as well as Soifer's own discoveries. With roughly 200 different problems, the reader is challenged to approach problems from different angles.

Mathematics as Problem Solving is aimed at students from high school through undergraduate levels and beyond, educators, and the general reader interested in the methods of mathematical problem solving.

Alexander Soifer is a Russian born and educated American mathematician, a professor of mathematics at the University of Colorado, an author of some 200 articles on mathematics, history of mathematics, mathematics education, film reviews, etc. He is Senior Vice President of the World Federation of National Mathematics Competitions, which in 2006 awarded him The Paul Erdös Award. 26 years ago Soifer founded has since chaired The Colorado Mathematical Olympiad, and served on both USSR and USA Mathematical Olympiads committees. Soifer's Erdös number is 1.   Springer has contracted his 7 books. "The Mathematical Coloring Book" is coming out in October 2008; 4 books will appear in 2009; followed by "Life and Fate: In Search of Van der Waerden", and a joint book with the late Paul Erdos "Problems of p.g.o.m. Erdos."   The author's previous books were self-published and received many positive reviews, below are excerpts from reviews of "How Does One Cut A Triangle?:   "Why am I urging you to read this? Mainly because it is such a refreshing book. Professor Soifer makes the problems fascinating, the methods of attack even more fascinating, and the whole thing is enlivened by anecdotes about the history of the problems, some of their recent solvers, and the very human reactions of the author to some beautiful mathematics. Most of all, the book has charm, somehow enhanced by his slightly eccentric English, sufficient to carry the reader forward without minding being asked to do rather a lot of work.   I am tempted to include several typical quotations but I'll restrain myself to two: From Chapter 8 "Here is an easy problem for your entertainment. Problem 8.1.2. Prove that for any parallelogram P, S(P)=5. Now we have a new problem, therefore we are alive! And the problem is this: what are all possible values of our newly introduced function S(F)? Can the function S(F) help us to classify geometry figures?"   Andfrom an introduction by Cecil Rousseau:   'There is a view, held by many, that mathematics books are dull. This view is not without support. It is reinforced by numerous examples at all levels, from elementary texts with page after page of mind-numbing drill to advanced books written in a relentless Theorem-Proof style. "How does one cut a triangle?" is an entirely different matter. It reads like an adventure story. In fact, it is an adventure story, complete with interesting characters, moments of exhilaration, examples of serendipity, and unanswered questions. It conveys the spirit of mathematical discovery and it celebrates the event as have mathematicians throughout history.'   And this isn't just publishers going over the top - it's all true!"   -- JOHN Baylis in The Mathematical Gazette   Soifer's work can rightly be called a "mathematical gem."   -- JAMES N. BOYD in Mathematics Teacher   This delightful bookconsiders and solves many problems in dividing triangles into n congruent pieces and also into similar pieces, as well as many extremal problems about placing points in convex figures. The book is primarily meant for clever high school students and college students interested in geometry, but even mature mathematicians will find a lot of new material in it. I very warmly recommend the book and hope the readers will have pleasure in thinking about the unsolved problems and will find new ones. -- PAUL ERDÖS It is impossible to convey the spirit of the book by merely listing the problems considered or even a number of solutions. The manner of presentation and the gentle guidance toward a solution and hence to generalizations and new problems takes this elementary treatise out of the prosaic and into the stimulating realm of mathematical creativity. Not only young talented people but dedicated secondary teachers and even a few mathematical sophisticates will findthis reading both pleasant and profitable. -- L. M. KELLY in Mathematical Reviews   We do not often have possibilities to look into a creative workshop of a m