The Art and Craft of Problem Solving

References and Further Reading

Further Reading

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Here are a few suggestions for expanding your horizons. This is by no means an

exhaus tive list, but just a few of our personal favorites.

General Problem Solving

For recreational math, read anything by Martin Gardner. Another great author is Ray­

mond Smullyan. If you would like to read about problem solving at a lower level than

this book (either for your own benefit or to teach), there are three excellent choices:

• Mathematical Circles (Russian Experience) [8] contains many very imaginative
  problems at an elementary level, with interesting teaching ideas. A must-have
  book for teachers of problem solving.


• The Art of Problem Solving, by Sandor Lehoczky and Richard Rusczyk [27] is
  a very comprehensive two-volume treatment of elementary tactics and tools. It
  is clear, fun to read, and has full solutions! These books are available from the
  web site http://www. artofproblemsolving. com.which.atthepresenttime.is
  the preeminent web site on problem solving. This site has a very active forum,
  sponsors online courses, etc.


• A Mathematical Mosaic, by Ravi Vakil [44], is short and not comprehensive,
  but it more than makes up for this with its exceptional taste in topics and its
  whimsical, fun-loving style. This book is really fun to dip into; it's superb
  bedside reading!
  At a more advanced level than this book, there are several superb books by Titu An­
  dreescu that showcase true art and elegance. The Wohascum County Problem Book
  [13] has many imaginative problems, and features many calculus problems that are
  not outrageously difficult; perfect for beginners. The Putnam Exam has three books
  of published solutions. The first two, covering the years 193 8-6^4 [14] and 1965-84
  [1] are good, but the latest one, covering the years 1985- 2000 [26], is magnificent,
  with many alternative solutions, mathematical digressions, and creative ideas about
  problem solving. It is an advanced book, but truly worth perusing.

Specific Topics

For algebra, we know of only two really notable books: Barbeau's superb guide to the

exploration of polynomials [3] and Steele's somewhat advanced text on inequalities

[4 0].

In contrast, there is a wealth of terrific combinatorics books. Our favorite choice

for beginners is by Slomson [38], but it is hard to find. Goodaire and Parmenter's