Mathematical Miniatures is a wonderful book, with a new approach to an old favorite topic: problem solving.
The book is comprised of 50 chapters, each one telling a mathematical story. Most of the problems come from Mathematical Olympiads, but what makes this book original is the way the problems are presented. The main mathematical idea — perhaps an interesting inequality, a geometric construction technique, or an unusual property — is presented in various disguises, as it has showed up either in contest problems or in math folklore.
The most clever solutions and applications are always accompanied by the name of the problem solver. One cannot help but be in awe at amazing insight demonstrated by high school students, under the restricted time frame of the Olympiad tests, in solving tremendously complicated problems. The proposers of these unusual problems are also named in the book, along with names of other mathematicians that made contributions in working related properties and statements. It is of great merit that the work of brilliant students and mathematicians from Bulgaria, Romania, Russia and other countries from Eastern Europe, less known to the western culture is thus brought forth to the mathematical world's attention.
The topics of the problems come mainly from algebra and geometry: inequalities, combinatorics, number theory, constructions, and occasional bits from analysis. This is reflective of the Math Olympiads topics, and will particularly appeal to the pure mathematician. But the beauty of the ingenious, albeit elementary mathematics will grab any passionate mathematician's attention. Some of my favorite chapters are: Trig Substitutions, The Three Jug Problem, Tetrahedra with a Point in Common, and Instead of an Afterword.
The sequence of 50 chapters — 50 mathematical topics is broken by nine "coffee breaks" and preceded by a warm up set. Each of the breaks proposes to the reader a handful of problems — some easy, some quite hard — all of them teasing and inviting. This is to say to the reader: don't just read about how others have cleverly solved problems, try your hand at some! I have to confess to falling for this trap several times and going through the house mumbling to myself "hmm... I used to get this in 5 minutes when I was in high school." Luckily all these problems come with solutions as well.
Overall, a delightful book, recommended not only for one's reading pleasure, but also for math clubs in either high school or college.
Ioana Mihaila (firstname.lastname@example.org) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is interested in mathematics competitions.