A Mathematical Mosaic: Patterns & Problem Solving (Revised Edition)
There are different books on my shelf. Some are large like dinosaurs: these are textbooks. Others are much smaller, but their educational value may be greater. For example, Kordemsky's book [3] contributed a lot to Russian childrens' interest in mathematics although its English edition fits in a hand. The book I am going to discuss fits in a coat pocket, but it speaks in an interesting and understandable way about number theory, combinatorics, game theory, geometry, and calculus, to say nothing about magic tricks, puzzles and other digressions. What is most important is that whenever Vakil starts to discuss something, he never leaves the reader without a piece of exact, rigorous knowledge. This is a book about mathematics, not about its fuzzy placebo. Ravi Vakil… tries to encourage curiosity, a sense of beauty, and the love of knowledge. This is a book I would like to have read as a boy. Why? Because it addresses the normal curiosity of children. It contains many good problems, facts, and stories. It is a mixture of just those ingredients which are most useful for children.  Vakil enjoys ideas that seem simple if you already know them, but may seem paradoxical if you don't. One of them is presented as a card trick (p. 44):   I ask you to shuffle a deck of cards thoroughly. Then I ask for them back (face down). Carefully examining the backs of the cards, I separate them into two piles. I then claim that, through the power of magic, I've made sure that the number of black cards in the first pile is the number of red cards in the second pile! Although Vakil's book is intended to be recreative and facultative, it contains many facts that are indispenable for mathematical literacy, including:    Criteria for divisibility. Those for 2, 3 and 7 are proved; other proofs are left for the reader (pp. 23, 29).    Let g(x) denote any polynomial in x. Then the remainder when g(x)  is divided by x   a is g(a).     2 is irrational (p. 121).    There are infinitely many primes (p. 124).    Heron's formula for the area of a triangle (p. 160)    The harmonic series diverges (p. 180). This book contains several personal profiles of gifted youngsters with whom Vakil became acquainted at olympiads. Vakil writes several lines about how they found their way into mathematics.  More than once Vakil stresses that mathematics is beautiful. In his preface he writes: Math is a uniquely aesthetic discipline; mathematicians use words like beauty, depth, elegance, and power to describe excellent ideas --American Mathematical Monthly-André ToomRavi Vakil has put together a collection of wonderful topics from number theory through combinatorics to game theory in a fashion that seventh- and eighth-grade students can handle yet high school students will find challenging. His book is divided into two parts. Part 1 introduces to the young reader a number of mathematics topics that will be very useful in part 2. For example, in the first section, Number Theory, such topics as calculating tricks, divisibility rules with proofs of why they work, and magic squares are investigated fully and clearly. In part 2, many of the earlier topics are revisited, but the level of difficulty is increased. In 'Number Theory Revisited,' an in-depth study of rational and irrational numbers, a fascinating painted-school-lockers problem, and other topics challenge students in an entertaining manner. Perhaps the best features of the book, however, are the historical digressions on great mathematicians and short personal profiles of contemporaries of the author. These glimpses into the lives of young male and female mathematicians make this book very much worth its price. Without a doubt, this book is a must for any library, teacher's reference, or student's amusement.- --The Mathematics Teacher, Vol 89, #7- John Cocharo