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Rating: 
Summary: Excellent --- for middle school to very early high school
Review: I understand that Ravi Vakil has a great record in the IMO and Putnam competitions. When I read the reviews and the back cover on a web site which explicitly stated: "Reveals some of [Vakil's] most powerful problem solving strategies," -- Well, of course I bought it immediately. Unfortunately most of the things in the book are quite elementary as far as problem solving goes. It is meant as an exposure for people who haven't seen these things so far(particularly talented jr high /middle school and early high school). Of course you will find some very interesting facts in here (and the rare mistakes-- Euler's Formula being called DeMoivre's Theorem) and will leave many young students in awe as Mr. Vakil shows off some beautiful mathematics. It is a good book, but I gave it only 3 stars because I expected much more( much more "powerful" stuff as the backcover said). If your a very experienced problem solver you will find many of the things in this book trivial(you will find some gems though), but if your a beginner with some talent, then you will love this book.
Rating: 
Summary: Insightful, interesting, excellent!
Review: Ravi Vakil gives a delightful introduction of some of the more interesting yet enigmatic areas of mathematics. Most of us are not too familiar with the cardinality of numbers or Galois Theory. However, the author presents these fascinating topics in such a lucid and insightful manner, that one becomes easily engrossed. Also, I particularly enjoyed reading the profiles of the young mathematicians - it was quite inspiring! Furthermore, the problems at the end of each section are thought provoking; many do not have complete solutions, and thus encourage the reader to think hard about them. As the title implies, it is NOT a textbook, but rather a compilation of scattered "lectures." It is a great book for the talented math reader, and students who want to explore some mathematics outside the general curriculum.
Rating:
Summary: A rare beauty !
Review: The author himself is renowned in math circles for his various achievements like gold medals at IMO ! and 4 times Putnam fellow ! No wonder this book is such beautiful rarety. I was totally impressed by the handling of the topics and the problem-solving approaches the author has chosen to explain. I recommend this book to anyone who is interested in accelerating his knowledge in maths for a competitive exam like ARML, USMO. IMO etc. Definitely not for fire-side reading so it aint worth your time if you have only casual interest in maths. This book is intended for serious kids.I'd given it 10 stars for such marvelous insight.
Rating:
Summary: This book is wonderful for what it attempts
Review: [...]This book does an incredibly admirable job of providing explanation of, and various methods of solving problems including, but not limited to, the irrationality of sqrt(2), the Fibbonacci Sequence and its relation to the Golden Ratio, Imaginary and Surreal numbers, and so on. Vakil is an engaging and sensitive writer who engages the reader who has no knowledge of thse problems, to the reader who has a fundamental understanding of the topics. Ravi Vakil was my professor of multivariate calculus, and I think his marvelous personality comes through in this superb introduction to some of mathematics' finest puzzles.
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