The Art and Craft of Problem Solving

Added the following on September 23rd of 2014. Today, I solved my first Geometry IMO problem (1st problem of IMO 1998). Maybe it is the first problem (usually, first problem at IMO are easier than the second and the third). This happened only because of this book that I am following. I highly recommend this book. Following review was written on August 18th of 2014. I am currently reading the geometry chapter. Obviously, I was not able to solve all the problems in the previous chapters as some of them are extremely difficult and I have not yet reached that level of expertise in problem solving. I can tell you that this is the best book ever. The tough ones are introduced with enough background about various techniques so that the reader has enough ideas to be able to solve the problem. I was previously not able to solve a single difficult problem but now, I have solved some including a few IMO problems. For me, it is a big achievement. I cannot thank Prof. Paul Zeitz enough for writing this masterpiece of a book. I wish I could be his student for ever since he knows a lot about Math and there is much more knowledge he has than what I can learn in ten lifetimes. As an example, the Number Theory chapter seemed to be a short one even though it did feel that I learn a lot. How much did I learn? I had no idea. Then I read Charles Vanden's book and Paul Zeitz pretty much covers entire Number Theory (the basics for sure, but vast majority of the advanced topics too). Similarly in Geometry, he covers all the way upto transformations even though he calls that chapter as "Geometry for Americans" as though he has tried to make it easy just because he assumes American audience for his book. It has a lot of information about each topic. What is best about his book is that after reading his techniques, you will feel like solving tough problems is a possibility. Atleast I did. I could not even think of being able to solve IMO or USAMO problems but I could solve some of them. I am very happy about it. I highly recommend this book to anyone who is a beginner to IMO level problem solving and has serious aspirations to get through to IMO. Just one advice from me. Please stick to one book. Just one. Once you have become an expert in solving problems form one book, move on to other books. If you are not there yet, it is better to stick to that one book till you become good at it. This is surely the best book to be that one book you will follow to get through to IMO. Following review was written during April of 2013. I am still in the second chapter of this book. I can tell you that if you solve the problems in this book, you are in great shape to solve problems in the contest. Some clarifications are in order. 1. This is not like "An introduction to solving mathematical 'exercises'." I say 'exercises' because pretty much everyone believes that a 'problem' and an 'exercise' are the same. They are not. A problem is way more tough than an exercise. A 10th grade text book has exercises while a book like this will have problems. You get exercise problems in the 10th grade exams but you get problems in the contests like Putnam. I used to solve exercise problems orally as a kid but I could not solve a single IMO problem. Problems are extremely tough in general, sometimes even for experienced problem solvers. 2. This book is about "Problem" solving. So please do not get into this book thinking that he has exercise type of questions. 3. Though not all the times, the standard procedure to solve a problem is to a) "see a pattern" b) form a conjecture c) try all the cases and make sure that the conjecture is true d) try to solve the problem using that conjecture, and e) prove that the conjecture is true. 4. This book is fantastic. No other book focuses on "Conjecture Making". He devotes Chapter two for conjecture making. He simply gives problems as practice to form conjectures and not necessarily prove them. Gradually, he builds you up to be able solve a problem which he does in the later chapter. 5. His chapter on Geometry may be small but is fantastic. It has enough for you to be able to get a good grasp of what is needed for IMO. Paul Zeitz was himself an IMO participant and he has trained a team once which I think scored a perfect score in one of the IMOs. He has immense experience and he is one of the few people who is truly capable of training kids so that they become excellent problem solvers. I highly recommend this book. There are other books tougher than this one but I think that the other books are tough not because the problems are any tougher but simply because they are not written as well as this book. I also suggest that in case you have time after reading this book, you must take a peak at Arthur Engel's "Problem Solving Strategies". He has a lot of problems with solutions but no material regarding conjecture making and psychology etc. Also, before reading this book, it will help to read the book "Mathematical Problem Solving" by Alan Schoenfeld. I feel that you must be very much aware of the psychology of problem solving before you get into the business of problem solving. Please read my review of this book for more details. One final remark is that contrary to what people say on the websites, I feel that doing atleast some of the problems from ONE book is much better than reading solutions from 10 books. I have heard that the IMO toppers have solved like 10 books on IMO level problem solving. I find it hard to believe. My experience with problem solving is that once you understand properly as to what is involved in problems solving, then solving more problems helps but it is hard to believe that there is enough time to solve all the problems from 10 different books. I can confidently say that this is one book that you can count on, stick to and if you solve it, will have a great shot at the IMO.

I really like this book. My son is interested in math and I bought it for him. But I was surprised that both he and I got a great deal of enjoyment from it. It's an entertainingly written and informative book that can be read by people with a wide range of abilities in math. If you or your child hasn't taken at least algebra yet, it's too advanced. But it could be enjoyed by someone who has just taken high school algebra or by a college math major.

A "cut to the chase" course in strategies for solving math problems of the kind found in the USAMO and IMO tests. The level of knowledge needed is up to a math major's sophomore year. I have yet to finish it but the book seems to offer an abundance of useful information along with example problems with step by step solutions. It would be nice, however, if access to solutions to the numerous problems and exercises was provided so people could check their work!

I have to admit i am not through reading this book but this book is what was and still is missing in my education :-) Why? Well, in my opinion the author understands why many people fear math - lack of proper method(s) + lack of confidence. And the author goes about tackling this problem by doing exactly that! This book provides many "problems" - i love the way the author phrased the word "problem" - plus many words of encouragement to push its readers to attempt the problems to 3 goals: 1) Have the courage to think out-of-the-box when it comes to solving problems; 2) Have the confidence to tackle them; 2.1) Building this confidence by providing the methods + the reader's willingness to get "dirty" 3) Never give up (Take a rest if you must, but never ever give up).

I bought this book for my kids. I have noticed offlate that they are quite puzzled with problems during examination and unit test. After discussing with them I realized that they are not aware of the facts the strategy and processes to be followed while encountering any problems. I have personally gone through this book at a glance and I find this is the best baseline for them to handle their current weakness of problem solving.

I recently got the 2nd edition and it seems to have some additional material compared to the first edition. There is a new chapter on Geometry and with expanded treatment of calculus. Seems like there are a few more problems in each chapter. This is a must have book for those interested in competetive mathematics. The presentation is very good -- but since the material covered is rather complex, its not easy to do self-study with this book. The book doesn't have a solution manual -- I tried contacting the publisher to get access to their instructor site but was turned down saying that I needed to be an instructor using this as a textbook in class and so on to get access to the solutions manual. It kind of sucks when you are doing self-study to not have a way to get help. I wish Wiley will reconsider this and give folks like me who are engaged in self-study a chance to use this book effectively. It is some consolation that the books web site has a "students" section providing hints for some problems. Overall, I would still give the book a 5 star rating because it is a class apart and covers a whole lot of ground. The first few chapters on strategies and tactics to solve problems are by themselves worth the price of the book. Definitely worth getting.

This was a required purchase for a Graduate Level Math Problem Solving class. The text is really hard to read and understand, but covers a plethra of topics. There are no worked out solutions or an answer key which makes checking you work very difficult.

This book is very useful for people who already know all the essentials but need to make the connections between various topics. I use it for math Olympiad prep for students with a lot of talent. Yet, they need help. It is much more advanced than the description says.

A must.

This is an excellent science book which is a helper to teachers, students and parents. However it is too expensive though.

Entrega super rápida e em muita qualidade.

For all those who aren't that good at problem solving I can only recommend this book. It helped me a lot and is full of useful tips. Every important topic is very well explained. The book gives lots of examples and problems to practice and it isn't too difficult. It is expensive, yes, but it's worth its money.