The Art and Craft of Problem Solving 2nd Edition by Paul Zeitz (PDF)

Description

The newly revised Second Edtion of this distinctive text uniquely blends interesting problems with strategies, tools, and techniques to develop mathematical skill and intuition necessary for problem solving. Readers are encouraged to do math rather than just study it. The author draws upon his experience as a coach for the International Mathematics Olympiad to give students an enhanced sense of mathematics and the ability to investigate and solve problems.

User’s Reviews

Editorial Reviews: Review “Overall, The Art and Craft of Problem Solving is an excellent gateway to the culture of problem solving. It is challenging and rewarding. Zeitz’s book shines a new light on mathematics and engages readers with its wonderful insights and problems.” (Mathematical Association of America 2016) From the Back Cover You’ ve got a lot of problems. That’s a good thing.Across the country, people are joining math clubs, entering math contests, and training to compete in the International Mathematical Olympiad. What’s the attraction? It’s simple—solving mathematical problems is exhilarating!This new edition from a self-described “missionary for the problem solving culture” introduces you to the beauty and rewards of mathematical problem solving. Without requiring a deep background in math, it arms you with strategies and tactics for a no-holds-barred investigation of whatever mathematical problem you want to solve. You’ll learn how to:get started and orient yourself in any problem.draw pictures and use other creative techniques to look at the problem in a new light.successfully employ proven techniques, including The Pigeonhole Principle, The Extreme Principle, and more.tap into the knowledge gained from folklore problems (such as Conway’s Checker problem).tackle problems in geometry, calculus, algebra, combinatorics, and number theory.Whether you’re training for the Mathematical Olympiad or you just enjoy mathematical problems, this book can help you become a master problem-solver!About the AuthorPaul Zeitz studied history at Harvard and received a Ph.D. in mathematics from the University of California, Berkeley. He currently is an associate professor at the University of San Francisco. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO and helped train several American IMO teams, most notably the 1994 “Dream Team” which, for the first time in history, achieved a perfect score. In 2003, he received the Deborah Tepper Haimo award, a national teaching award for college and university math, given by the Math Association of America. About the Author Paul Zeitz studied history at Harvard and received a Ph.D. in mathematics from the University of California, Berkeley. He currently is an associate professor at the University of San Francisco. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO and helped train several American IMO teams, most notably the 1994 “Dream Team” which, for the first time in history, achieved a perfect score. In 2003, he received the Deborah Tepper Haimo award, a national teaching award for college and university math, given by the Math Association of America. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Strengths:1) Conjecture making in chapter 2: making a formula and the algorithmic approach.2) Investigation process of problems in chapter 2: The subchapter ‘2.2 Strategies for Getting Started’ has excellent examples of long mathematical investigations in great detail. Frequently used procedure (simplify, analyze, generalize and prove):(i) Simplify the problem: use small numbers.(ii) Analyze the problem with numerical experimentation.(iii) Generalize: make a formula that can apply for all cases of the original problem.(iv) Prove that the generalization is true for all cases of the original problem.A big fan of this classic procedure is the mathematician Carl Friedrich Gauss. The heuristics in the procedure are the most used heuristics in the next chapters.3) For some readers: No solutions of the problems in the book can be an advantage. This is a simulation of reality. In reality, there exist no solutions for problems at this moment. You are the first to find a solution of the problem.4) Beautiful problems with asking questions, experimentation, conjecture making, proving and generalizations.5) Chapter Eulerian mathematics: Write proofs without rigor like Euler and Ramanujan. Write the same proofs with rigor like Niels Abel and Augustin Louis Cauchy. Advantages and disadvantages of proofs with rigor and without rigor.Weaknesses:1) Diffusion of heuristics in the book. There is no list of all heuristics in chapter 1 like the book Problem-Solving Through Problems by Loren C. Larson. The heuristics in the book of Larson are explained in the next chapters.2) No chapter of the heuristic ‘generalize’ and no chapter of the heuristic ‘work backwards’. The term generalize means in this book conjecture making. There is the tool about generalize: ‘define a function’ in this book. For example in page 90. But not in a separated chapter. The book Problem-Solving Through Problems by Loren C. Larson contains a full chapter of the heuristic generalize. In this book the term generalize means: Replace a constant by the variable x to enable the use of techniques of Calculus.3) For some readers: No solutions of the problems.4) There are no standard names in most heuristics. Most heuristics have names with an analogy to mountaineering. For example: penultimate step, get your hands dirty and crux move. These can be annoying for the reader. Instead of technical/formal math flavor of naming those heuristics. Good examples of names already used: Invariants, extreme principle, pigeonhole principle, algebraic/geometric symmetry, PIE, ….Conclusion: The book The Art and Craft of Problem-Solving has excellent examples of mathematical investigations. The classifications of problem solving techniques in strategies (problem solving techniques to achieve long term goals), tactics (problem solving techniques to achieve short term goals) and tools are good. Suggestion: Instead of tools, I would use the name algebraic tricks. And integrate this technique in the list of tactics. To learn research in pure math, this book is not complete. You must master the skills of mathematical problem solving AND rigorous argumentation. You learn rigorous argumentation in a math major without problem solving. You need to learn problem solving implicitly in a math major on your own. Because argumentation is easy to grade and problem solving is hard to grade for a math professor. The book ‘How to Solve It’ by G. Polya has better general problem solving strategies but without good details: 1. understanding the problem, 2. make a plan, 3. carrying out the plan and 4. looking back (reflection) and are applicable outside math. I would use the problem solving strategies by Polya first combined with the details of the steps in the book of Paul Zeitz combined with rigourous argumentation by Ted Sundstrom (book: mathematical reasoning: writing and proof version 2.1). The book Fundamentals of mathematics: An introduction to proofs, logic, sets and numbers of Bernd S.W. Schröder, publisher Wiley, has good explanation of the synthesis of clean math proofs see pages 61-65.Suggestion and wider scope of problem solving. To solve a math problem I would use the steps:1. Problem solving with 2 subfases:1.1 Analyze ( understand the problem and more general). With more general: resolving problems by methods which, at first sight, seem to have nothing to do with the problem at hand for example change the problem. Another technique is find the invariant. The invariant is the abstraction of guessing a mathematical formula/conjecture, parity, a numerical constant, algebraic and geometric symmetry.1.2 Plan without guidelines of proof writing (proof strategy/draft version of the proof/2-column proof).2. Argumentation:2.1 Plan with guidelines of proof writing.2.2 Synthesis: summary/expert proof/the condensed form of the analysis and the plan of problem solving/polishing the solution: abstract the details of the solution.3. Reflection (optional): Reflection is the abstraction of problem solving and argumentation.3.1 Generalize the problem and solve this problem.3.2 Make an alternative solution.The details of the analysis of the problem, the intermediate steps of calculations, a similar case of the proof technique case analysis, a similar direction of ‘if and only if’, the details of mathematical induction and other things are omitted in a formal proof. The most extreme form of condensed proof is an expert proof. The details of a proof must be filled in by the reader. Carl Friedrich Gauss used the principle of condensation in clean math proofs. Gauss writing style was terse, polished, and devoid of motivation. Abel said: `He is like the fox, who effaces his tracks in the sand with his tail’. Gauss:`no self-respecting architect leaves the scaffolding in place after completing the building’. A mathematical proof is similar to the summary of a literary genre such as the summary of the fairy tale “Alice in Wonderland” on the back of the book. The details of problem solving are omitted (abstraction) in a clean math proof because abstraction is important in pure mathematics and the proof is dense full of mathematical proza. A common trait of a clean math proof in pure mathematics and a (software) algorithm in computer science is the argument. The difference: Details in algorithms are important so that a computer can exexute the instructions. In a clean math proof, the details are omitted for the abstraction AND for the brevity. A human reads the proof not a machine. The structure of a clean math proof is always linear: from the hypothesis P to the conclusion Q (one of the guidelines of proof writing. Another guideline is a Halmos symbol to show the end of a proof). The problem solving technique working backwards is frequently used for writing epsilon-delta proofs in the draft version. In a clean version the proof must be arranged in linear order. The plan in problem solving is a detailed plan without guidelines of proof writing and is a possible solution. The argument is an abstracted plan with guidelines of proof writing. With the application of the rule of synthesis (polished solution). You see symmetry when you compare problem solving and argumentation in mathematics: analysis in problem solving is the opposite of synthesis in argumentation. A plan without guidelines in problem solving is the opposite of a plan with guidelines in the argumentation.A technique of the plan that cannot solve the mathematical problem is still a technique of analyzing. Hence the division of analyzing and making a plan can be a blurred. When the solution is found, it’s clear which techniques have been used for analyzing and which techniques have been applied for creating a plan. After the solution is found, it’s clear which technique of the plan is a strategy or a tactic.The suggestion to solve mathematical problems of math olympiads to learn problem solving in math is useless. You must first learn the fundamentals of proof techniques, follow the guidelines of rigourous proof writing and learn commonly used techniques of problem-solving among mathematicians (techniques in this book). The next step is solving mathematical problems.Problem solving is the non-rigorous phase. Argumentation is the rigorous phase. Another book of making mathematical proofs I would like to recommend you is Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand (Author), Albert D. Polimeni (Author), Ping Zhang (Author) and buy the latest edition. The content of The Art and Craft of Problem Solving is similar with the book How to think like a mathematician (Kevin Houston). The most important technique for a problem solver is: make examples or find a counterexample. I agree with this fact. You create MANY examples for insight in the abstraction or for a hint for proof writing. You can’t use examples for a proof but: Examples give a hint to use a existing mathematical fact in a proof. Abstract the concrete hint and use the abstract hint as fact in a proof. Applied math is pure calculation. Pure math contains abstract concepts you must perform with little to no calculations. Or where you will not be able to solve the problem with just calculation. An example of solving an abstract concept is to prove a statement of set theory. Creating examples or a counterexample is the most important technique for the problem solver and it’s the basis for mathematical explorations in pure mathematics.You can download the book in PDF-format from the Internet Archive free at this link: https://archive.org/details/the-art-and-craft-of-problem-solvingand choose the download option: PDF.Every day, hundreds of papers on research of pure mathematics are uploaded to arXiv from math departments of universities worldwide. Go to this link: https://arxiv.org/archive/math to read the math papers.

⭐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 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).

⭐Used this book for my proof based math class to prepare for the Putnam exam. Problem set provided a variety of difficulty, and the introduction to each proof technique was good. It’s not gonna specialized as a book solely devoted to number theory, per se, but this book gives you a well rounded selection of math topics. It was easy to read, relative to other math books, yet not too dumbed down neither. There are analogies between mathematical problem solving and climbing a mountain in this book… which I could do without.

⭐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!

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

⭐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.

⭐A must.

⭐Entrega super rápida e em muita qualidade.

⭐

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