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The Art And Craft of Problem Solving
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.
366 pages, Paperback
First published February 23, 1999
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Displaying 1 - 19 of 19 reviews
September 6, 2015
This book should definitely be on the shelf of every problem solver or olympic student. The main purpose is to teach thought process and a whole arsenal of tools to students. The target audience would be high-school/undergrad but every problem solver should read it (at least once).
The book focus on teaching three things: strategies, tactics, tools. It helps you to develop the line of reasoning in order to extract information from problems in order to solve them. Similarly to other books, it describes techniques like
* Work in smaller cases
* Use symmetry
* Draw a (nice) figure
* Get your hands dirty - input some values, etc.
* Make it easier - Change the rules in your advantage
but the real differential of the book are the frequent mention of:
* Penultimate step - Thinking backwards, what would be the last thing to prove? What tools you have for that
* Crux move - Comes from Mountaneiring, and represents to the key obstacle.
* Wishful thinking - What if the problem was solved?
In many cases Zeitz brilliantly not only shows the solution but also the tought process to arrive at it, which is always enlightening. He also makes you focus on the difference between How x Why. After solving a problem you should search for Moving curtain arguments, that reminds you why a mathematical argument is true.
A few other lessons mentioned on the book:
- don't attack the problem immediately. Think about it on a less focused level first.
- anything that furthers your investigation is worth doing.
- Stick your butt out - master as many tactical ideas
There are a couple of mistakes towards the end, but they don't obfuscate the brightness of this masterpiece.
Chapter 1
Zeitz starts with the difference between Exercises and Problems. In general, problems take longer and are composed of two parts: investigation and argument. Strategies, tactics and tools are also defined.
Chapter 2
Psychological strategies:
* Mental Toughness: Don't give up (Polya's Mouse).
* Confidence and concentration are essential.
* Creativity
* Toughen up, loosen up and practice!
Steps
* Read the problem carefully
* Identify the type of problem, hypothesis and conclusion.
* Brainstorm
* get your hands dirty, penultimate step, wishful thinking, draw a picture, recast the problem, change the point of view , etc.
* argument ( induction, deduction, contradiction, etc.)
Chapter 3
Tactics
* look for order, to simplify - monotonize when you can, etc
* Symmetry - you might impose it too ( e.g. gaussian pairing tool)!
* extreme principle
* pigeonhole
* invariants
Chapter 4
Crossover tactics:
* Graph theory - connectivity , cycles, eulerian , hamiltonian paths, etc,
* Complex numbers - basic operations, polar form, conjugation,de moivre, exponential form, roots of unity,
* generating functions - local/global polynomial duality, recurrence, partitions, etc.
Chapter 5
Algebra
* add zero creatively
* extract squares
* simplify when possible
* define a function - name things
* substitution
* telescope sums, geometric sums , binomial sums
** massage tool - fiddle with an expression to make it manageable
Chapter 6
Combinatorics
* binomial theorem
** Combinatorial arguments - they give the "why" some statements are true.Algebra will give you how.
* Crux move => Count in two ways
* Flexible point of view.
** partition + encode
* Information Management - use proper encoding to precise information management
* Principle of Inclusion-Exclusion
Chapter 7
Number theory - very basic chapter
* simplify ( divide common factors,etc)
*
Chapter 8
Geometry
* Power of a point
* Similar triangles
* Angle chasing
* phantom points method
* angle bisector theorem
* euler line
* area ratios
impose symmetry and make a transformation keeping something invariant.
Chapter 9
* Eulerian Mathematics - use informal arguments
* Interchange sum and integral
* Generating functions
* riemman sums
* power series
The book focus on teaching three things: strategies, tactics, tools. It helps you to develop the line of reasoning in order to extract information from problems in order to solve them. Similarly to other books, it describes techniques like
* Work in smaller cases
* Use symmetry
* Draw a (nice) figure
* Get your hands dirty - input some values, etc.
* Make it easier - Change the rules in your advantage
but the real differential of the book are the frequent mention of:
* Penultimate step - Thinking backwards, what would be the last thing to prove? What tools you have for that
* Crux move - Comes from Mountaneiring, and represents to the key obstacle.
* Wishful thinking - What if the problem was solved?
In many cases Zeitz brilliantly not only shows the solution but also the tought process to arrive at it, which is always enlightening. He also makes you focus on the difference between How x Why. After solving a problem you should search for Moving curtain arguments, that reminds you why a mathematical argument is true.
A few other lessons mentioned on the book:
- don't attack the problem immediately. Think about it on a less focused level first.
- anything that furthers your investigation is worth doing.
- Stick your butt out - master as many tactical ideas
There are a couple of mistakes towards the end, but they don't obfuscate the brightness of this masterpiece.
Chapter 1
Zeitz starts with the difference between Exercises and Problems. In general, problems take longer and are composed of two parts: investigation and argument. Strategies, tactics and tools are also defined.
Chapter 2
Psychological strategies:
* Mental Toughness: Don't give up (Polya's Mouse).
* Confidence and concentration are essential.
* Creativity
* Toughen up, loosen up and practice!
Steps
* Read the problem carefully
* Identify the type of problem, hypothesis and conclusion.
* Brainstorm
* get your hands dirty, penultimate step, wishful thinking, draw a picture, recast the problem, change the point of view , etc.
* argument ( induction, deduction, contradiction, etc.)
Chapter 3
Tactics
* look for order, to simplify - monotonize when you can, etc
* Symmetry - you might impose it too ( e.g. gaussian pairing tool)!
* extreme principle
* pigeonhole
* invariants
Chapter 4
Crossover tactics:
* Graph theory - connectivity , cycles, eulerian , hamiltonian paths, etc,
* Complex numbers - basic operations, polar form, conjugation,de moivre, exponential form, roots of unity,
* generating functions - local/global polynomial duality, recurrence, partitions, etc.
Chapter 5
Algebra
* add zero creatively
* extract squares
* simplify when possible
* define a function - name things
* substitution
* telescope sums, geometric sums , binomial sums
** massage tool - fiddle with an expression to make it manageable
Chapter 6
Combinatorics
* binomial theorem
** Combinatorial arguments - they give the "why" some statements are true.Algebra will give you how.
* Crux move => Count in two ways
* Flexible point of view.
** partition + encode
* Information Management - use proper encoding to precise information management
* Principle of Inclusion-Exclusion
Chapter 7
Number theory - very basic chapter
* simplify ( divide common factors,etc)
*
Chapter 8
Geometry
* Power of a point
* Similar triangles
* Angle chasing
* phantom points method
* angle bisector theorem
* euler line
* area ratios
impose symmetry and make a transformation keeping something invariant.
Chapter 9
* Eulerian Mathematics - use informal arguments
* Interchange sum and integral
* Generating functions
* riemman sums
* power series
September 13, 2019
If you are looking for problem-solving inspiration/strategies, this is a good book to read. I personally read the first half of the book. The first few chapters mostly discuss not about the math itself specifically but setting the right mindset to solve the problem and how to approach them.
December 3, 2018
This book could be considered the catalyst for many of the events in my life. I was lucky I stumbled upon it at such an early age.
In highschool, this book was the closest I got to an influence of real math. US math tends to favor memorizing how to take the derivative of x^2 and get 2x instead of teaching math correctly. Memorizing algebra, trig, and calculus rules are mostly a waste of time if one's goal is to use math to solve problems that pay the big bucks.
I would guess <5% of US students who passed calculus could actually solve a simple related rates problem with calculus, the #1 use for it.
Recommended if you dislike memorizing in place of understanding. Although I've come full circle and have learned to appreciate memorization as another useful tool, you certainly don't need to memorize very much at all if you want to understand calculus, number theory, graph theory, complex numbers, combinatorics, set theory, and other mathematical topics frequently visited by the Putnam.
In highschool, this book was the closest I got to an influence of real math. US math tends to favor memorizing how to take the derivative of x^2 and get 2x instead of teaching math correctly. Memorizing algebra, trig, and calculus rules are mostly a waste of time if one's goal is to use math to solve problems that pay the big bucks.
I would guess <5% of US students who passed calculus could actually solve a simple related rates problem with calculus, the #1 use for it.
Recommended if you dislike memorizing in place of understanding. Although I've come full circle and have learned to appreciate memorization as another useful tool, you certainly don't need to memorize very much at all if you want to understand calculus, number theory, graph theory, complex numbers, combinatorics, set theory, and other mathematical topics frequently visited by the Putnam.
May 11, 2017
This is the best book on mathematical problem solving I've yet read, although I have neither read Larson's nor Engel's book, which are purported to be classics. Since reading it, I have been able to solve several Putnam and IMO problems with relative ease. If you take this book seriously, you will become a better mathematician.
May 20, 2009
Yay for math! And yay for a math book with so much variety! And yay for a math book with so much approachability!
Boo for problems without an answer key. I don't want to have to sit at my computer while I do math just so I can check my answers.
Boo for problems without an answer key. I don't want to have to sit at my computer while I do math just so I can check my answers.
October 23, 2022
Lots of good problems, and the author encourages people (esp Americans) to learn math the right way. Would be a great book for a precocious high school student, not as good for someone who did a STEM-related degree in college, as a lot of the problems are less hard after having gone through discrete math and advanced math technique classes in engineering.
But I still really like the way the book introduces concepts, encourages certain styles of thinking, and helps students learn a very important skill, which is: "if you are alone with pen and paper and faced with a difficult problem with no consultation, what do you do?"
Surprisingly, the book has an answer for how you can make progress despite starting with nothing. That is a very powerful concept.
But I still really like the way the book introduces concepts, encourages certain styles of thinking, and helps students learn a very important skill, which is: "if you are alone with pen and paper and faced with a difficult problem with no consultation, what do you do?"
Surprisingly, the book has an answer for how you can make progress despite starting with nothing. That is a very powerful concept.
July 22, 2025
Used this book while self-studying math in 11th grade. The problems were difficult for me, but doable. This book really got me grinding my gears. I used to take a problem or two and work on it at random points in the day. Very fun. There still are a lot of problems in it that I haven't solved yet. I'm still working on those :)
September 4, 2018
I book I'll carry with myself.
September 15, 2018
I'll carry it with my hands wherever I go.
January 25, 2022
Amazing book love
April 3, 2025
The go to book if you want to learn how to think mathematically. Helped me a lot, and reveals what mathematical beauty is
May 7, 2020
Just reading 50 pages of the book. and now I can say if you're looking forward to a problem solving books. Just read it. and try to solve every page of the book and remember what you solved .
December 24, 2011
Se amate risolvere i problemi matematici, non lasciatevi spaventare dal prezzo di questo libro: Amazon americano al momento te lo vende anche a metà prezzo. Non preoccupatevi nemmeno della mancanza delle risposte nel libro stesso: se cercate bene in giro si trova il pdf "per gli insegnanti" (file che non è in vendita, quindi non si può nemmeno parlare di mancato introito per l'editore). Io ho apprezzato sia lo stile dell'autore, che molto onestamente spiega come il mondo dei problemi matematici non è il mondo reale e quindi la strategia del wishful thinking spesso paga, che le tecniche più "serie" per affrontare i problemi: alcune le usavo già inconsciamente, altre sono state una piacevole scoperta. In definitiva, un testo indubbiamente di nicchia, ma favoloso per chi nella nicchia ci sta già!
September 28, 2012
Este libro es genial y está de forma permanente en mi currently-reading. Me gusta mucho cómo presentan cada tema y la selección de problemas es muy buena. Lo mejor es que no es un libro de matemáticas, o al menos no sólo de eso, sino de resolver problemas. Y resolver problemas es, para mí, la mejor razón para estudiar.
June 3, 2011
This is a lovely book about sharpening your mind to tackle tougher and tougher problems. Unfortunately it is so focused on math that it doesn't lend itself readily to being adapted into my courses. It was still fun to try some of the problems, though.
December 11, 2014
I'm a bit new to Olympiad maths, but this book has given me a decent overview. I especially liked the combinatorics but all of it was pretty good. We'll see how I'll do in BMO1, but it's not fair to blame my failures on this book! Fnarr fnarr...
June 18, 2007
Great collection of problems covering good ranges of topics and difficulty levels.
January 10, 2021
I read this in high school. Definitely changed my perspective of problem solving for the better.
December 3, 2024
Undoubtedly the single best resource for any high schooler looking to get into olympiad math.
Displaying 1 - 19 of 19 reviews