How to Solve It: A New Aspect of Mathematical Method
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- Create Date：2022-05-25 09:54:33
- Update Date：2025-09-06
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- Author：George Pólya
- ISBN：069116407X
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Summary
A perennial bestseller by eminent mathematician G。 Polya， How to Solve It will show anyone in any field how to think straight。 In lucid and appealing prose， Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be reasoned out--from building a bridge to winning a game of anagrams。 Generations of readers have relished Polya's deft--indeed， brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem。
Reviews
Philippe Fanaro
，
For its time， it was probably an amazing book， but it seems very limited nowadays。 At any rate， many of its problems are incredibly useful and fascinating。
Minotaurs
，
I flunked an exam， asked for book recs to get good at math， and was recommended this book。 I have now gotten （slightly） gooder at math and this book helped。 Mainly because it slapped me in the face by exposing how/where I was falling short in mathematical reasoning。 "But why must I ponder the god damn question statement for five minutes？" I said。 "Because you are a dummy and will make silly little mistakes like assuming what you have to prove by induction :）" said the book， and I felt very seen。 I flunked an exam， asked for book recs to get good at math， and was recommended this book。 I have now gotten （slightly） gooder at math and this book helped。 Mainly because it slapped me in the face by exposing how/where I was falling short in mathematical reasoning。 "But why must I ponder the god damn question statement for five minutes？" I said。 "Because you are a dummy and will make silly little mistakes like assuming what you have to prove by induction :）" said the book， and I felt very seen。Highlights:- Ponder！ （Strategise your 'plan of attack' for the problem。）- Attack + sanity check every step- If you get stuck， start running the questions （have I seen proofs like these？ What do the definitions mean？ What theorems does this rely on？）- Finish and Ponder （understand how you messed up -- what are the logic mistakes you keep on making and what are the gaps in your knowledge。） 。。。more
Charles Mendelson PitchBook
，
This is the only book I universally recommend。While it’s written for mathematicians and math students， the principles of problem solving are applicable to any problem solving domain。In particular I recommend it to software engineers and developers to give a framework for approaching any problem they might encounter。
Madhusa K
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This was once a great book， and in many respects still is。 But books published over the last two decades have already surpassed it in quality i。e "The Art and Craft of Problem Solving" by Paul Zeitz and "Solving Mathematical Problems" by Terence Tao。I still managed to sang a copy for my personal library but only for sentimental reasons。 It has its place in history， but it isn't worth reading if you have access to better material。 This was once a great book， and in many respects still is。 But books published over the last two decades have already surpassed it in quality i。e "The Art and Craft of Problem Solving" by Paul Zeitz and "Solving Mathematical Problems" by Terence Tao。I still managed to sang a copy for my personal library but only for sentimental reasons。 It has its place in history， but it isn't worth reading if you have access to better material。 。。。more
Aaron Daniel
，
In my opinion this should be mandatory reading for any math teacher！ If you’ve been teaching or doing math for some time， most of the stuff in this book will not be really new to you but it will help you better understand and teach the process of solving problems。 Basically the book is a long list of tipps and tricks to help you crack complex problems with some great advice on how to help someone else during that process。
Jeff Hui
，
There’s a lot of useful value here - even though the writing style and target audience is a bit too narrow:- has some assumption of a teacher/student context- assumes a more mathematical context- generally assumes an academic context is most partsBut it’s a good articulation of how to solve problems。
Alessio De Monte
，
This the best book I have read about this topic。 I think the main ideas should be taught to every student or even every person for that matter， because solving problems is arguably the main activity of our species。 Of course it has a mathematical tone， which may or may not suited to you， but the principles in their generality are applicable to any category of life in which problems arise。
Nico Grassetto
，
This review has been hidden because it contains spoilers。 To view it， click here。 Context: I am currently enrolled in a bachelor's degree in data science and artiticial intelligence。 I bought this book a while ago as a supplementary material for my discrete mathematics class as I have always been bad at proofs and solving mathematical problems。 I 。。。 didn't read it and it languished gathering dust on my shelf for now more than 2 years。 I am now almost graduated and I have to say that this book despite from having some sections here and there that are of interest， is pretty ba Context: I am currently enrolled in a bachelor's degree in data science and artiticial intelligence。 I bought this book a while ago as a supplementary material for my discrete mathematics class as I have always been bad at proofs and solving mathematical problems。 I 。。。 didn't read it and it languished gathering dust on my shelf for now more than 2 years。 I am now almost graduated and I have to say that this book despite from having some sections here and there that are of interest， is pretty basic。This book looks almost like a student-teacher guide。 Most sections are about "How to be a good student" or "How to be a good teacher"。 Though I have no doubt this will please mathematics geeks and they are likely to enjoy the author's style。 It's grown to become a classic but I wouldn't lust for actual skills here。 This will be a pleasant reading for anyone interested in mathematics （for the beauty of it） and less so for someone seeking a pragmatical approach to mathematics。 The title lured me I have to confess and I wasn't the target audience。 。。。more
Aaron Schumacher
，
The first rule of style is to have something to say。I wish I had read How to Solve It when I was a fairly young student of mathematics。 I wish also that I had read it when I was becoming a teacher of mathematics。 It has been recommended many times， and now I will recommend it also。 Really first rate stuff。 Everyone's excited these days about Thinking， Fast and Slow， but if you're moving into real problem-solving that necessitates "slow" mental work， it's Polya who really has something to say。 Hi The first rule of style is to have something to say。I wish I had read How to Solve It when I was a fairly young student of mathematics。 I wish also that I had read it when I was becoming a teacher of mathematics。 It has been recommended many times， and now I will recommend it also。 Really first rate stuff。 Everyone's excited these days about Thinking， Fast and Slow， but if you're moving into real problem-solving that necessitates "slow" mental work， it's Polya who really has something to say。 His comments extend beyond the core problem-solving theses as well。 I include here some quotes of particular note。> Mathematics is interesting in so far as it occupies our reasoning and inventive powers。> Can our knowledge in mathematics be based on formal proofs alone？ 。。。 It is certain that your knowledge， or my knowledge， or your students' knowledge in mathematics is not based on formal proofs alone。 If there is any solid knowledge at all， it has a broad experimental basis， and this basis is broadened by each problem whose result is successfully tested。> Definitions in dictionaries are not very much different from mathematical definitions in the outward form but they are written in a different spirit。 The writer of a dictionary is concerned with the current meaning of the words。 He accepts， of course， the current meaning and states it as neatly as he can in form of a definition。 The mathematician is not concerned with the current meaning of his technical terms， at least not primarily concerned with that。 What "circle" or "parabola" or other technical terms of this kind may or may not denote in ordinary speech matters little to him。 The mathematical definition creates the mathematical meaning。> Teaching to solve problems is education of the will。> Another "problem to prove" is to "prove the theorem of Pythagoras。" We do not say: "Prove or disprove the theorem of Pythagoras。" It would be better in some respects to include in the statement of the problem the possibility of disproving， but we may neglect it， because we know that the chances for disproving the theorem of Pythagoras are rather slight。> If the student failed to get acquainted with this or that particular geometric fact， he did not miss so much; he may have little use for such facts in later life。 But if he failed to get acquainted with geometric proofs， he missed the best and simplest examples of true evidence and he missed the best opportunity to acquire the idea of strict reasoning。 Without this idea， he lacks a true standard with which to compare alleged evidence of all sorts aimed at him in modern life。> Not all mathematical theorems can be split naturally into hypothesis and conclusion。 Thus， it is scarcely possible to split so the theorem: "There are an infinity of prime numbers。"That last is one place where I may disagree with Polya。 The split is easy and natural: "Assuming our usual definitions （hypothesis） there are an infinity of prime numbers （conclusion）。" It's easy to forget that what we consider natural， as "the natural numbers" and so on， is just a made-up mathematical system when considered formally。 Where mathematics comes from is a good exposition on this sort of thing， I think。 And here， in a bit of a contrast， is perhaps my favorite quasi-philosophical quote from Polya's book:> Do not believe anything but doubt only what is worth doubting。 。。。more
Sophie
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What is the unknown？ What is the condition？ Where are the data？ What I have gained from this book is that reflections of what we have already know when facing a problem is a way to start planning solving the problem。 As the author has been repeated his principles of solving problem: thinking about where should I start？ What can I do？ What can I gain by doing so？ After carrying out a plan， it is extremely important to look back if there the solution can be more simple therefore to find a better s What is the unknown？ What is the condition？ Where are the data？ What I have gained from this book is that reflections of what we have already know when facing a problem is a way to start planning solving the problem。 As the author has been repeated his principles of solving problem: thinking about where should I start？ What can I do？ What can I gain by doing so？ After carrying out a plan， it is extremely important to look back if there the solution can be more simple therefore to find a better solution。 。。。more
Chris Esposo
，
Polya’s book on the meta-theory of analyzing mathematical problems， and （hopefully） solving them is a rare （semi） technical text that can be read fairly well both on audio and by eye， and still get something out of both。 Of course， to really get the full instructional value， you must read this text visually， on account of the mathematical notation。 Still Polya’s commentary on the nature of heuristics and how they are applied to problem solving shines through even with just the audio。 In many way Polya’s book on the meta-theory of analyzing mathematical problems， and （hopefully） solving them is a rare （semi） technical text that can be read fairly well both on audio and by eye， and still get something out of both。 Of course， to really get the full instructional value， you must read this text visually， on account of the mathematical notation。 Still Polya’s commentary on the nature of heuristics and how they are applied to problem solving shines through even with just the audio。 In many ways， this book covers the standard material that many other modern textbooks aimed at introduction to proofs and logic also survey， which includes the standard construction techniques of indirect proofs， the reductio ad-absurdum， the workhorse of direct proofs （of well-ordered objects）， the induction， as well as a smattering of examples taken from what we would now call “school mathematics”， the fields of geometry， counting， and （classical） algebra。 However， unlike modern textbooks designed for the standard bridge-course into formalism， there is much less focus in the practice of procedure （solving problems via exercises， to which there are none in this book， at least numbered ones）， and more focus on the method of procedure， that is a commentary in english on why we are doing what we are doing in solutioning/analysis， and how to think about it as a human being。 This is a critical piece of knowledge that is often missing in modern text for this subject matter， and is an essential prerequisite to that thing we call “mathematical maturity” i。e。 the ability to think， reason， and extend mathematical objects to our purposes。 Many modern textbooks have a tendency to routinize this process of discovery/solutioning and this is a mistake。 Although there may be a prescription that yields solutions “most of the time” for many classes of problems， to learn the process of interacting with mathematics in this way is to learn to be a computer。 That is， it is to learn to divorce oneself with their ‘humanness’ in the way of thinking about mathematical things， and it is practically diminishing one’s capacity to efficiently solve problems as a human vis-a-vis the machine。 That is， it is unlikely a human will be able to compete with a machine in traversing the problem in that avenue of approach。 Instead， the way many mathematicians have approached these abstract objects is to leverage their intuition。 Just as a Chess player does not literally ken all of the enumerations of steps that can take place， but instead has a meta-idea of how to think of the board， and deconstructs it using that “theory” of the board to inform their next move， a human mathematician will have to leverage their intuition to devise heuristics to assess the object at hand， then attack the problem with that assessment。 Poyla teaches it’s readers how to go about this process methodically。 I found re-visiting this subject matter after having left it for a few years to do professional work， it has awakened many of my dormant capacities， and more importantly， helped me to extend previously weaker parts of my skill-sets。 This is a great example of a supremely well-written （somewhat） technical book， which rivals （possibly exceeds） the few others that also do this intersection well， such as Richard Feynman’s more technical instructional writings。 Highly recommended to anyone learning （or relearning） this subject as a more holistic commentary on craft in mathematics。 。。。more
Divakaran
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Best book for beginners to frame systematic thinking to approach math
Alex
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Overall quite useful as a general guide to "heuristic problem solving，" though the examples focus specifically on geometry for the most part， and the language is at this juncture fairly dated， occasionally to the point of obscuring some of the author's ideas。 Structurally interesting， as the bulk of the text is laid out as a "dictionary" that nevertheless keeps a （mostly） linear conceptual flow if read in order。 Overall quite useful as a general guide to "heuristic problem solving，" though the examples focus specifically on geometry for the most part， and the language is at this juncture fairly dated， occasionally to the point of obscuring some of the author's ideas。 Structurally interesting， as the bulk of the text is laid out as a "dictionary" that nevertheless keeps a （mostly） linear conceptual flow if read in order。 。。。more
Bogdan Alexa
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It's quite a good book for the average person。 I picked it up because I've found it as a recommendation in a stackoverflow comment on a topic regarding learning programming algorithms。 It's a good book because it introduces you in the right mental framework to approach the problem of learning something complex （such as algorithms）。 It's great for those who are in a teaching proffesion， especially mathematics， and especially with children。 George Polya must have been a great teacher。 Not being in It's quite a good book for the average person。 I picked it up because I've found it as a recommendation in a stackoverflow comment on a topic regarding learning programming algorithms。 It's a good book because it introduces you in the right mental framework to approach the problem of learning something complex （such as algorithms）。 It's great for those who are in a teaching proffesion， especially mathematics， and especially with children。 George Polya must have been a great teacher。 Not being in that niche， 3 stars from me。 。。。more
Chelsea Eaton
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This book is the perfect book for anyone aspiring to become a teacher or even current teachers， it simply and in great detail explains the mind of the teacher and student and how to use the students thought process to be able to answer their questions and help them solve the problem。 This book may also be good for anyone looking to per-sue mathematics as the maths within the book are simple and explained well。 In addition to this the book gives a great insight in understanding computational thin This book is the perfect book for anyone aspiring to become a teacher or even current teachers， it simply and in great detail explains the mind of the teacher and student and how to use the students thought process to be able to answer their questions and help them solve the problem。 This book may also be good for anyone looking to per-sue mathematics as the maths within the book are simple and explained well。 In addition to this the book gives a great insight in understanding computational thinking and could be a good recommendation for future computer scientists needing to better understand computational thinking。 For me however this book is only 2 stars， it wasn’t relevant to anything i needed and i didn’t enjoy the writing style of the author。 。。。more
Eugene
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I've finally read Polya's "How to Solve It。" Polya describes methods for solving math problems， yet I can relate that it works in software development， too。One of the surprising things is that Polya mentioned the sense of progress。 Some could miss this sense and give up， but a successful problem-solver feels progress。 It encourages to keep up and solve it eventually。I want everyone to develop this sense of progress during solving problems and to succeed at problem-solving and creating software。 I've finally read Polya's "How to Solve It。" Polya describes methods for solving math problems， yet I can relate that it works in software development， too。One of the surprising things is that Polya mentioned the sense of progress。 Some could miss this sense and give up， but a successful problem-solver feels progress。 It encourages to keep up and solve it eventually。I want everyone to develop this sense of progress during solving problems and to succeed at problem-solving and creating software。 。。。more
Chetan Vashisht
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A very verbose book。 It some absolute gems in it！ Can't recommend it enough for those little nuggets。 Good book overall， not for everyone。 A very verbose book。 It some absolute gems in it！ Can't recommend it enough for those little nuggets。 Good book overall， not for everyone。 。。。more
Daniel Santos
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Comecei a ler porque este livro， apesar do teor extremamente matemático， é indicado por muitos como contendo técnicas que podem ser aplicadas à resolução de problemas de modo geral。 Alguns trechos são de fato interessantes， mas o livro não possui nada de excepcional。
Mohsen
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Seems the author tries to treat the reader as a really slow learner student and keeps repeating the same concept over and over again in different phrasings。 If you could run a deduplication algorithm on the content of the book， it shouldn't be more than 50 pages， and the half of that would be the example problems。I am surprised why John Conway has admired the book in the foreword。You would learn little after reading the first two parts。 Skip the dictionary part。 Seems the author tries to treat the reader as a really slow learner student and keeps repeating the same concept over and over again in different phrasings。 If you could run a deduplication algorithm on the content of the book， it shouldn't be more than 50 pages， and the half of that would be the example problems。I am surprised why John Conway has admired the book in the foreword。You would learn little after reading the first two parts。 Skip the dictionary part。 。。。more
Uriel Vidal
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Este es uno de esos libros que sí solo se evaluara una parte estaría mejor rankeado， y es que las primeras dos partes son fantásticas， son bastantes resumidas y muy específicas para poder resolver problemas de forma adecuada y se deben tener a la mano para poder enfrentar cualquier adversidad a la que nos enfrentemos。 La segunda parte es algo más tediosa， pero no deja de ser de mucha utilidad， el gran problema que es un diccionario y si alguien ha tratado de leer un diccionario de la A a la Z sa Este es uno de esos libros que sí solo se evaluara una parte estaría mejor rankeado， y es que las primeras dos partes son fantásticas， son bastantes resumidas y muy específicas para poder resolver problemas de forma adecuada y se deben tener a la mano para poder enfrentar cualquier adversidad a la que nos enfrentemos。 La segunda parte es algo más tediosa， pero no deja de ser de mucha utilidad， el gran problema que es un diccionario y si alguien ha tratado de leer un diccionario de la A a la Z sabrá lo complicado que resulta debido a que no se tiene una linealidad en lo que se lee y en muchas ocasiones se repiten conceptos， lo que resulta que las ideas estén constantemente apareciendo hasta el hartazo， sin embargo， la finanlidad de este diccionario es tener al alcance de la mano una forma rápida de acceder a este conocimiento mucho tiempo después de la lectura y tener una referencia muy rápida de todo este método que plantea。 Todo el libro es una gran lectura， que se le debe tener paciencia y que a las personas que les interese mucho la hurística es una lectura obligada。Sin duda es recomendable para maestros， personas que les gusten las matemáticas， para emprendedores y para cualquier persona que en la vida se encuentre atorada por un problema irresoluble。 。。。more
Matthew Emery
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A truly bizarre book。 Part reference manual for reasoning about logic puzzles and part philosophical treatise on the nature of education。 It won't surprise you that some parts of this book are dry， but after finishing it I've found it on my desk multiple times as I thought about problems at my work。 A truly bizarre book。 Part reference manual for reasoning about logic puzzles and part philosophical treatise on the nature of education。 It won't surprise you that some parts of this book are dry， but after finishing it I've found it on my desk multiple times as I thought about problems at my work。 。。。more
Henry Cooksley
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Good， but I was expecting a bit more。
Shijing
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This is the kind of book that makes the reader realize why mathematics is philosophical and the fundamental of the natural science。 It would have changed my point of view on approaching the math problems， if I had read it in grad school。 But I was able to apply several methods in the book to the work project。
Marco
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This is definitely a book that would have been super useful some years ago。 Polya explains how to tackle mathematical （and non） problems in a rigid but intuitive way。 Would suggest this to every future math teacher for sure。
Martin Hruska
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Pólya možná není nejzábavnější spisovatel a popularizátor vědy， ale jeho kniha je nesmírně cenná nejen pro matematiky。 Sada heuristik a návodů pro řešení matematických problému je výbornou pomůckou ne protože vás vždy dovede k řešení （to ani není možné）， ale protože je to způsob jak nějak postupovat při řešení a ne jen sedět a smutně koukat na prázdný papír。 Samotný technický styl Pólyova psaní nakonec není ke škodě， ale naopak nutí čtenáře k přemýšlení a soustředění。 Jeho kniha také může být ob Pólya možná není nejzábavnější spisovatel a popularizátor vědy， ale jeho kniha je nesmírně cenná nejen pro matematiky。 Sada heuristik a návodů pro řešení matematických problému je výbornou pomůckou ne protože vás vždy dovede k řešení （to ani není možné）， ale protože je to způsob jak nějak postupovat při řešení a ne jen sedět a smutně koukat na prázdný papír。 Samotný technický styl Pólyova psaní nakonec není ke škodě， ale naopak nutí čtenáře k přemýšlení a soustředění。 Jeho kniha také může být obecný návod， který lze aplikovat i do jiných oborů。 Přemýšlejte o tom， jak postupujete při řešení úloh， sepište si obecný návod a sadu konkrétních heuristik。 Vnuknutí správného řešení je jistě unikátní nepřenositelný jev， ale pomocí knih jako tahle se o tom alespoň pokusíme nějak mluvit a při řešení problému pak tzv。 udržet balón ve hře。 。。。more
Naveen Arun
，
It's a useful dictionary of problem-solving heuristics - however it's important to note that it applies specifically to problems in pure mathematics and doesn't generalize well beyond that。 Some of the heuristics can generalize to engineering or general problem-solving， but not all of them。 The problems + hints at the end were good practice in applying the heuristics listed in the book。 I think I can solve a slightly wider range of math problems and proofs after reading this， but I haven't teste It's a useful dictionary of problem-solving heuristics - however it's important to note that it applies specifically to problems in pure mathematics and doesn't generalize well beyond that。 Some of the heuristics can generalize to engineering or general problem-solving， but not all of them。 The problems + hints at the end were good practice in applying the heuristics listed in the book。 I think I can solve a slightly wider range of math problems and proofs after reading this， but I haven't tested this belief 😂 。。。more
Alb85
，
Questo libro è rivolto a chi vuole risolvere problemi matematici， in particolare problemi di determinazione e di dimostrazione。Nella prima parte vengono proposte delle strategie per coinvolgere lo studente nella risoluzione del problema。 La chiave è fornire le domande giuste， senza dare implicitamente la soluzione。 Nella seconda parte， la più corposa， vengono presentate diverse metodologie per risolvere un problema matematico， come ad esempio la risoluzione per analogia （cioè si può partire ad e Questo libro è rivolto a chi vuole risolvere problemi matematici， in particolare problemi di determinazione e di dimostrazione。Nella prima parte vengono proposte delle strategie per coinvolgere lo studente nella risoluzione del problema。 La chiave è fornire le domande giuste， senza dare implicitamente la soluzione。 Nella seconda parte， la più corposa， vengono presentate diverse metodologie per risolvere un problema matematico， come ad esempio la risoluzione per analogia （cioè si può partire ad esempio dalle proprietà di un segmento per risolvere un problema che riguarda un triangolo， o partire dalle proprietà di un triangolo per risolvere il problema di un tetraedro）， la dimostrazione per assurdo， la dimostrazione per analogia。L’autore sottolinea più volte nel testo l’importanza di porsi le seguenti domande: Qual è l’incognita？ Quali sono i dati？ Qual è la condizione？Vengono poi menzionati altri aspetti di secondaria importanza come l’importanza dei simboli da usare， la differenza tra problema matematico e problema pratico。Il libro non è di facile lettura， ha molte informazioni ridondanti。 Non ha soddisfatto le mie aspettative。 。。。more
Maria
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pues un libro un poco aburrido donde la mayoria de la segunda parte es completamente bla bla 。 la parte importante。 Basicamente intenta enseñar al lector un método para solve problems， split en cuatro partes: understanding the problem， devising a plan， carrying out the plan y looking back。what is the unkown， what are the data， what is the condition，draw a figuredo you know a related problem， look at the unknown， here ias a problem related to yours and solved before， could you use itcheck each st pues un libro un poco aburrido donde la mayoria de la segunda parte es completamente bla bla 。 la parte importante。 Basicamente intenta enseñar al lector un método para solve problems， split en cuatro partes: understanding the problem， devising a plan， carrying out the plan y looking back。what is the unkown， what are the data， what is the condition，draw a figuredo you know a related problem， look at the unknown， here ias a problem related to yours and solved before， could you use itcheck each stepcheck the result if you can 。。。more
Marcelo
，
Pólya explained in a clear and concise way what mind operations we do when solving a problem。 He presented systematic questions we should remember to solve a problem when we got stuck。 Most of the tips are intuitive and simple and it is very easy to apply。 But， as humans， we have a lot of other problems can interfere in our concentration and learning。 I think simple questions may help us to change our point of view， Polya did a great job to sum up those questions。
Brad Stieber
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