We will keep fighting for all libraries - stand with us!

Internet Archive Audio

how to solve a problem george polya

  • This Just In
  • Grateful Dead
  • Old Time Radio
  • 78 RPMs and Cylinder Recordings
  • Audio Books & Poetry
  • Computers, Technology and Science
  • Music, Arts & Culture
  • News & Public Affairs
  • Spirituality & Religion
  • Radio News Archive

how to solve a problem george polya

  • Flickr Commons
  • Occupy Wall Street Flickr
  • NASA Images
  • Solar System Collection
  • Ames Research Center

how to solve a problem george polya

  • All Software
  • Old School Emulation
  • MS-DOS Games
  • Historical Software
  • Classic PC Games
  • Software Library
  • Kodi Archive and Support File
  • Vintage Software
  • CD-ROM Software
  • CD-ROM Software Library
  • Software Sites
  • Tucows Software Library
  • Shareware CD-ROMs
  • Software Capsules Compilation
  • CD-ROM Images
  • ZX Spectrum
  • DOOM Level CD

how to solve a problem george polya

  • Smithsonian Libraries
  • FEDLINK (US)
  • Lincoln Collection
  • American Libraries
  • Canadian Libraries
  • Universal Library
  • Project Gutenberg
  • Children's Library
  • Biodiversity Heritage Library
  • Books by Language
  • Additional Collections

how to solve a problem george polya

  • Prelinger Archives
  • Democracy Now!
  • Occupy Wall Street
  • TV NSA Clip Library
  • Animation & Cartoons
  • Arts & Music
  • Computers & Technology
  • Cultural & Academic Films
  • Ephemeral Films
  • Sports Videos
  • Videogame Videos
  • Youth Media

Search the history of over 866 billion web pages on the Internet.

Mobile Apps

  • Wayback Machine (iOS)
  • Wayback Machine (Android)

Browser Extensions

Archive-it subscription.

  • Explore the Collections
  • Build Collections

Save Page Now

Capture a web page as it appears now for use as a trusted citation in the future.

Please enter a valid web address

  • Donate Donate icon An illustration of a heart shape

How To Solve It

Bookreader item preview, share or embed this item, flag this item for.

  • Graphic Violence
  • Explicit Sexual Content
  • Hate Speech
  • Misinformation/Disinformation
  • Marketing/Phishing/Advertising
  • Misleading/Inaccurate/Missing Metadata

plus-circle Add Review comment Reviews

Download options, in collections.

Uploaded by TheDeaderzOne on July 26, 2023

SIMILAR ITEMS (based on metadata)

How to Solve It

Before you purchase audiobooks and ebooks.

Please note that audiobooks and ebooks purchased from this site must be accessed on the Princeton University Press app. After you make your purchase, you will receive an email with instructions on how to download the app. Learn more about audio and ebooks .

Support your local independent bookstore.

  • United States
  • United Kingdom

Mathematics

How to Solve It: A New Aspect of Mathematical Method

  • John H. Conway

The bestselling book that has helped millions of readers solve any problem

  • Princeton Science Library

how to solve a problem george polya

  • Download Cover

A must-have guide by eminent mathematician G. Polya, How to Solve It shows 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 help you attack any problem that can be reasoned out—from building a bridge to winning a game of anagrams. How to Solve It includes a heuristic dictionary with dozens of entries on how to make problems more manageable—from analogy and induction to the heuristic method of starting with a goal and working backward to something you already know. This disarmingly elementary book explains how to harness curiosity in the classroom, bring the inventive faculties of students into play, and experience the triumph of discovery. But it’s not just for the classroom. Generations of readers from all walks of life have relished Polya’s brilliantly deft instructions on stripping away irrelevancies and going straight to the heart of a problem.

"Every mathematics student should experience and live this book"— Mathematics Magazine

"[ How to Solve It ] shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected."—Herman Weyl, Mathematical Review

"Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art."—A. C. Schaeffer, American Journal of Psychology

"I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it."— Scientific Monthly

"Every prospective teacher should read [ How to Solve It ]. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.'"—E. T. Bell, Mathematical Monthly

"In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come."—A. Bultheel, European Mathematical Society

“Every mathematics student should experience and live this book.”— Mathematics Magazine

“[ How to Solve It ] shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected.”—Herman Weyl, Mathematical Review

“Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher’s art.”—A. C. Schaeffer, American Journal of Psychology

“I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it.”— Scientific Monthly

“Every prospective teacher should read [ How to Solve It ]. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: ‘He writes a, he says b, he means c; but it should be d.’”—E. T. Bell, Mathematical Monthly

“In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let’s hope this classic will remain a source of inspiration for several generations to come.”—A. Bultheel, European Mathematical Society

Stay connected for new books and special offers. Subscribe to receive a welcome discount for your next order. 

Book Lover Sale  - Feb 5-26.  75% off  with code 75OFF  at checkout. See our FAQ for more details.

  • ebook & Audiobook Cart
  • My Dashboard

OPEPP Logo

  • Content: Polya’s Problem-Solving Method

Back to: Module: Helping Students Do Math

The purpose of this tool for the field is to help paraprofessionals become more familiar with, and practice using, Polya’s four-step problem-solving method.

how to solve a problem george polya

  • Read the example below about Mrs. Byer’s class, and then look over the example of how Polya’s method was used to solve the problem.

how to solve a problem george polya

Every person at a party of 12 people said hello to each of the other people at the party exactly once. How many “hellos” were said at the party?           

how to solve a problem george polya

A new burger restaurant offers two kinds of buns, three kinds of meats, and two types of condiments. How many different burger combinations are possible that have one type of bun, one type of meat, and one condiment type?

A family has five children. How many different gender combinations are possible, assuming that order matters? (For example, having four boys and then a girl is distinct from having a girl and then four boys.)

Hillary and Marco are both nurses at the city hospital. Hillary has every fifth day off, and Marco has off every Saturday (and only Saturdays). If both Hillary and Marco had today off, how many days will it be until the next day when they both have off?

Reflect on your experience.

  • In which types of situations do you think students would find Polya’s method helpful?
  • Are there types of problems for which students would find the method more cumbersome than it is helpful?
  • Can you think of any students who would particularly benefit from a structured problem-solving approach such as Polya’s?

                           Background Information

how to solve a problem george polya

Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our lives where we encounter problems—not just math. Although the method appears to be a straightforward method where you start at Step 1, and then go through Steps 2, 3, and 4, the reality is that you will often need to go back and forth through the four steps until you have solved and reflected on a problem.

Polya’s Problem-Solving Chart: An Example

A version of Polya’s problem-solving chart can be found below, complete with descriptions of each step and an illustration of how the method can be used systematically to solve the following problem:

Scenario 

There are 22 students in Mrs. Byer’s third grade class. Every student is required to either play the recorder or sing in the choir, although students have the option of doing both. Eight of Mrs. Byer’s students chose to play the recorder, and 20 students sing in the choir. How many of Mrs. Byer’s students both play the recorder and sing in the choir?

Module: Helping Students Do Math

  • Introductory Scenario
  • Content: Does Anyone Know What Math Is?
  • Content: The Fennema-Sherman Attitude Scales
  • Content: Past Experience with Math
  • Content: Learning About Math
  • Content: What is it like to teach math?
  • Content: Using a Frayer Model
  • Content: Helping a Child Learn from a Textbook
  • Content: Using Online Math Resources
  • Content: Helping a Student Learn to use a Calculator
  • Links for More Information
  • Content: Better Questions
  • Content: Practice Asking Good Questions
  • Content: Applying Poly’s Method to a Life Decision
  • Content: Learning Progression Activities
  • Content: Connecting Concepts and Procedures
  • Content: Resources
  • Activity: The Old Guy’s No-Math Test
  • Open All · Close All

Contact:  [email protected]

The Ohio Partnership for Excellence in Paraprofessional Preparation is primarily supported through a grant with the Ohio Department of Education, Office for Exceptional Children. Opinions expressed herein do not necessarily reflect those of the Ohio Department of Education or Offices within it, and you should not assume endorsement by the Ohio Department of Education.

how to solve a problem george polya

 MacTutor

George pólya.

... diligence and good behaviour.
I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.
I was greatly influenced by Fejér , as were all Hungarian mathematicians of my generation, and, in fact, once or twice in small matters I collaborated with Fejér . In one or two papers of his I have remarks and he made remarks in one or two papers of mine, but it was not really a deep influence.
On Christmas 1913 I travelled by train from Zürich to Frankfurt and at that time I had a verbal exchange - about my basket that had fallen down - with a young man who sat across from me in the train compartment. I was in an overexcited state of mind and I provoked him. When he did not respond to my provocation, I boxed his ear. Later on it turned out that the young man was the son of a certain Geheimrat; he was a student, of all things, in Göttingen. After some misunderstandings I was told to leave by the Senate of the University.
I was... deeply influenced by Hurwitz . In fact I went to Zürich in order to be near Hurwitz and we were in close touch for about six years, from my arrival in Zürich in 1914 to his passing in ... 1919 . And we have one joint paper, but that is not the whole extent. I was very much impressed by him and edited his works. I was also impressed by his manuscripts.
I came very late to mathematics. ... as I came to mathematics and learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered?
... a mathematical masterpiece that assured their reputations.
Pólya was arguably the most influential mathematician of the 20 th century. His basic research contributions span complex analysis, mathematical physics, probability theory , geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career.
For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Pólya.
The aim of heuristic is to study the methods and rules of discovery and invention .... Heuristic, as an adjective, means 'serving to discover'. ... its purpose is to discover the solution of the present problem. ... What is good education? Systematically giving opportunity to the student to discover things by himself.
If you can't solve a problem, then there is an easier problem you can solve: find it.
Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. ... However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. ... But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems.
Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there is great latitude and much possibility for personal differences. ... let me tell you what my idea of teaching is. Perhaps the first point, which is widely accepted, is that teaching must be active, or rather active learning. ... the main point in mathematics teaching is to develop the tactics of problem solving.
... a remarkable theorem in a remarkable paper, and a landmark in the history of combinatorial analysis.
The whole work displays the taste of the authors for the concrete and explicit result, for elegance and ingenious methods.
With no hesitation, George Pólya is my personal hero as a mathematician. ... [ he ] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

References ( show )

  • G L Alexanderson, The Polya picture album ( Basel, 1987) .
  • G L Alexanderson, The random walks of George Pólya ( Washington, DC, 2000) .
  • H Taylor and L Taylor, George Pólya : Master of Discovery ( Palo Alto, CA, 1993) .
  • D J Albers and G L Alexanderson ( eds. ) , Mathematical People: Profiles and Interviews ( Boston, 1985) , 245 - 254 .
  • G L Alexanderson and L H Lange, Obituary: George Pólya, Bull. London Math. Soc. 19 (6) (1987) , 559 - 608 .
  • G L Alexanderson and J Pedersen, George Pólya : his life and work ( Hungarian ) , Mat. Lapok 33 (4) (1982 / 86) , 225 - 233 .
  • R P Boas, Selected topics from Pólya's work in complex analysis, Math. Mag. 60 (5) (1987) , 271 - 274 .
  • R P Boas, Pólya's work in analysis, Bull. London Math. Soc. 19 (6) (1987) , 576 - 583 .
  • H Cartan, La vie et l'oeuvre de George Pólya, C. R. Acad. Sci. Sér. Gén. Vie Sci. 3 (6) (1986) , 619 - 620 .
  • K L Chung, Pólya's work in probability, Bull. London Math. Soc. 19 (6) (1987) , 570 - 576 .
  • F Harary, Homage to George Pólya, J. Graph. Theory 1 (4) (1977) , 289 - 290 .
  • P Hilton and J Pedersen, The Euler characteristic and Pólya's dream, Amer. Math. Monthly 103 (2) (1996) , 121 - 131 .
  • J-P Kahane, The grand figure of George Pólya ( Czech ) , Pokroky Mat. Fyz. Astronom. 35 (4) (1990) , 177 - 191 .
  • J Kilpatrick, George Pólya's influence on mathematics education, Math. Mag. 60 (5) (1987) , 299 - 300 .
  • D H Lehmer, Comments on number theory, Bull. London Math. Soc. 19 (6) (1987) , 584 - 585 .
  • A Pfluger, George Pólya, J. Graph Theory 1 (4) (1977) , 291 - 294 .
  • R C Read, Pólya's theorem and its progeny, Math. Mag. 60 (5) (1987) , 275 - 282 .
  • R C Read, Pólya's enumeration theorem, Bull. London Math. Soc. 19 (6) (1987) , 588 - 590 .
  • P C Rosenbloom, Studying under Pólya and Szegö at Stanford, in A century of mathematics in America II ( Providence, RI, 1989) , 279 - 281 .
  • D Schattschneider, The Pólya-Escher connection, Math. Mag. 60 (5) (1987) , 293 - 298 .
  • D Schattschneider, Pólya's geometry, Bull. London Math. Soc. 19 (6) (1987) , 585 - 588 .
  • M M Schiffer, George Pólya (1887 - 1985) , Math. Mag. 60 (5) (1987) , 268 - 270 .
  • M M Schiffer, Pólya's contributions in mathematical physics, Bull. London Math. Soc. 19 (6) (1987) , 591 - 594 .
  • A H Schoenfeld, Pólya, problem solving, and education, Math. Mag. 60 (5) (1987) , 283 - 291 .
  • A H Schoenfeld, George Pólya and mathematicvs education, Bull. London Math. Soc. 19 (6) (1987) , 594 - 596 .
  • Y S Tseng, On Pólya's Mathematical discovery ( Chinese ) , J. Math. Res. Exposition 3 (1) (1983) , 213 - 216 .
  • Y S Tseng, Correction: 'On Pólya's Mathematical discovery' ( Chinese ) , J. Math. Res. Exposition 3 (2) (1983) , 22 .
  • A A Wieschenberg, A conversation with George Pólya, Math. Mag. 60 (5) (1987) , 265 - 268 .
  • I M Yaglom, George Pólya ( on the 100 th anniversary of his birth ) ( Russian ) , Mat. v Shkole (3) (1988) , 67 - 70 .

Additional Resources ( show )

Other pages about George Pólya:

  • Pólya on Fejér
  • Pólya and Szegö's Problems and Theorems in Analysis
  • Hardy's reference for Pólya at ETH
  • Some of Pólya's favourite quotes
  • Preface to Pólya's How to solve it
  • Heinz Klaus Strick biography

Other websites about George Pólya:

  • Australia Mathematics Trust
  • Mathematical Genealogy Project
  • MathSciNet Author profile
  • zbMATH entry

Honours ( show )

Honours awarded to George Pólya

  • LMS Honorary Member 1956
  • Popular biographies list Number 44

Cross-references ( show )

  • Societies: Canadian Mathematical Society
  • Societies: Society for Industrial and Applied Mathematics
  • Societies: Zurich Scientific Research Society
  • Other: 1936 ICM - Oslo
  • Other: 2009 Most popular biographies
  • Other: Earliest Known Uses of Some of the Words of Mathematics (C)
  • Other: Earliest Known Uses of Some of the Words of Mathematics (E)
  • Other: Earliest Known Uses of Some of the Words of Mathematics (M)
  • Other: Earliest Known Uses of Some of the Words of Mathematics (P)
  • Other: Earliest Uses of Symbols of Number Theory
  • Other: London Learned Societies
  • Other: Most popular biographies – 2024
  • Other: Popular biographies 2018

Logo for UH Pressbooks

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Problem Solving

Problem Solving Strategies

Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills.  He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities).  He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

George Pólya ca 1973

 In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

  • First, you have to understand the problem.
  • After understanding, then make a plan.
  • Carry out the plan.
  • Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

  • What if the picture was different?
  • What if the numbers were simpler?
  • What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

  • Describe all of the patterns you see in the table.
  • Can you explain and justify any of the patterns you see? How can you be sure they will continue?
  • What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

how to solve a problem george polya

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

  • What is the sum of all the numbers on the clock’s face?
  • Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
  • How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

how to solve a problem george polya

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

  • Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

George Pólya & problem solving ... An appreciation

  • General / Article
  • Published: 06 May 2014
  • Volume 19 , pages 310–322, ( 2014 )

Cite this article

  • Shailesh A. Shirali 1  

841 Accesses

3 Citations

3 Altmetric

Explore all metrics

George Pólya belonged to a very rare breed: he was a front-rank mathematician who maintained an extremely deep interest in mathematics education all through his life and contributed significantly to that field. Over a period of several decades he returned over and over again to the question of how the culture of problem solving could be nurtured among students, and how mathematics could be experienced ‘live’. He wrote many books now regarded as masterpieces: Problems and Theorems in Analysis (with Gábor Szegö), How to Solve It, Mathematical Discovery , among others. This article is a tribute to Pólya and a celebration of his work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Suggested Reading

http://www-history.mcs.st-and.ac.uk/Biographies/Polya.html

T Gowers, The Two Cultures of Mathematics , https://www.dpmms.cam.ac.uk/~wtg102cultures.pdf

http://en.wikipedia.org/wiki/George_Polya

T Passmore, Polya’s legacy: fully forgotten or getting a new perspective in theory and practice , http://eprints.usq.edu.au/3625/1/Passmore.pdf

G Pólya, Mathematics and Plausible Reasoning , Princetron University Press, Vols 1&2, 1954.

G Pólya, Mathematical Discovery , Vols 1&2, 1965.

G Pólya, How To Solve It , Princeton University Press, 1973.

Google Scholar  

G Pólya, Teaching us a Lesson (MAA), http://vimeo.com/48768091 (video recording of an actual lecture by Polya).

http://www.math.utah.edu/~pa/math/polya.html

Geoffrey Howson, Review of Mathematical Discovery, The Mathematical Gazette , Vol. 66, No. 436, pp.162–163, June 1982.

Article   Google Scholar  

Download references

Author information

Authors and affiliations.

Sahyadri School, Tiwai Hill, Rajgurunagar, Pune, 410 513, India

Shailesh A. Shirali

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Shailesh A. Shirali .

Additional information

Shailesh Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP). He has been in the field of mathematics education for three decades, and has been closely involved with the Math Olympiad movement in India. He is the author of many mathematics books addressed to high school students, and serves as an editor for Resonance and for At Right Angles . He is engaged in many outreach projects in teacher education.

Rights and permissions

Reprints and permissions

About this article

Shirali, S.A. George Pólya & problem solving ... An appreciation. Reson 19 , 310–322 (2014). https://doi.org/10.1007/s12045-014-0037-7

Download citation

Published : 06 May 2014

Issue Date : April 2014

DOI : https://doi.org/10.1007/s12045-014-0037-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

  • problem solving
  • Basel problem
  • Find a journal
  • Publish with us
  • Track your research

how to solve a problem george polya

  • Science & Math
  • Mathematics

Amazon prime logo

Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery

Amazon Prime includes:

Fast, FREE Delivery is available to Prime members. To join, select "Try Amazon Prime and start saving today with Fast, FREE Delivery" below the Add to Cart button.

  • Cardmembers earn 5% Back at Amazon.com with a Prime Credit Card.
  • Unlimited Free Two-Day Delivery
  • Streaming of thousands of movies and TV shows with limited ads on Prime Video.
  • A Kindle book to borrow for free each month - with no due dates
  • Listen to over 2 million songs and hundreds of playlists
  • Unlimited photo storage with anywhere access

Important:  Your credit card will NOT be charged when you start your free trial or if you cancel during the trial period. If you're happy with Amazon Prime, do nothing. At the end of the free trial, your membership will automatically upgrade to a monthly membership.

Buy new: $69.98 $69.98 FREE delivery: Thursday, Feb 22 Ships from: Amazon Sold by: GLOBALIXIR

  • Free returns are available for the shipping address you chose. You can return the item for any reason in new and unused condition: no shipping charges
  • Learn more about free returns.
  • Go to your orders and start the return
  • Select the return method

Buy used: $22.97

Fulfillment by Amazon (FBA) is a service we offer sellers that lets them store their products in Amazon's fulfillment centers, and we directly pack, ship, and provide customer service for these products. Something we hope you'll especially enjoy: FBA items qualify for FREE Shipping and Amazon Prime.

If you're a seller, Fulfillment by Amazon can help you grow your business. Learn more about the program.

Other Sellers on Amazon

Kindle app logo image

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

QR code to download the Kindle App

Image Unavailable

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library, 34)

  • To view this video download Flash Player

how to solve a problem george polya

Follow the author

Georg Polya

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library, 34)

Purchase options and add-ons.

  • Book Description
  • Editorial Reviews

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.

About the Author

  • ISBN-10 069111966X
  • ISBN-13 978-0691119663
  • Publisher Princeton University Press
  • Publication date September 25, 2015
  • Part of series Princeton Science Library
  • Language English
  • Dimensions 5.25 x 0.75 x 8 inches
  • Print length 288 pages
  • See all details

how to solve a problem george polya

Frequently bought together

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library, 34)

Similar items that may ship from close to you

All the Math You Missed: (But Need to Know for Graduate School)

Product details

  • Publisher ‏ : ‎ Princeton University Press (September 25, 2015)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 288 pages
  • ISBN-10 ‏ : ‎ 069111966X
  • ISBN-13 ‏ : ‎ 978-0691119663
  • Item Weight ‏ : ‎ 10.2 ounces
  • Dimensions ‏ : ‎ 5.25 x 0.75 x 8 inches
  • #66 in Mathematical Logic
  • #125 in Mathematics History
  • #4,008 in Unknown

Important information

To report an issue with this product or seller, click here .

About the author

how to solve a problem george polya

Georg Polya

Discover more of the author’s books, see similar authors, read author blogs and more

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

  • Sort reviews by Top reviews Most recent Top reviews

Top reviews from the United States

There was a problem filtering reviews right now. please try again later..

how to solve a problem george polya

Top reviews from other countries

how to solve a problem george polya

  • Amazon Newsletter
  • About Amazon
  • Accessibility
  • Sustainability
  • Press Center
  • Investor Relations
  • Amazon Devices
  • Amazon Science
  • Start Selling with Amazon
  • Sell apps on Amazon
  • Supply to Amazon
  • Protect & Build Your Brand
  • Become an Affiliate
  • Become a Delivery Driver
  • Start a Package Delivery Business
  • Advertise Your Products
  • Self-Publish with Us
  • Host an Amazon Hub
  • › See More Ways to Make Money
  • Amazon Visa
  • Amazon Store Card
  • Amazon Secured Card
  • Amazon Business Card
  • Shop with Points
  • Credit Card Marketplace
  • Reload Your Balance
  • Amazon Currency Converter
  • Your Account
  • Your Orders
  • Shipping Rates & Policies
  • Amazon Prime
  • Returns & Replacements
  • Manage Your Content and Devices
  • Recalls and Product Safety Alerts
  • Conditions of Use
  • Privacy Notice
  • Your Ads Privacy Choices

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

2.1: George Polya's Four Step Problem Solving Process

  • Last updated
  • Save as PDF
  • Page ID 132871

Step 1: Understand the Problem

  • Do you understand all the words?
  • Can you restate the problem in your own words?
  • Do you know what is given?
  • Do you know what the goal is?
  • Is there enough information?
  • Is there extraneous information?
  • Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)

AIP Publishing Logo

  • Previous Article
  • Next Article

Students’ Polya problem solving skills on system of linear equations with two variables bases on mathematical ability

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

  • Split-Screen
  • Article contents
  • Figures & tables
  • Supplementary Data
  • Peer Review
  • Open the PDF for in another window
  • Reprints and Permissions
  • Cite Icon Cite
  • Search Site

Pathuddin , Dewi Hamidah , Silvana Panggalo , Zun Azizul Hakim , Muhammad Suwardin , Bakri , Evie Awuy; Students’ Polya problem solving skills on system of linear equations with two variables bases on mathematical ability. AIP Conf. Proc. 16 February 2024; 3046 (1): 020018. https://doi.org/10.1063/5.0194577

Download citation file:

  • Ris (Zotero)
  • Reference Manager

This study seeks students’ abilities to solve Polya problems involving systems of linear equations with two variables based on their mathematical prowess. Three junior high school students with high, average, and low math aptitudes were chosen as participants. Research data were collected using a written test on the system of linear equations with two variables problem in the form of word questions and interviews. The time triangulation technique is used to establish the credibility of the research data. The findings demonstrated that students with strong mathematical aptitudes could comprehend problems, plan solutions, and carry out problem-solving strategies. Still, they lacked the knowledge necessary to confirm the accuracy of the outcomes, leading students to doubt the proper solution. Students with moderate math abilities can only understand problems, plan solutions and carry out problem-solving methods. Students won’t believe the correct answer since they don’t know how to double-check the results. Meanwhile, students with low math ability can only understand the problem and fail to make problem-solving plans. Students with weak math skills make problem-solving mistakes due to this incompetence. The research’s findings can be used to create learning models that will motivate students to develop their ability to solve mathematical problems, particularly those involving the system of linear equations with two variables material, by emphasizing the accuracy of information conversion into variables, understanding of the fundamental ideas underlying the use of procedures, and the significance of double-checking the outcomes of system of linear equations with two variables problem-solving.

Citing articles via

Publish with us - request a quote.

how to solve a problem george polya

Sign up for alerts

  • Online ISSN 1551-7616
  • Print ISSN 0094-243X
  • For Researchers
  • For Librarians
  • For Advertisers
  • Our Publishing Partners  
  • Physics Today
  • Conference Proceedings
  • Special Topics

pubs.aip.org

  • Privacy Policy
  • Terms of Use

Connect with AIP Publishing

This feature is available to subscribers only.

Sign In or Create an Account

  How to Solve It by George Polya

George Polya(1887-1985) was a famous mathematician, who got importance results in probability, analysis, number theory, geometry, combinatorics and mathematical physics. His book How to Solve It was probably the most significant contribution to heuristic since Descartes' Discourse on Method. And its value is not limited to mathematic. For Unix sysadmins this is a perfect books about principles of troubleshooting as a problem solving activity

In this book he proposed structuring the problem-solving process into four stages (see How To Solve It Step-by-step Plan ).

  • Understanding the Problem
  • Devising a Plan
  • Carrying out the Plan
  • Looking Back

For each stage George Polya supplied a series of questions that help to solve the problem. Some of them are:

  • What is the unknown?
  • What are the data?
  • Could you restate the problem?
  • Is there a related problem that has been solved before?

For complete list questions in a form of step-by-step plan see How To Solve It Step-by-step Plan

The examples in the book are drawn mostly from elementary math, but the method is quite general and applies to nearly every problem one might encounter. It is especially valuable for solving programming problems.  I read it when I was 15 years old and this was the only mathematical book that I really liked. I thought that it  contains amazing insights into problem solving skills.  That perception was wrong as for my mathematical successes but later my intensive study of the book paid off somewhat in programming.  Or at least I tend to think so.

BTW  it was George Polya, who said "There are many questions which fools can ask that wise men cannot answer." (See H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988).

IMHO the book is much more valuable to programmers than all modern "pattern" stuff (although the latter might be useful too, but only in great moderation ;-).  The only review of the book in programming literature that I found was a review in DDJ.   BTW Microsoft used to give this book to all of its new programmers. Probably not any more ;-).

See also: Synthesis of Research on Problem Solving .

27-190 Discrete Structures in Computer Science Techniques of Proof and Problem Solving

Famous quotations

For every problem you can't solve there exists an easier problem that you can: find it.

"Geometry is the science of correct reasoning on incorrect figures."

"My method to overcome a difficulty is to go round it."

Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.

- George Pólya How to Solve it: A New Aspect of Mathematical Method (1957), 223.

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

D. J. Albers and G. L. Alexanderson, Mathematical People, Boston: Birkhäuser, 1985.

Mathematics consists of proving the most obvious thing in the least obvious way.

In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

The American Mathematical Monthly, v. 100, no. 3.

The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation:

"In order to solve this differential equation you look at it till a solution occurs to you."

"This principle is so perfectly general that no particular application of it is possible."

"What is the difference between method and device? A method is a device which you used twice."

How to Solve It. Princeton: Princeton University Press. 1945.

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

How to Solve It. Princeton: Princeton University Press. 1945

NEWS CONTENTS

  • 190317 : Problem Solving Island Polyas How To Solve It ( Problem Solving Island Polya's How To Solve It , )
  • 190317 : Review of George Polya's How to Solve it? ( Review of George Polya's How to Solve it? , )

Old News ;-)

Problem solving island polya's how to solve it.

In How To Solve It , G. Polya describes four steps for solving problems and outlines them at the very beginning of the book for easy reference. The steps outline a series of general questions that the problem solving student can use to successfully write resolutions. Without the questions, common sense goes through the same process; the questions simply allow students to see the process on paper. Polya designed the questions to be general enough that students could apply them to almost any problem. The four steps are: understanding the problem, devising a plan, carrying out the plan, and looking back. This method is very similar to the method in Thinking Mathematically by John Mason, except Polya separates devising a plan, and carrying out the plan. This may seem silly at first, but Polya argues that it does make a difference. By first devising a plan, students can eliminate mistakes they might make by rushing into the actual execution of the plan. When they plan it out first and then do the math, it is possible to check their work as they go along. In the first step, students should be able to state the unknown, or the thing they want to find to answer the question, the data the question gives them to work with, and the condition, or limiting circumstances they must work around. If they can identify all of these, and explain the question to other people, then they have a good understanding of what the problem is asking. Polya suggests that students draw a picture if possible, or introduce some kind of notation to visualize the question. To devise a plan, students can start by trying to think of a related problem they have solved before to help them. If the student can think of a problem they have solved before that had a similar unknown, it could also be helpful. Students can also try to restate the problem in an easier or different way, and try to solve that. By looking at these related problems, students may be able to use the same method, or other part of the plan used. After students have decided which calculations, computations, or constructions that they need, and have made sure that all data and conditions were used, they can try out their plan. To carry out the plan, they must do all the calculations, and check them as they go along. Then they should ask themselves, "Can I see it is right?" and then, "Can I prove it is right?" When students look back on the problem and the plan they carried out, they can increase their understanding of the solution. It is always good to recheck the result and argument used, and to make sure that it is possible to check them. Then students should ask, "Can I get the result in a different way?" and "Can I use this for another problem?" The last chapter of the book is a very helpful encyclopedia of the terms used in the explanation of the first chapter. By Beth Nuckolls . A few links: Another review of How to Solve It . A review of another book by Polya

Review of George Polya's How to Solve it? McGill Mathematics Magazine

A recurring theme throughout the book is that if you can not solve a problem, then you should find an easier but similar one. 'Do you know a related problem?' Polya would ask. For example, suppose the student has just learned the Pythagorean theorem, and is now asked to find the length of the spatial diagonal of a parallelepiped. A small amount of ingenuity is required to make the jump from the plane to the space, and the student is naturally stumped. 'Do you know a problem with a similar unknown?' the teacher asks. The student gets a brilliant insight – a previously solved problem. 'Good! Here's a problem related to yours and solved before. Can you use it?' the teacher presses on. Eventually the solution is found, and all is well. Of course, the point of that passage is not to introduce to the readers the Pythagorean theorem in higher dimensions, but to show the readers the process with which one can use to find an 'auxiliary problem' that has been solved and use it to solve the harder problem. Polya himself is often accredited the quote, 'For every problem you can't solve there exists an easier problem that you can: find it. Unfortunately, the effectiveness of the book is debatable. As Feynman said, 'You can't learn to solve problems by reading about it.' There is no other way of gaining problem-solving experience save for actually solving problems. Hence, ironically, it is hard to appreciate the book unless you no longer need it.

How To Solve It Step-by-step Plan

  • How To Solve It - G. Polya
  • [PDF] George Polya
  • George Polya, How to Solve It

Recommended Links

  • How to Solve It - Wikipedia, the free encyclopedia
  • Pólya, George (1945). How to Solve It . Princeton University Press. ISBN 0-691-08097-6 .
  • Quotations by Polya
  • Diagrammatic Reasoning site
  • Minsky, Marvin , Steps Toward Artificial Intelligence , http://web.media.mit.edu/~minsky/papers/steps.html .
  • Schoenfeld, Alan H.; D. Grouws (Ed.) (1992). " Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics ". Handbook for Research on Mathematics Teaching and Learning (New York: MacMillan): pp. 334–370. http://gse.berkeley.edu/faculty/ahschoenfeld/Schoenfeld_MathThinking.pdf . .
  • More information on Pólya can be found here.
  • An uneven summary of the whole book.
Groupthink : Two Party System as Polyarchy : Corruption of Regulators : Bureaucracies : Understanding Micromanagers and Control Freaks : Toxic Managers :   Harvard Mafia : Diplomatic Communication : Surviving a Bad Performance Review : Insufficient Retirement Funds as Immanent Problem of Neoliberal Regime : PseudoScience : Who Rules America : Neoliberalism  : The Iron Law of Oligarchy : Libertarian Philosophy
War and Peace : Skeptical Finance : John Kenneth Galbraith : Talleyrand : Oscar Wilde : Otto Von Bismarck : Keynes : George Carlin : Skeptics : Propaganda   : SE quotes : Language Design and Programming Quotes : Random IT-related quotes :  Somerset Maugham : Marcus Aurelius : Kurt Vonnegut : Eric Hoffer : Winston Churchill : Napoleon Bonaparte : Ambrose Bierce :  Bernard Shaw : Mark Twain Quotes
Vol 25, No.12 (December, 2013) Rational Fools vs. Efficient Crooks The efficient markets hypothesis : Political Skeptic Bulletin, 2013 : Unemployment Bulletin, 2010 :  Vol 23, No.10 (October, 2011) An observation about corporate security departments : Slightly Skeptical Euromaydan Chronicles, June 2014 : Greenspan legacy bulletin, 2008 : Vol 25, No.10 (October, 2013) Cryptolocker Trojan (Win32/Crilock.A) : Vol 25, No.08 (August, 2013) Cloud providers as intelligence collection hubs : Financial Humor Bulletin, 2010 : Inequality Bulletin, 2009 : Financial Humor Bulletin, 2008 : Copyleft Problems Bulletin, 2004 : Financial Humor Bulletin, 2011 : Energy Bulletin, 2010 : Malware Protection Bulletin, 2010 : Vol 26, No.1 (January, 2013) Object-Oriented Cult : Political Skeptic Bulletin, 2011 : Vol 23, No.11 (November, 2011) Softpanorama classification of sysadmin horror stories : Vol 25, No.05 (May, 2013) Corporate bullshit as a communication method   : Vol 25, No.06 (June, 2013) A Note on the Relationship of Brooks Law and Conway Law
Fifty glorious years (1950-2000): the triumph of the US computer engineering : Donald Knuth : TAoCP and its Influence of Computer Science : Richard Stallman : Linus Torvalds   : Larry Wall  : John K. Ousterhout : CTSS : Multix OS Unix History : Unix shell history : VI editor : History of pipes concept : Solaris : MS DOS :  Programming Languages History : PL/1 : Simula 67 : C : History of GCC development :  Scripting Languages : Perl history   : OS History : Mail : DNS : SSH : CPU Instruction Sets : SPARC systems 1987-2006 : Norton Commander : Norton Utilities : Norton Ghost : Frontpage history : Malware Defense History : GNU Screen : OSS early history

Classic books:

The Peter Principle : Parkinson Law : 1984 : The Mythical Man-Month :  How to Solve It by George Polya : The Art of Computer Programming : The Elements of Programming Style : The Unix Hater’s Handbook : The Jargon file : The True Believer : Programming Pearls : The Good Soldier Svejk : The Power Elite

Most popular humor pages:

Manifest of the Softpanorama IT Slacker Society : Ten Commandments of the IT Slackers Society : Computer Humor Collection : BSD Logo Story : The Cuckoo's Egg : IT Slang : C++ Humor : ARE YOU A BBS ADDICT? : The Perl Purity Test : Object oriented programmers of all nations : Financial Humor : Financial Humor Bulletin, 2008 : Financial Humor Bulletin, 2010 : The Most Comprehensive Collection of Editor-related Humor : Programming Language Humor : Goldman Sachs related humor : Greenspan humor : C Humor : Scripting Humor : Real Programmers Humor : Web Humor : GPL-related Humor : OFM Humor : Politically Incorrect Humor : IDS Humor : "Linux Sucks" Humor : Russian Musical Humor : Best Russian Programmer Humor : Microsoft plans to buy Catholic Church : Richard Stallman Related Humor : Admin Humor : Perl-related Humor : Linus Torvalds Related humor : PseudoScience Related Humor : Networking Humor : Shell Humor : Financial Humor Bulletin, 2011 : Financial Humor Bulletin, 2012 : Financial Humor Bulletin, 2013 : Java Humor : Software Engineering Humor : Sun Solaris Related Humor : Education Humor : IBM Humor : Assembler-related Humor : VIM Humor : Computer Viruses Humor : Bright tomorrow is rescheduled to a day after tomorrow : Classic Computer Humor

The Last but not Least Technology is dominated by two types of people: those who understand what they do not manage and those who manage what they do not understand ~Archibald Putt. Ph.D

Copyright © 1996-2021 by Softpanorama Society . www.softpanorama.org was initially created as a service to the (now defunct) UN Sustainable Development Networking Programme ( SDNP ) without any remuneration. This document is an industrial compilation designed and created exclusively for educational use and is distributed under the Softpanorama Content License . Original materials copyright belong to respective owners. Quotes are made for educational purposes only in compliance with the fair use doctrine.

FAIR USE NOTICE This site contains copyrighted material the use of which has not always been specifically authorized by the copyright owner. We are making such material available to advance understanding of computer science, IT technology, economic, scientific, and social issues. We believe this constitutes a 'fair use' of any such copyrighted material as provided by section 107 of the US Copyright Law according to which such material can be distributed without profit exclusively for research and educational purposes.

This is a Spartan WHYFF (We Help You For Free) site written by people for whom English is not a native language. Grammar and spelling errors should be expected. The site contain some broken links as it develops like a living tree...

Disclaimer:

The statements, views and opinions presented on this web page are those of the author (or referenced source) and are not endorsed by, nor do they necessarily reflect, the opinions of the Softpanorama society. We do not warrant the correctness of the information provided or its fitness for any purpose. The site uses AdSense so you need to be aware of Google privacy policy. You you do not want to be tracked by Google please disable Javascript for this site. This site is perfectly usable without Javascript.

Last modified: March 12, 2019

How to Solve It

  • Published: 21 April 2022
  • ISBN: 9780140124996
  • Imprint: Penguin Press
  • Format: Paperback
  • RRP: $35.00

How to Solve It

A New Aspect of Mathematical Method

  • George Polya

how to solve a problem george polya

The defintive guide to mathemathical problem solving from one of the great teachers of the twentieth century

How to Solve It offers something unique: a tried and tested set of strategies for overcoming any maths dilemma. Based on decades of analysis of the methods and rules of problem solving and invention, acclaimed mathematician George Pólya's approach brilliantly demonstrates how to create useful analogies, tackle problems from unusual angles and how to squeeze a little more information from the data. While maths can often seem nothing more than a process of dry deduction, Pólya wonderfully conveys the challenge and excitement at the heart of any problem, and lays out the blueprint for unpicking the riddles they pose. With more than a million copies sold around the world, How to Solve It has inspired generations of teachers and students to look afresh at maths - and continues to be the definitive guide to mathematical problem solving.

Praise for How to Solve It

The outcome of careful and informed deliberation by one of the great teachers among the ranks of research mathematicians Ian Stewart, author of Do Dice Play God?
Walk inside Polya's mind as he builds up maxims on how to comprehend a problem, how to build up a strategy, and then how to test it . . . a superb book on how to think fresh thoughts Guardian

Related titles

Our top books, exclusive content and competitions. straight to your inbox..

Sign up to our newsletter using your email.

By clicking subscribe, I acknowledge that I have read and agree to Penguin Books Australia’s Terms of Use and Privacy Policy .

Thank you! Please check your inbox and confirm your email address to finish signing up.

George Pólya

1 george pólya.

American mathematician, Born: György Pólya in Budapest, Hungary in 1887, ( d. 1985 in Palo Alto, USA)

“ His first job was to tutor the young son, Gregor, of a Hungarian baron. Gregor struggled due to his lack of problem solving skills. ” Thus, according to Long ( [ 1 ] ), Polya insisted that the skill of “ solving problems was not an inborn quality but, something that could be taught ”.

In 1940, George Polya and his wife, Stella, (the only daughter of Swiss Dr. Weber, in Zurich) moved to the United States because of their justified fear of Nazism in Germany ( [ 1 ] ).

Understand the Problem

Devise a Plan on how to approach the Problem; such a plan may include one or several of the following:

Make a first guess to begin with, and then verify the answer

Solve a simpler problem

Consider special cases that are much easier to solve

Look for a pattern

Draw a picture

Use a model

Use direct reasoning but double-check your results

Eliminate possibilities

Carry out the Plan, as modified by partial solutions

If plan doesn’t work, make an improved plan but do not give up

Last-but-not-least, look back and examine critically your solution(s):

Does the solution make sense? Does it check out in particular cases?

Make sure there are no gaps and no steps missing.

He published also a two-volume book, “ Mathematics and Plausible Reasoning ” in 1954, and Mathematical Discovery in 1962.

  • 1 Long, C. T., & DeTemple, D. W., Mathematical reasoning for elementary teachers . (1996). Reading MA: Addison-Wesley
  • 2 Reimer, L., & Reimer, W. Mathematicians are people too . (Volume 2). (1995) Dale Seymour Publications
  • 3 Polya, G. How to solve it . (1957) Garden City, NY: Doubleday and Co., Inc.
  • 4 A. Motter,, http://www.math.wichita.edu/history/men/polya.html “A Biography of George Polya”

Cart

  • SUGGESTED TOPICS
  • The Magazine
  • Newsletters
  • Managing Yourself
  • Managing Teams
  • Work-life Balance
  • The Big Idea
  • Data & Visuals
  • Reading Lists
  • Case Selections
  • HBR Learning
  • Topic Feeds
  • Account Settings
  • Email Preferences

Find the AI Approach That Fits the Problem You’re Trying to Solve

  • George Westerman,
  • Sam Ransbotham,
  • Chiara Farronato

how to solve a problem george polya

Five questions to help leaders discover the right analytics tool for the job.

AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better decisions, regardless of how fast technology changes. You can work with your technical experts to use the right tool for the right job. Then each solution today becomes a foundation to build further innovations tomorrow. But without the right questions, you’ll be starting your journey in the wrong place.

Leaders everywhere are rightly asking about how Generative AI can benefit their businesses. However, as impressive as generative AI is, it’s only one of many advanced data science and analytics techniques. While the world is focusing on generative AI, a better approach is to understand how to use the range of available analytics tools to address your company’s needs. Which analytics tool fits the problem you’re trying to solve? And how do you avoid choosing the wrong one? You don’t need to know deep details about each analytics tool at your disposal, but you do need to know enough to envision what’s possible and to ask technical experts the right questions.

  • George Westerman is a Senior Lecturer in MIT Sloan School of Management and founder of the Global Opportunity Forum  in MIT’s Office of Open Learning.
  • SR Sam Ransbotham is a Professor of Business Analytics at the Boston College Carroll School of Management. He co-hosts the “Me, Myself, and AI” podcast.
  • Chiara Farronato is the Glenn and Mary Jane Creamer Associate Professor of Business Administration at Harvard Business School and co-principal investigator at the Platform Lab at Harvard’s Digital Design Institute (D^3). She is also a fellow at the National Bureau of Economic Research (NBER) and the Center for Economic Policy Research (CEPR).

Partner Center

IMAGES

  1. How to Solve It by George Polya

    how to solve a problem george polya

  2. how to solve a problem george polya

    how to solve a problem george polya

  3. how to solve a problem george polya

    how to solve a problem george polya

  4. polya's 4 method of problem solving

    how to solve a problem george polya

  5. george polya 4 steps to problem solving

    how to solve a problem george polya

  6. Polya's Four Steps in Problem Solving

    how to solve a problem george polya

VIDEO

  1. WHAT IS PUZZLES

  2. Polya's Problem Solving

  3. CHP 3 Lesson 2, Polya's 4 steps in problem solving

  4. CTET MASTER:GEORGE POLYA/जॉर्ज पोल्या:#PROBLEM SOLVING MODEL समस्या समाधान तकनीक।CTET EXAM 2022-23

  5. POLYAS PROBLEM SOLVING STRATEGY

  6. 3 Utilities Problem

COMMENTS

  1. PDF Polya's Problem Solving Techniques

    Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will nd choosing a strategy increasingly easy. A partial list of strategies is included:

  2. How to Solve It

    How to Solve It (1945) is a small volume by mathematician George Pólya, describing methods of problem solving. [1] This book has remained in print continually since 1945. Four principles [ edit] How to Solve It suggests the following steps when solving a mathematical problem : First, you have to understand the problem. [2]

  3. 2.3.1: George Polya's Four Step Problem Solving Process

    Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1. Guess and test. 11. Solve an equivalent problem. 2.

  4. How To Solve It : George Polya

    How To Solve It Bookreader Item Preview ... How To Solve It by George Polya. Publication date 1973 Topics Math Collection opensource Language English. Math Strategy. Addeddate 2023-07-26 17:59:45 Identifier polya-how-to-solve-it Identifier-ark ark:/13960/s22kp01bvm9 Ocr tesseract 5.3.0-3-g9920 Ocr_detected_lang en Ocr_detected_lang_conf

  5. Polya's Problem Solving Process

    1) Understand the Problem-Make sure you understand what the question is asking and what information will be used to solve the problem. 2) Devise a Plan-Figure out what method you will use to...

  6. PDF Polya's Four Phases of Problem Solving

    1. Understanding the Problem. You have to understand the problem. What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation.

  7. How to Solve It

    Mathematics How to Solve It : A New Aspect of Mathematical Method G. Polya Foreword by The bestselling book that has helped millions of readers solve any problem Collections: Princeton Science Library Paperback Price: $19.95/£16.99 ISBN: 9780691164076 Published: Oct 27, 2014 Copyright: 1945 Pages: 288 Size: 5.5 x 8.5 in. ebook Price:

  8. Tool for the Field: Polya's Problem-Solving Method

    Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our ...

  9. Intermediate Algebra Tutorial 8

    Learning Objectives After completing this tutorial, you should be able to: Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. Introduction

  10. George Pólya (1887

    If you can't solve a problem, then there is an easier problem you can solve: find it. Pólya published further books on the art of solving mathematical problems. For example Mathematics and plausible reasoning (1954), and Mathematical discovery which was published in two volumes (1962, 1965).

  11. Problem Solving Strategies

    George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). ... Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing ...

  12. Polya's Problem Solving Process

    This video walks you through using Polya's Problem Solving Process to solve a word problem.

  13. George Pólya & problem solving ... An appreciation

    George Pólya belonged to a very rare breed: he was a front-rank mathematician who maintained an extremely deep interest in mathematics education all through his life and contributed significantly to that field. Over a period of several decades he returned over and over again to the question of how the culture of problem solving could be nurtured among students, and how mathematics could be ...

  14. How to Solve It: A New Aspect of Mathematical Method (Princeton Science

    Amazon.com: How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library, 34): 9780691119663: Polya, G., Conway, John H.: Books Books › Science & Math › Mathematics Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery eTextbook $9.99

  15. (PDF) George Pólya & problem solving ... An appreciation

    ... The research steps are as follows. Firstly, researchers will give 27 students multiplication questions then they will check students answer based on the Polya theory and analyze the reason...

  16. 2.1: George Polya's Four Step Problem Solving Process

    2.1: George Polya's Four Step Problem Solving Process Expand/collapse global location 2.1: George Polya's Four Step Problem Solving Process ... Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1. Guess ...

  17. G. Polya, How to Solve It.

    Summary taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957, ISBN -691-08097-6. UNDERSTANDING THE PROBLEM First. You have to understand the ... If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem?

  18. Polya explains the problem solving technique

    I dont have any copyright over this video. Just found this on web and surprised this was not on youtube. Hope this helps.

  19. Solving Any Problem in 4 Steps

    Step 2: Devise a Plan. Once you've understood the problem, the next step is to devise a plan. This involves finding connections between the data and the unknown. Consider what strategies you can ...

  20. George Pólya

    In How to Solve It, Pólya provides general heuristics for solving a gamut of problems, including both mathematical and non-mathematical problems. The book includes advice for teaching students of mathematics and a mini-encyclopedia of heuristic terms. It was translated into several languages and has sold over a million copies.

  21. Students' Polya problem solving skills on system of linear equations

    This study seeks students' abilities to solve Polya problems involving systems of linear equations with two variables based on their mathematical prowess. ... Bakri, Evie Awuy; Students' Polya problem solving skills on system of linear equations with two variables bases on mathematical ability. AIP Conf. Proc. 16 February 2024; 3046 (1 ...

  22. How to Solve It by George Polya

    In this book he proposed structuring the problem-solving process into four stages (see How To Solve It Step-by-step Plan. Understanding the Problem. Devising a Plan. Carrying out the Plan. For each stage George Polya supplied a series of questions that help to solve the problem. Some of them are:

  23. How to Solve It by George Polya

    The defintive guide to mathemathical problem solving from one of the great teachers of the twentieth century. How to Solve It offers something unique: a tried and tested set of strategies for overcoming any maths dilemma.Based on decades of analysis of the methods and rules of problem solving and invention, acclaimed mathematician George Pólya's approach brilliantly demonstrates how to create ...

  24. George Pólya

    He designed a complete strategy for problem solving that can help both the beginner and the advanced mathematician to solve both mathematical and physical problems. ... In 1940, George Polya and his wife, Stella, (the only daughter of Swiss Dr. Weber, in Zurich) moved to the United States because of their justified fear of Nazism in Germany ().

  25. Did Pólya say, "can" or "cannot"?

    However, I did find a section on "If you cannot solve the proposed problem", p. 114. There Polya advises to "try to solve first some related problem". He suggests inventing such a problem. Besides this, he suggests working a related problem (p. 98), Generalization (p. 108), A problem related to yours and solved before (p. 119), Variation of the ...

  26. Find the AI Approach That Fits the Problem You're Trying to Solve

    Summary. AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better ...