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How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya 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 find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.


  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.

From Yahoo Answers

Question:what is the value of x? on a transversal angle 2 and 4 are vertical to each other* angle 2= 2x+3 and angle 4= 4x-7 it says in the back of the book that its 5 but idk how please help

Answers:It's important to know that vertical angles are congruent. Therefore angles 2 and 4 are =. That means: 2x + 3 = 4x - 7 solve for x: -2x = -10 x = 5 that's it! ;)

Question:9th grade math. Please and thank you. I need help with this, its giving me a hard time. If u have a pair of vertical angles, and one angle equals 5x-20 and the other angle equals 3x-2 , how do u find the measure of each angle and what is the answer? PLEASE and THANK YOU! I appreciate it greatly (:

Answers:the measure of vertical angles are equal so you have an equation: 5x - 20 = 3x - 2 solve for x: 2x = 18 x = 9 plug in the 9 for x: 5(9) - 20 = 45 - 20 = 25 or 3(9) - 2 = 27 - 2 = 25 The measure of each angle is 25 that's it! ;)

Question:http://education.yahoo.com/homework_help/math_help/solutionimages/minigeogt/1/1/1/minigeogt_1_1_1_17_1/f-50-21-oyo-1.gif That's an image of one. Does anyone know what the (6x-16) and the (4x+6) equals together? Like i know what the degree of a straight angle[180] but what is the degree of these?

Answers:The vertical angle theorem tells us that opposite angles are equal. So, 6x - 16 = 4x + 6 so 2x = 22 => x = 11. Once you plug it in, you get that the angle is 50 degrees. Then you can say that 4x + 6 + (8x + 7y) = 180 (since they form a straight line). Since you know that 4x + 6 is 50 degrees you can plug that in and solve for y. Also plug in x = 11 for x. 50 + 88 + 7y = 180 138 + 7y = 180 7y = 42 y = 6 So you get that 8x + 7y = 130. 8x + 7y = 130 6x - 16 = 4x + 6 = 50

Question:if you were to draw a Venn Diagram about Vertical and Adjacent angles how would you draw it? i know what they mean but im not sure how to draw a venn diagram about them =( can someone tell me how it looks like?

Answers:Since adjacent angles can never be vertical angles, the two circles representing then would not overlap.

From Youtube

Find a Missing Vertical Angle :www.mathproblemgenerator.com - How to find a missing vertical angle in a triangle. For more practice and to create math worksheets, visit Davitily Math Problem Generator at www.mathproblemgenerator.com

Linear Pairs & Vertical Angles Diagram Problem :Solve a problem from a diagram given expressions for various angles and utilizing linear pairs and vertical angles.