How to Solve It
Encyclopedia
How to Solve It is a small volume by mathematician George Pólya
George Pólya
George Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...

describing methods of problem solving
Problem solving
Problem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Consideredthe most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of...

.

## Four principles

How to Solve It suggests the following steps when solving a mathematical problem
Mathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...

:
1. First, you have to understand
Understanding
Understanding is a psychological process related to an abstract or physical object, such as a person, situation, or message whereby one is able to think about it and use concepts to deal adequately with that object....

the problem
Problem
A problem is an obstacle, impediment, difficulty or challenge, or any situation that invites resolution; the resolution of which is recognized as a solution or contribution toward a known purpose or goal...

.
2. After understanding, then make a plan
Plan
A plan is typically any diagram or list of steps with timing and resources, used to achieve an objective. See also strategy. It is commonly understood as a temporal set of intended actions, through which one expects to achieve a goal...

.
3. Carry out the plan.
4. Look back
Review
A review is an evaluation of a publication, a product or a service, such as a movie , video game, musical composition , book ; a piece of hardware like a car, home appliance, or computer; or an event or performance, such as a live music concert, a play, musical theater show or dance show...

on your work. How could it be better
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

?

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

### 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.

## Heuristics

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
Lateral thinking
Lateral thinking is solving problems through an indirect and creative approach, using reasoning that is not immediately obvious and involving ideas that may not be obtainable by using only traditional step-by-step logic...

, mind mapping, brainstorming
Brainstorming
Brainstorming is a group creativity technique by which a group tries to find a solution for a specific problem by gathering a list of ideas spontaneously contributed by its members...

, and creative problem solving
Creative problem solving
Creative problem solving is the mental process of creating a solution to a problem. It is a special form of problem solving in which the solution is independently created rather than learned with assistance.Creative problem solving always involves creativity....

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## Influence

• 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
Marvin Minsky
Marvin Lee Minsky is an American cognitive scientist in the field of artificial intelligence , co-founder of Massachusetts Institute of Technology's AI laboratory, and author of several texts on AI and philosophy.-Biography:...

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 physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

Zhores I. Alfyorov
Zhores Ivanovich Alferov
Zhores Ivanovich Alferov is a Soviet and Russian physicist and academic who contributed significantly to the creation of modern heterostructure physics and electronics. He is an inventor of the heterotransistor and the winner of 2000 Nobel Prize in Physics. He is also a Russian politician and has...

, (Nobel laureate
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

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
TRIZ
TRIZ is "a problem-solving, analysis and forecasting tool derived from the study of patterns of invention in the global patent literature". It was developed by the Soviet inventor and science fiction author Genrich Altshuller and his colleagues, beginning in 1946...

, which in many aspects reproduces or parallels Pólya's work.