How to Solve It

Encyclopedia

**is a small volume by mathematician George Pólya**

*How to Solve It*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...

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## 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...

:

- First, you have to
*understand*.UnderstandingUnderstanding 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 problemProblemA 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... - After understanding, then
*make a plan*.PlanA 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... -
*Carry out the plan*. -
*Look back*on your work. How could it be betterReviewA 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...Scientific methodScientific 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...

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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.## Heuristics

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

Analogy Analogy Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process... |
Can you find a problem analogous to your problem and solve that? | Map Map (mathematics) In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a... |

Generalization Generalization A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it... |
Can you find a problem more general than your problem? | Generalization Generalization A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it... |

Induction | Can you solve your problem by deriving a generalization from some examples? | Induction |

Variation of the Problem | Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem? | Search Search algorithm In computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots... |

Auxiliary Problem | Can you find a subproblem or side problem whose solution will help you solve your problem? | Subgoal |

Here is a problem related to yours and solved before | Can you find a problem related to yours that has already been solved and use that to solve your problem? | Pattern recognition Pattern recognition In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes... Pattern matching Pattern matching In computer science, pattern matching is the act of checking some sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact. The patterns generally have the form of either sequences or tree structures... Reduction Reduction (complexity) In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. Depending on the transformation used this can be used to define complexity classes on a set of problems.... |

Specialization | Can you find a problem more specialized? | Specialization |

Decomposing Decomposition (computer science) Decomposition in computer science, also known as factoring, refers to the process by which a complex problem or system is broken down into parts that are easier to conceive, understand, program, and maintain.- Overview :... and Recombining |
Can you decompose the problem and "recombine its elements in some new manner"? | Divide and conquer Divide and conquer algorithm In computer science, divide and conquer is an important algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same type, until these become simple enough to be solved directly... |

Working backward | Can you start with the goal and work backwards to something you already know? | Backward chaining Backward chaining Backward chaining is an inference method that can be described as working backward from the goal... |

Draw a Figure | Can you draw a picture of the problem? | Diagrammatic Reasoning Diagrammatic reasoning Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.... |

Auxiliary Elements | Can you add some new element to your problem to get closer to a solution? | Extension |

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 MinskyMarvin MinskyMarvin 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 physicistPhysicistA 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. AlfyorovZhores Ivanovich AlferovZhores 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 laureateNobel Prize in PhysicsThe 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 TRIZTRIZTRIZ 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.