How to Solve It

Template:Infobox book How to Solve It (1945) is a small volume by mathematician George Pólya, describing methods of problem solving.<ref name="Pólya,1945"> Template:Cite book </ref>

This book has remained in print continually since 1945.

Four principles

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

1. First, you have to understand the problem.<ref>Template:Harvnb pp. 6–8</ref>
2. After understanding, make a plan.<ref name="Pólya 1957">Template:Harvnb pp. 8–12</ref>
3. Carry out the plan.<ref>Template:Harvnb pp. 12–14</ref>
4. Look back on your work.<ref>Template:Harvnb pp. 14–15</ref> How could it be better?

If this technique fails, Pólya advises:<ref>Template:Harvnb p. 114</ref> "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,<ref>Template:Harvnb p. 33</ref> depending on the situation, such as:

• What are you asked to find or show?<ref>Template:Harvnb p. 214</ref>
• 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.<ref name="Pólya 1957"/> 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<ref>Template:Harvnb p. 99</ref>
• Make an orderly list<ref>Template:Harvnb p. 2</ref>
• Eliminate possibilities<ref>Template:Harvnb p. 94</ref>
• Use symmetry<ref>Template:Harvnb p. 199</ref>
• Consider special cases<ref>Template:Harvnb p. 190</ref>
• Use direct reasoning
• Solve an equation<ref>Template:Harvnb p. 172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable.</ref>

Also suggested:

• Look for a pattern<ref>Template:Harvnb p. 108</ref>
• Draw a picture<ref>Template:Harvnb pp. 103–108</ref>
• Solve a simpler problem<ref>Template:Harvnb p. 114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly'</ref>
• Use a model<ref>Template:Harvnb p. 105, pp. 29–32, for example, Pólya discusses the problem of water flowing into a cone as an example of what is required to visualize the problem, using a figure.</ref>
• Work backward<ref>Template:Harvnb p. 105, p. 225</ref>
• Use a formula<ref>Template:Harvnb pp. 141–148. Pólya describes the method of analysis</ref>
• Be creative<ref>Template:Harvnb p. 172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).)</ref>
• Applying these rules to devise a plan takes your own skill and judgement.<ref>Template:Harvnb pp. 148–149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first'</ref>

Pólya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them.

Third principle: Carry out the plan

This step is usually easier than devising the plan.<ref>Template:Harvnb p. 35</ref> 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. <ref name="Pólya 1957" />

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 did not, and with thinking about other problems where this could be useful.<ref>Template:Harvnb p. 36</ref><ref>Template:Harvnb pp. 14–19</ref> 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 Template:Original research inline
Analogy	Can you find a problem analogous to your problem and solve that?	Map
Auxiliary Elements	Can you add some new element to your problem to get closer to a solution?	Extension
Generalization	Can you find a problem more general than your problem?	Generalization
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
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 matching Reduction
Specialization	Can you find a problem more specialized?	Specialization
Decomposing and Recombining	Can you decompose the problem and "recombine its elements in some new manner"?	Divide and conquer
Working backward	Can you start with the goal and work backwards to something you already know?	Backward chaining
Draw a Figure	Can you draw a picture of the problem?	Diagrammatic Reasoning<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>

Influence

• The book 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 paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}.</ref>

• Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.<ref>Template:Cite journal.</ref>
• 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.
• How to Solve it by Computer is a computer science book by R. G. Dromey.<ref>Template:Cite book</ref> It was inspired by Pólya's work.

See also

• Inventor's paradox

References

• Template:Cite book

External links

• More information on Pólya can be found here. Template:Webarchive
• Softpanorama page about the value of the book in programming