How to Solve It


How to Solve It 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, make a plan.
  3. Carry out the plan.
  4. Look back on 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:
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:
Also suggested:
Polya 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. 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 did not, and with thinking about other problems where this could be useful. 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:
HeuristicInformal DescriptionFormal analogue
AnalogyCan you find a problem analogous to your problem and solve that?Map
GeneralizationCan you find a problem more general than your problem?Generalization
InductionCan you solve your problem by deriving a generalization from some examples?Induction
Variation of the ProblemCan you vary or change your problem to create a new problem whose solution will help you solve your original problem?Search
Auxiliary ProblemCan 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 beforeCan you find a problem related to yours that has already been solved and use that to solve your problem?Pattern recognition
Pattern matching
Reduction
SpecializationCan you find a problem more specialized?Specialization
Decomposing and RecombiningCan you decompose the problem and "recombine its elements in some new manner"?Divide and conquer
Working backwardCan you start with the goal and work backwards to something you already know?Backward chaining
Draw a FigureCan you draw a picture of the problem?Diagrammatic Reasoning
Auxiliary ElementsCan you add some new element to your problem to get closer to a solution?Extension

Influence