How to Solve It
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George Pólya's 1945 book How to Solve It (ISBN 0-691-08097-6) is a small volume describing methods of problem solving. It suggests the following steps when solving a mathematical problem:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
If this technique fails, Polya 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?"
His 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 | can you find a problem analogous to your problem and solve that? | Map |
| 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 |
| 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 [1] |
| 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, mind mapping and brainstorming.
[edit] Influence
Marvin Minsky said in his influential paper "Steps Toward Artificial Intelligence": "And everyone should know the work of George Polya [87] on how to solve problems." [2]
Polya's book has had a large influence on mathematics textbooks. "Most formulations of a problem solving framework in U. S. textbooks attribute some relationship to Polya's problem solving stages (1945)." [3]
