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Everyone must have felt at least once in his or her life how wonderful it would be if we could solve a problem at hand preferably without much difficulty or even with some difficulties. Unfortunately the problem solving is an art at this point and there are no universal approaches one can take to solving problems. Basically one must explore possible avenues to a solution one by one until one comes across a right path to a solution. Thus generally speaking, there is guessing and hence an element of luck involved in problem solving. However, in general, as one gains experience in solving problems, one develops one's own techniques and strategies, though they are often intangible. Thus the guessing is not an arbitrary guessing but an educated one. In this chapter we are going to learn a framework for problem solving and get a glimpse of strategies that are often used by experts. They are based on the work of Polya. For further study, his book, and others such as Larson are recommended (but not required).
The following four phases can be identified in the process of solving problems: (1) Understanding the problem
(2) Making a plan of solution
(3) Carrying out the plan
(4) Looking back i.e. verifying
Each of the first two phases is going to be explained below a little more in detail. Phases (3) and (4) should be self-explanatory.
Needless to say that if you do not understand the problem you can never solve it. It is also often true that if you really understand the problem, you can see a solution. Below are some of the things that can help us understand a problem. (1) Extract the principal parts of the problem.
The principal parts are:
For "find" type problems, such as "find the principal and return for a given investment", UNKNOWNS, DATA and CONDITIONS, and for "proof" type problems HYPOTHESIS and CONCLUSION. For examples that illustrate these, see examples below. Be careful about hidden assumptions, data and conditions.
(2) Consult definitions for unfamiliar (often even familiar) terminologies.
(3) Construct one or two simple example to illustrate what the problem says.
Where to start?
Start with the consideration of the principal parts: unknowns, data, and conditions for "find" problems, and hypothesis, and conclusion for "prove" problems.
What can I do?
Once you identify the principal parts and understand them, the next thing you can do is to consider the problem from various angles and seek contacts with previously acquired knowledge. The first thing you should do is to try to find facts that are related to the problem at hand. Relevant facts usually involve words that are the same as or similar to those in the given problem. It is also a good idea to try to recall previously solved similar problems.
What should I look for?
Look for a helpful idea that shows you the way to the end. Even an incomplete idea should be considered. Go along with it to a new situation, and repeat this process.
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