Strategies in Problem Solving in Mathematics

Posted January 25, 2019 10:20am

The journey of a thousand miles begins with one step. -Lao Tzu

Like and unlike the proverb above, the solution to a problem begins (and continues, and ends) with simple, logical steps. But as long as one steps in a firm, clear direction, with long strides and sharp vision, one would need far, far less than the millions of steps needed to journey a thousand miles. And mathematics, being abstract, has no physical constraints; one can always restart from scratch, try new avenues of attack, or backtrack at an instant’s notice.

One does not always have these luxuries in other forms of problem-solving (e.g. trying to go home if you are lost). Of course, this does not necessarily make it easy. There are several general strategies and perspectives to solve a problem correctly; (Polya 1957) is a classic reference for many of these. Some of these strategies are discussed below:

What kind of problem is it? There are three main types of problems:

• ‘Show that ...’ or ‘Evaluate ...’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out. These are problems start with given data and the objective is to deduce some statement or find the value of an expression; this type of problem is generally easier than the other two types because there is a clearly visible objective, one that can be deliberately approached.

• ‘Find a ...’ or ‘Find all ...’ questions, which requires one to find something (or everything) that satisfies certain requirements. These problems are more hit-and-miss; generally one has to guess one answer that nearly works, and then tweak it a bit to make it more correct; or alternatively one can alter the requirements that the object-to-find must satisfy, so that they are easier to satisfy.

• ‘Is there a ...’ questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem). The type of problem is important because it determines the basic method of approach. This problems are typically the hardest, because one must first make a decision on whether an object exists or not, and provide a proof on one hand, or a counter-example on the other.

Of course, not all questions fall into these neat categories; but the general format of any question will still indicate the basic strategy to pursue when solving a problem.

What is given in the problem? Usually, a question talks about a number of objects which satisfy some special requirements. To understand the data, one needs to see how the objects and requirements react to each other. This is important in focusing attention on the proper techniques and notation to handle the problem.

What do we want? One may need to find an object, prove a statement, determine the existence of an object with special properties, or whatever. Like the flip side of this strategy, ‘understand the data’, knowing the objective helps focus attention on the best weapons to use. Knowing the objective also helps in creating tactical goals which we know will bring us closer to solving the question.

Putting everything down on paper helps in three ways:

- you have an easy reference later on;
- the paper is a good thing to stare at when you are stuck;
- the physical act of writing down of what you know can trigger new inspirations and connections. Be careful, though, of writing superfluous material, and do not overload your paper;one compromise is to highlight those facts which you think will be most useful, and put more questionable, redundant, or crazy ideas in another part of your scratch paper.

There are many ways to vary a problem into one which may be easier to deal with:

- Consider a special case of the problem, such as extreme or degenerate cases.
- Solve a simplified version of the problem.
- Formulate a conjecture which would imply the problem, and try to prove that first.
- Derive some consequence of the problem, and try to prove that first.
- Reformulate the problem (e.g. take the contrapositive, prove by contradiction, or try some substitution).
- Examine solutions of similar problems.
- Generalize the problem. This is useful when you cannot even get started on a problem, because solving for a simpler related problem sometimes reveals the way to go on the main problem. Similarly, considering extreme cases and solving the problem with additional assumptions can also shed light on the general solution. But be warned that special cases are, by their nature, special, and some elegant technique could conceivably apply to them and yet have absolutely no utility in solving the general case. This tends to happen when the special case is too special. Start with modest assumptions first, because then you are sticking as closely as possible to the spirit of the problem.

In this more aggressive type of strategy, we perform major modifications to a problem such as removing data, swapping the data with the objective, or negating the objective (e.g. trying to disprove a statement rather than prove it). Basically, we try to push the problem until it breaks, and then try to identify where the breakdown occurred; this identifies what the key components of the data are, as well as where the main difficulty will lie. These exercises can also help in getting an instinctive feel of what strategies are likely to work, and which ones are likely to fail.

Data is there to be used, so one should pick up the data and play with it. Can it produce more meaningful data? Also, proving small results could be beneficial later on, when trying to prove the main result or to find the answer. However small the result, do not forget it—it could have bearing later on. Besides, it gives you something to do if you are stuck.

Now we have set up notation and have a few equations, we should seriously look at attaining our tactical goals that we have established. In simple problems, there are usually standard ways of doing this. (For example, algebraic simplification is usually discussed thoroughly in high-school level textbooks.) Generally, this part is the longest and most difficult part of the problem: however, once can avoid getting lost if one remembers the relevant theorems, the data and how they can be used, and most importantly the objective.

It is also a good idea to not apply any given technique or method blindly, but to think ahead and see where one could hope such a technique to take one; this can allow one to save enormous amounts of time by eliminating unprofitable directions of inquiry before sinking lots of effort into them, and conversely to give the most promising directions priority.

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