Wickelgren - How to Solve Mathematical Problems

This book is primarily a practical guide to how to solve a certain class of problems, specifically, what I call formal problems or just problems (with the adjective formal being understood in later contexts). Formal problems include all mathematical problems of either the to find or the to prove character but do not include problems of defining mathematically interesting axiom systems. A student taking mathematics courses will hardly be aware of the practical significance of this exclusion, since defining interesting axiom systems is a problem not typically encountered except in certain areas of basic research in mathematics. Similarly, the problem of constructing a new mathematical theory in any field of science is not a formal problem, as I use the term, and I will not discuss it in this book. However, any other mathematical problem that comes up in any field of science, engineering, or mathematics is a formal problem in the sense of this book.

Problems such as what you should eat for breakfast, whether you should marry x or y, whether you should drop out of school, or how can you get yourself to spend more time studying are not formal problems. These problems are virtually impossible at the present time to turn into formal problems because we have no good ways of restricting our thinking to a specified set of given information and operations (courses of action we might take), nor do we often even know how to specify precisely what our goals are in solving these problems. Understanding formal problems can undoubtedly make some contributions to your thinking in regard to these poorly specified personal problems, but the scope of the present book does not include such problems. Even if it did, it would be extremely difficult to specify any precise methods for solving them.

However, formal problems include a large class of practical problems that people might encounter in the real world, although they usually encounter them as games or puzzles presented by friends or appearing in magazines. A practical problem such as how to build a bridge across a river is a formal problem if, in solving the problem, one is limited to some specified set of materials (givens), operations, and, of course, the goal of getting the bridge built.

In actuality, you might limit yourself in this way for a while and, if no solution emerged, decide to consider the use of some additional materials, if possible. Expanding the set of given materials (by means other than the use of acceptable operations) is not a part of formal problem solving, but often the situation presents certain givens in sufficiently disguised or implicit form that recognition of all the givens is an important part of skill in formal problem solving. That skill will be discussed later.

Practical problems or puzzles of the type we will consider differ from problems in mathematics, science, or engineering in that to pose them requires less background information and training. Thus, puzzle problems are especially suitable as examples of problem-solving methods in this book, because they communicate the workings of the methods most easily to the widest range of readers. For this reason, puzzle problems will constitute a large proportion of the examples used in this bookat least prior to the last chapter.

In principle, it might seem that most important problem-solving methods would be unique to each specialized area of mathematics, science, or engineering, but this is probably not the case. There are many extremely general problem-solving methods, though, to be sure, there are also special methods that can be of use in only a limited range of fields.

It may be quite difficult to learn the special methods and knowledge required in a particular field, but at least such methods and knowledge are the specific object of instruction in courses. By contrast, general problem-solving methods are rarely, if ever, taught, though they are quite helpful in solving problems in every field of mathematics, science, and engineering.