Every professional software developer is sure to have one of the great books on algorithms close at hand, such as Knuth's mother-of-all-programming-references Art of Computer Programming or Sedgewick's kinder, gentler (if considerably less encyclopedic) Algorithms, Second Edition. This is as it should be--books are tools too, and knowing where to look up a technique when it is needed is a perfectly honorable alternative to brute-force memorization and application of rules and recipes. Besides, our field, though young, is already vast, and has progressed far beyond the point where it's feasible for the average practitioner to master every one of its many subdivisions.
Unfortunately, the algorithm cook-book approach short-circuits one of programming's most intense emotional rewards and certainly its most profound learning experience: the analysis of an unfamiliar problem, the search for a solution, the realization of the solution in code, and finally the satisfaction of seeing the solution in use by others. Polya, a Stanford professor of mathematics who died in 1985, recognized the enormous significance of finding one's own answers. He set out to teach a problem-solving method, rather than a set of solutions--a sort of systematic approach to invention--that would heighten the interest of his students and serve their needs well in any situation.
How to Solve It has a high name-recognition factor. I daresay almost every serious programmer has at least heard of the book, and most have a general idea of its theme. And I venture to say hardly any--other than those who made their way into the computer field via a major in mathematics--have ever actually held a copy in their hands, let alone read it. I believe this to be true because none of the scores of references to the book I've heard or seen over 20 years gave me any feeling for the book that matched the book's true personality. And what is that personality?
First and foremost, you must remember that Polya was a mathematician, and was not concerned with computers at all. As a matter of fact, when the book was originally written in 1944, general-purpose computers as we know them today didn't even exist. So How to Solve It is written from the point of view of a mathematician teaching other mathematicians how to teach mathematics to students, and the style of writing is an odd admixture--sometimes direct, fluent, and a tad ironic; sometimes stuffy in a typically mathematical sort of way, with carefully numbered lists; sometimes Socratic, with imaginary dialogs between teacher and student; sometimes pedantic; and sometimes exasperatingly quaint. Nevertheless, when all is said and done, the book lives up to its reputation.
Polya's "method" has four main stages or elements, which may be summarized as follows:
Polya opts out, however, when faced with one of the topics that most intrigues me: the phenomenon of subconscious work. Maybe once every year or two I run into a program bug that is, at first, totally baffling. You know the kind of bug I'm talking about--when a program's behavior simply can't be reconciled with its source code, despite the most determined disk checking, execution profiling, and runtime instrumentation. The first few times I ran into such problems, I harbored the usual dark suspicions of hardware errors and even, I must admit, entertained some paranoid (if temporary) fantasies about the integrity of physical reality. But I've now learned the best approach to such bugs is to scrutinize them with all my energy for a couple of days, learn everything I can about the symptoms, and then put the problem completely aside and go on to something else. In a few days, or at most a few weeks, an explanation will suddenly pop into my mind--and the explanation is nearly always right.
In such cases, the solution is welcome, but the nagging question remains: Who found the solution? Polya says, "The fact is, that a problem, after prolonged absence, may return into consciousness essentially clarified, much nearer to its solution than it was when it first dropped out of consciousness. Who clarified it, who brought it nearer to the solution? Obviously, oneself, working at it subconsciously." This is, however, not a true answer but an appeal to a higher (or lower) authority. It's disconcerting to realize there's a more powerful problem-solving engine beneath the surface which is not under our direct control. Could our conscious selves, our sense of time, and our sensitive egos be little more than standing waves on a far deeper current, ephemeral side-effects of a complex but secretive undermind going about its unknown business? Polya remains adamantly practical and refuses to speculate, noting only that:
Only such problems come back improved whose solution we passionately desire, or for which we have worked with great tension; conscious effort and tension seem to be necessary to set the subconscious work going. At any rate, it would be too easy if it were not so; we could solve difficult problems just by sleeping and waiting for a bright idea. Past ages regarded a sudden good idea as an inspiration, a gift of the gods. You must deserve such a gift by work, or at least by a fervent wish.
Copyright © 1992, Dr. Dobb's Journal
Having set forth this list in the very first few pages of his book, Polya proceeds to beat on the list repeatedly from every possible angle, using various examples, analogies, anecdotes, and even startlingly modern-sounding quotations from ancient philosophers. In short, he states and restates his thesis until he gets his points across, just as he encourages the reader to state and restate a problem until it has been massaged into a form more easily understood. His thought experiments are all mathematical, of course, but the mathematics are not complex and can be followed easily by anyone with a dim recollection of algebra and solid geometry; the most sophisticated mathematics in the book involve a simple differential.