How to Solve It (Princeton Science Library)

What
is the unknown?

Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice.

Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
These questions are often useful at an early stage when they do not need a final answer but just a provisional answer, a guess. For examples, see sections 8, 18.
It is good to foresee any feature of the result for which we work. When we have some idea of what we can expect, we know better in which direction we should go. Now, an important feature of a problem is the number of solutions of which it admits. Most interesting among problems are those which admit of just one solution; we are inclined to consider problems with a uniquely determined solution as the only “reasonable” problems. Is our problem, in this sense, “reasonable”? If we can answer this question, even by a plausible guess, our interest in the problem increases and we can work better.
Is our problem “reasonable”? This question is useful at an early stage of our work if we can answer it easily. If the answer is difficult to obtain, the trouble we have in obtaining it may outweigh the gain in interest. The same is true of the question “Is it possible to satisfy the condition?” and the allied questions of our list. We should put them because the answer might be easy and plausible, but we should not insist on them when the answer seems to be difficult or obscure.
The corresponding questions for “problems to prove” are: Is it likely that the proposition is true? Or is it more likely that it is false? The way the question is put shows clearly that only a guess, a plausible provisional answer, is expected

Inventor’s paradox. The more ambitious plan may have more chances of success.
This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new, more ambitious problem is easier to handle than the original problem. More questions may be easier to answer than just one question. The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.
The paradox disappears if we look closer at a few examples (GENERALIZATION, 2; INDUCTION AND MATHEMATICAL INDUCTION, 7). The more ambitious plan may have more chances of success provided it is not based on mere pretension but on some vision of the things beyond those immediately present

Examine your guess. Your guess may be right, but it is foolish to accept a vivid guess as a proven truth—as primitive people often do. Your guess may be wrong. But it is also foolish to disregard a vivid guess altogether—as pedantic people sometimes do. Guesses of a certain kind deserve to be examined and taken seriously: those which occur to us after we have attentively considered and really understood a problem in which we are genuinely interested. Such guesses usually contain at least a fragment of the truth although, of course, they very seldom show the whole truth. Yet there is a chance to extract the whole truth if we examine such a guess appropriately

If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.
Could you imagine a more accessible related problem? You should now invent a related problem, not merely remember one; I hope that you have tried already the question: Do you know a related problem?
The remaining questions in that paragraph of the list which starts with the title of the present article have a common aim, the VARIATION OF THE PROBLEM. There are different means to attain this aim as GENERALIZATION, SPECIALIZATION, ANALOGY, and others which are various ways of DECOMPOSING AND RECOMBINING

Do you know a related problem? We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but, if such a problem could exist, it would be insoluble. In fact, when solving a problem, we always profit from previously solved problems, using their result, or their method, or the experience we acquired solving them. And, of course, the problems from which we profit must be in some way related to our present problem. Hence the question: Do you know a related problem?
There is usually no difficulty at all in recalling formerly solved problems which are more or less related to our present one. On the contrary, we may find too many such problems and there may be difficulty in choosing a useful one. We have to look around for closely related problems; we LOOK AT THE UNKNOWN, or we look for a formerly solved problem which is linked to our present one by GENERALIZATION, SPECIALIZATION, or ANALOGY.
The question we discuss here aims at the mobilization of our formerly acquired knowledge (PROGRESS AND ACHIEVEMENT, 1). An essential part of our mathematical knowledge is stored in the form of formerly proved theorems. Hence the question: Do you know a theorem that could be useful? This question may be particularly suitable when our problem is a “problem to prove,” that is, when we have to prove or disprove a proposed theorem

Determination, hope, success. It would be a mistake to think that solving problems is a purely “intellectual affair”; determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments

When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should, first of all, stir up his curiosity, give him some desire to solve the problem. The teacher should also allow some time to the student to make up his mind, to settle down to his task.
Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears. If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point