Polya’s Problem Solving Techniques

[https://math.berkeley.edu/~gmelvin/polya.pdf]

In 1945 George Polya published the book How To Solve It which quickly became
his most prized publication. It sold over one million copies and has been translated
into 17 languages. In this book he identifies four basic principles of problem solving.

Polya’s First Principle: Understand the problem

This seems so obvious that it is often not even mentioned, yet students are often
stymied in their efforts to solve problems simply because they don’t understand it
fully, or even in part. Polya taught teachers to ask students questions such as:
	•	Do you understand all the words used in stating the problem?
	•	What are you asked to find or show?
	•	Can you restate the problem in your own words?
	•	Can you think of a picture or diagram that might help you understand the problem?
	•	Is there enough information to enable you to find a solution?

Polya’s Second Principle: Devise a plan

Polya mentions that there are many reasonable ways to solve problems. The skill
at choosing an appropriate strategy is best learned by solving many problems. You
will find choosing a strategy increasingly easy. A partial list of strategies is included:
	•	Guess and check
	•	Look for a pattern
	•	Make an orderly list
	•	Draw a picture
	•	Eliminate possibilities
	•	Solve a simpler problem
	•	Use symmetry
	•	Use a model
	•	Consider special cases
	•	Work backwards
	•	Use direct reasoning
	•	Use a formula
	•	Solve an equation
	•	Be ingenious

Polya’s Third Principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is
care and patience, given that you have the necessary skills. Persist with the plan that
you have chosen. If it continues not to work discard it and choose another. Don’t be
misled, this is how mathematics is done, even by professionals.

Polya’s Fourth Principle: Look back

Polya mentions that much can be gained by taking the time to reflect and look
back at what you have done, what worked, and what didn’t. Doing this will enable
you to predict what strategy to use to solve future problems.

So starting on the next page, here is a summary, in the master’s own words, on strategies for attacking problems in mathematics class. This is taken from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957

1. UNDERSTAND THE PROBLEM
	•	First. You have to understand the problem.
	•	What is the unknown? What are the data? What is the condition?
	•	Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
	•	Draw a figure. Introduce suitable notation.
	•	Separate the various parts of the condition. Can you write them down?

2. DEVISING A PLAN
	•	Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
	•	Have you seen it before? Or have you seen the same problem in a slightly different form?
	•	Do you know a related problem? Do you know a theorem that could be useful?
	•	Look at the unknown! Try to think of a familiar problem having the same or a similar unknown.
	•	Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
	•	Could you restate the problem? Could you restate it still differently? Go back to definitions.
	•	If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
	•	Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

3. CARRYING OUT THE PLAN
	•	Third. Carry out your plan.
	•	Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

4. LOOKING BACK
	•	Fourth. Examine the solution obtained.
	•	Can you check the result? Can you check the argument?
	•	Can you derive the solution differently? Can you see it at a glance?
	•	Can you use the result, or the method, for some other problem?

Polya’s quotes:

	⁃	“If you can’t solve a problem, then there is an easier problem you can’t solve: find it”
	⁃	“If the student is not able to do much, the teacher should leave him at lease some illusion of independent work. In order to do so, the teacher should help the student discreetly, unobtrusively.”
	⁃	“Look at the unknown!”
	⁃	“There are two aims which the teacher may have in view when addressing to his students a question or suggestion of the list: First, to help the student to solve the problem at hand. Second, to develop the student’s ability so that he may solve future problems by himself.”

Main divisions, main questions:

Trying to find the solution, we may repeatedly change out point of view. We have to shift out position again and again. Our concept of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution. 

Four phases: 
	1.	We have to understand the problem - see clearly what’s required
	2.	Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain idea of the solution, to make a plan
	3.	We carry out our plan.
	4.	We look back at the complete solution, we review and discuss it.

Understanding the problem:
It is foolish to answer a question that you do not understand.

	⁃	What is the unknown?
	⁃	What are the data?
	⁃	What is the condition?

If there is a figure connected with a problem he should draw a figure and point out on the unknown and the data. Introduce suitable notation if needed.
	⁃	Is it possible to satisfy the condition?

Devising a plan:
We have a plan when we know, or know at least in outline, which calculations, computation, or constructions we have to perform in order to obtain the unknown. The way from understanding the problem to conceiving a plan may be long and tortuous. In fact, the main achievement in the solution of a problem is to conceive the idea of a plan.

It is hard to have a good idea if we have little knowledge of the subject and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge. Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts; materials alone are not enough for constructing a house but we cannot construct a house without collecting the necessary materials. The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems or formerly proved theorems. Thus, it is often appropriate to start the work with the question:

	⁃	Do you know a related problem?
	⁃	Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
	⁃	Here is a problem related to yours and solved before. Could you use it?
	⁃	Could you restate problem?
	⁃	If you cannot solve the proposed problem try to solve first some related problem.
	⁃	Did you use all the data? Did you use the whole condition?

Carrying out the plan:
To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patients.
	⁃	Can you see clearly that the step is correct?
	⁃	But can you also prove that the step is correct?

Looking back. Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of work. By looking back at the completed solution, by reconsidering and re-examining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve the problems.
	⁃	Can you check the result?
	⁃	Can you check the argument?

In order to convince ourselves of presence or the quality of an object, we like to see and to touch it. And as we prefer perception through two different senses, so we prefer conviction by two different proofs: 
	⁃	Can you derive result differently? 
	⁃	Can you see it a a glance?
	⁃	Can you use the result or the method, for some other problem?

PART II. HOW TO SOLVE IT A DIALOG:

Getting acquainted:

Where should I start? Start from the statement of the problem. 

What can I do? Visualize the problem as a whole as clearly and as vividly as you can. Do not concern yourself with details for the moment.

What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevant points.

Working for better understanding:

Where should I start? Start again from the statement of the problem. Start when this statement is so clear to you and so well impressed on your mind that you may lose sight of it for a while without fear of losing it altogether.

What can I do? Isolate the principal parts of your problem. The hypothesis and the conclusion are the principal parts of the “problem to prove”; the unknown, the data, and the conditions are principal parts of “problem to find”. Go through the principal parts of your problem, consider them one by one, consider them in turn, consider them in various combinations, relating each detail to other details and each to the whole of the problem.

What can I gain by doing so? You should prepare and clarify details which are likely to play a role afterwards.

Hunting for the helpful idea:

Where should I start? Start from the consideration of the principle parts of your problem. Start when these principle parts are distinctly arranged and clearly conceived, thanks to your previous work, and then your memory seems responsive.
What can I do? Consider your problem from various sides and seek contacts with your formerly acquired knowledge. 	Consider your problem from various sides. Emphasize different parts, examine different details, examine the same details repeatedly but in different ways, combine the details repeatedly but in different ways, combine the details differently, approach them from different sides. Try to see some new meaning in each detail, some new interpretation of the whole.
	Seek contacts with your formerly acquired knowledge. Try to think of what helped you in similar situations in the past. Try to recognize something familiar in what you are examine, try to perceive something useful in what you recognize.

What could I perceive? A helpful idea, perhaps a decisive idea that shows you at a glance the way to the very end.

How can an idea be helpful? It shows you the whole of the way or a part of the way; it suggests to you more or less distinctly how you can proceed. Ideas are more or less complete. You are lucky if you have any idea at all.

What can I do with an incomplete idea? You should consider it. If it looks advantageous you should consider it longer. If it looks reliable you should ascertain how far it leads you, and reconsider the situation. The situation has changed, thanks to your helpful idea. Consider the new situation from various sides and seek contacts with your formerly acquired knowledge.

What can I gain by doing so again? You may be lucky and have another idea. Perhaps your next idea will lead you to the solution right away. Perhaps you need a few more helpful ideas after the next. Perhaps you will be led astray by some of your ideas. Nevertheless you should be grateful for all new ideas, also for the lesser ones, also for hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction of a less fortunate one. Even if you do not have any appreciable new ideas for a while you should be grateful if your conception of the problem becomes more complete or more coherent, more homogeneous or better balanced.

Carrying out the plan:

Where should I start? Start from the lucky idea that led you to the solution. Start when you feel sure of you grasp of the main connection and you fell confident that you can supply the minor details that may be wanting.

What can I do? Make your grasp quite secure. Carry through in detail all the algebraic or geometric operations which you have recognized previously as feasible. Convince yourself of the correctness of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish “great” steps and “small” steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards. 

What can I gain by doing so? A presentation of the solution each step of which is correct beyond doubt. 

Looking back:

Where should I start? From the solution, complete and correct in each detail. 

What can I do? Consider the solution from various sides and seek contacts with your formerly acquired knowledge.
	Consider the details of the solution and try to make them as simple as you can; survey more extensive parts of the solution and try to make them shorter; try to see the whole solution a a glance. Try to modify to their advantage smaller or larger parts of the solution, try to improve the whole solution, to make it intuitive, to fit it into your formerly acquired knowledge as naturally as possible. Scrutinize the method that led you to the solution, try to see its point, and try to make use of it for other problems. Scrutinize the result and try to make use of it for other problems.

What can I gain by doing so? You may find a new and better solution, you may discover new and interesting facts. In any case, if you get into the habit of surveying and scrutinizing your solutions in this way, you will acquire some knowledge well ordered and ready to use, and you will develop your ability of solving problems.

PART III. SHORT DICTIONARY OF HEURISTIC

Auxiliary elements:

There are various kinds of auxiliary elements. Solving a geometric problem, we mat introduce new lines into our figures, auxiliary lines. Solving an algebraic problem, we may introduce and auxiliary unknown. An auxiliary theorem is a theorem whose proof we undertake in the hope of promoting the solution of our original problem.

There are various reasons for introducing auxiliary elements. We are glad when we have succeeded in recollecting a problem related to ours and solved before. It is probable that we can use such a problem but we do not know yet how to use it. For instance, the problem which we are trying to solve is a geometric problem, and the related problem which we have solved before and have now succeeded in recollecting is a problem about triangles. Yet there is no triangle in our figure; in order to make any use of the problem recollected we must have a triangle; therefore, we have to introduce one, by adding suitable auxiliary lines to our figure. In general, having recollected a formerly solved related problem and wishing to use it for our present one, we must often ask: Should we introduce some auxiliary element in order to make its use possible? 

Going back to definitions, we have another opportunity to introduce auxiliary elements. For instance, explicating the definition of a circle we should not only mention its centre and its radius, but we should also introduce these geometric elements into our figure. Without introducing them, we could not make any concrete use of the definition; stating the definition without drawing something is mere lip-service.

Trying to use known results and going back to definition are among the best reasons for introducing auxiliary elements;  but they are not the only ones. We may add auxiliary elements to the conception of our problem in order to make it fuller, more suggestive, more familiar although we scarcely know yet explicitly how we shall be able to use the elements added. We may just feel that it is a “bright idea” to conceive the problem that way with such and such elements added. 

We may have this or that reason for introducing an auxiliary element, but we should have some reason. We should not introduce auxiliary elements wantonly.

Teachers and authors of textbooks should not forget that the intelligent student and the intelligent reader are not satisfied by verifying that the steps of a reasoning are correct but also want to know the motive and the purpose of the various steps. The introduction of an auxiliary element is a conspicuous step. If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated. Mathematics is interesting in so far as it occupies out reasoning and inventive powers. But there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible. To make such steps comprehensible by suitable remarks or by carefully chosen questions and suggestions takes a lot of time and effort; but it may be worth while.

Auxiliary problem:

Auxiliary problem is a problem which we consider, not for its own sake, but because we hope that its consideration may help up to solve another problem, our original problem. The original problem is the end we wish to attain, the auxiliary problem a means by which we try to attain out end.

To devise an auxiliary problem is an important operation of the mind. To raise a clear-cut new problem subservient to another problem, to conceive distinctly as an end what is means to another end, is a refined achievement of the intelligence. It is an important task to learn (or to teach) how to handle auxiliary problem intelligently.

Risk. We take away from the original problem the time and the effort that we devote to the auxiliary problem. If our investigation of the auxiliary problem fails, the time and effort we devoted to it may be lost. Therefore, we should exercise our judgement in choosing an auxiliary problem. We may have various good reasons for out choice. The auxiliary problem may appear more accessible that the original problem; or it may appear instructive; or it may have some sort of aesthetic appeal. Sometimes the only advantage of the auxiliary problem is that it is new and offers unexplored possibilities; we choose it because we are tired of the original problem all approaches to which seem to be exhausted.

How to find one. The discovery of the solution of the proposed problem often depends on the discovery of a suitable auxiliary problem. Unhappily, there is no infallible method of discovering suitable auxiliary problems as there is no infallible method of discovering the solution. There are, however, questions and suggestions which are frequently helpful, as LOOK AT THE UNKNOWN. We are often led to useful auxiliary problems by VARIATION OF THE PROBLEM.

Equivalent problems. Two problems are equivalent if solution of each involves the solution of the other. 

The passage from the original problem to the auxiliary problem is called convertible reduction, or bi-lateral reduction, or equivalent reduction if these two problems, the original and the auxiliary, are equivalent. Convertible reductions are in a certain respect, more important and more desirable than other ways to introduce auxiliary problems, but auxiliary problems which are not equivalent to the original problem may also be very useful.

Chain of equivalent auxiliary problems are frequent in mathematical reasoning. We are required to solve a problem A; we cannot see the solution, but we may find that A is equivalent to another problem B. Considering B we may run into a third problem C equivalent to B. Proceeding in the same way, we reduce C to D, and so on, until we come upon a last problem L whose solution is known or immediate. Each problem being equivalent to the preceding, the last problem L must be equivalent to out original problem A. Thus we are able to infer the solution of the original problem A from the problem L which we attained as the last link in a chain of auxiliary problems. 

Chain of problems of this kind were noticed by the Greek mathematicians as we may see from an important passage of PAPPUS.

Unilateral reduction. We have two problems, A and B, both unsolved. If we could solve A we could hence derive the full solution of B. But not conversely; if we could solve B, we would obtain, possibly, some information about A, but we would not know how to derive the full solution of A from that of B. IN such a case, more is achieved by the solution of A than by the solution of B. Let us call A the more ambitious, and B the less ambitious of two problems.

If, from a proposed problem, we pass either to a more ambitious or to a less ambitious auxiliary problem we call the step a unilateral reduction. There are two kinds of unilateral reduction, and both are, in some way or other, more risky than a bilateral or convertible reduction.

Unilateral reduction to a more ambitious problem may also be successful. In fact, the more ambitious problem may be more accessible; this is the INVENTOR’S PARADOX.

Bright idea, or “good idea”, or “being the light,” is colloquial expression describing a sudden advance toward the solution. The coming of a bright idea is an experience familiar to everybody but difficult to describe and so it may be interesting to notice that a very suggestive description of it has been incidentally given by an authority as old as Aristotle.

Most people will agree that conceiving a bright idea is an “act of sagacity.” Aristotle defines “sagacity” as follows: “Sagacity is a hitting by guess upon the essential connection in an inappreciable time. As for example, if you see a person talking with a rich man in a certain way, you may instantly guess that that person is trying to borrow money. Or observing that the bright side of the moon is always toward the sun, you may suddenly perceive why this is; namely, because the moon shines by the light of the sun.”

The first example is not bad but rather trivial; not much sagacity is needed to guess things of this sort about rich men and money, and the idea is not very bright. The second example, however, is quite impressive if we make a little effort of imagination to see it in its proper setting.

We should realize that a contemporary of Aristotle has to watch the sun and the stars if he wished to know the time since there were no wristwatches, and had to observe the phases of the moon if he planned traveling by night since there were no street lights. He was much better acquainted with the sky than the modern city dweller, and his natural intelligence was not dimmed by undigested fragments of journalistic presentations of astronomical theories. He saw the full moon as a flat disk, similar to the disc of the sun but much less bright. He must have wondered at the incessant changes in the shape and position of the moon. He observed the moon occasionally also at daytime, about sunrise or sunset, and found out “that the bight side of the moon is always towards the sun” which was in itself a respectable achievement. And now he perceives that the varying aspects of the moon are like the various aspects of a ball which is illuminated from one side so that one half of it is shiny and the other half dark. He conceives the sun and the moon not as flat discs but as around bodies, one giving and the other receiving the light. He understands the essential connection, he rearranges his former conception instantly, “in an inappreciable time”: there is a sudden leap of the imagination, a bright idea, a flash of genius.

Can you check the result? Can you check the argument? A good answer to these question strengthens our trust in the solution and contributes to the solidity of our knowledge.

Numerical results of mathematical problems can be tested by comparing them to observed numbers, or to a common sense estimate of observable numbers. As problems arising from practical needs or natural curiosity almost always aim at facts it could be expected that such comparisons with observable facts are seldom omitted. Yet every teacher knows that students are not disturbed at all when they find 16,130 ft. for the length of the boat and 8 years, 2 months for the age of the captain who is, by the way, known to be a grandfather. Such neglect of the obvious does not show necessarily stupidity but rather indifference toward artificial problem.

Can you check the argument? Checking the argument step by step, we should avoid mere repetition. First, mere repetition is apt to become boring, uninstructive, a strain on the attention. Second, where we stumbled once, there we are likely to stumble again if the circumstances are the same as before. If we feel that it is necessary to go again through the whole argument step by step, we should at least change the order of the steps, or their grouping, to introduce some variation.

It requires less exertion and is more interesting to pick out the weakest point of the argument and examine it first. A question very useful in picking out points of the argument that are worth while examining is: DID YOU USE ALL THE DATA?

It is clear that our non mathematical knowledge cannot be based entirely on formal proofs. The more solid part of our everyday knowledge is continually tested and strengthened by our everyday experience. Tests by observation are more systematically conducted in the natural sciences. Such tests take the form of careful experiments and measurements, and are combined with mathematical reasoning in the physical sciences. Can our knowledge in mathematics be based on formal proofs alone?

This is a philosophical question which we cannot debate here. It is certain that your knowledge, or my knowledge, or your students’ knowledge in mathematics is not based on formal proofs alone. If there is any solid knowledge at all, it has a broad experimental basis, and this basis is broadened by each problem whose result is successfully tested.

Can you derive the result differently? When the solution that we have finally obtained is long and involved, we naturally suspect that there is some cleaner and less roundabout solution: Can you derive the result differently? Can you see it at a glance? Yet even if you we have succeeded in finding another solution. We desire to convince ourselves of the validity of a  theoretical result by two different derivations as we desire to perceive a material object through two different sense. Having found a proof, we wish to find another proof as we wish to touch an object after having seen it.

Two proofs are better that one. “It is safe riding at two anchors.”

Can you use the result? TO find the solution of a problem by our own means is a discovery. If the problem not difficult, the discovery is not so momentous, but it is a discovery nevertheless. Having made some discovery, however modest, we should not fail to inquire whether there is something more behind it, we should not miss the possibilities opened up by the new result, we should try to use again the procedure used. Export your success! Can you use the result, or the method for some other problem?

We can easily imagine new problems if we are somewhat familiar with the principal means of varying a problem, as GENERALIZATION, SPECIALIZATION, ANALOGY, DECOMPOSING AND RECOMBINING. We start from a proposed problem, we derive from it others by the means we just mentioned, from the problems we obtained we derive still others, and so on. The process is unlimited in theory but, in practice, we seldom carry it very far, because the problems that we obtain so are apt to be inaccessible.

On the other hand we can construct new problems which we can easily solve using the solution of a problem previously solved; but these easy new problems are apt to be uninteresting.

To find a new problem which is both interesting and accessible, is not so easy; we need experience, taste, and good luck. Yet we should not fail to look around for more good problems when we have succeeded in solving one. Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.

Carrying out. To conceive a plan and to carry it through are two different things. 

We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself. In the same way, when the solution is sufficiently advanced, we brush aside all kinds of provisional and merely plausible arguments, and the result should be supported by rigorous argument alone.

Devising the plan of the solution, we should not be too afraid of merely plausible, heuristic reasoning. Anything is right that leads to the right idea. But we have to change this standpoint when we start carrying out the plan and then we should accept only conclusive, strict arguments. Carrying out your plan of the solution check each step. Can you see clearly that the step is correct?

The more painstakingly we check out steps when carrying out the plan, the more freely we may use heuristic reasoning when devising it.

We should give some consideration to the order in which we work out the details of our plan, especially if our problem is complex. We should not omit any detail, we should understand the relation of the detail before us to the whole problem, we should not lose sight of the connection of the major steps. Therefore, we should proceed in proper order. 

In particular, it is not reasonable to check minor details before we have a good reasons to believe that the major steps of the argument are sound. If there is a break in the main line of the argument, checking this or that secondary detail would be useless anyhow.

The order in which we work out the details of the argument may be very different from the order in which we invented them; and the order in which we write down the details in a definitive exposition may be still different.

In Euclid’s exposition all arguments proceed in the same direction: from the data toward the unknown in “problem to find,” and from the hypothesis toward the conclusion in “problems to prove.” Any new element, point, line, etc., has to be correctly proved from the hypothesis or from assertions correctly proved in foregoing steps. Each new element, each new assertion is examined when it is encountered first, and so it has to be examined just once; we may concentrate all out attention upon the present step, we need to look behind us, or look ahead. The very last new element whose derivation we have to check, is the unknown. They very last assertion whose proof we have to examine, is the conclusion. If each step is correct, also the last one, the whole argument is correct.

Lets us sum up. Euclid’s manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect from making understandable the main line of the argument.

It is highly desirable that the student should examine their own arguments in the Euclidean manner, proceeding from the data to the unknown, and checking each step although nothing of this kind should be too rigid
day enforced. It is not so desirable that the teacher should present many proofs in the pure Euclidean manner, although the Euclidean presentation may be very useful after a discussion in which, as is recommended by the present book, students guided by the teacher discover the main idea of the solution as independently as possible. Also desirable seems to be the manner adopted by some netbooks in which a intuitive sketch of the main idea is presented first and the details in the Euclidean order of exposition afterwards.

Wishing to satisfy himself that his proposition is true, the conscientious mathematician tries to see it intuitively and to give a formal proof. Can you see clearly that it is correct? Can you prove that it is correct? Intuitive insight and formal proof are two different ways of perceiving the truth, comparable to the perception of a material object through two different senses, sight and touch.

Intuitive insight may rush far ahead of formal proof. Any intelligent student, without any systematic knowledge of solid geometry, can see as soon as he has clearly understood the terms that two straight lines parallel to the same straight line are parallel to each other (the three lines may or may not be in the same plane). Yet the proof of this statement, as given in proposition 9 of the 11th book of Euclid’s Elements, needs a long, careful, and ingenious preparation.

Formal manipulation of logical rules and algebraic formulas may get far ahead of intuition. Almost everybody can see at once that 3 straight lines, taken at random, divide the plane into 7 parts (look at the only finite part, the triangle included by the 3 lines). Scarcely anybody is able to see, even straining his attention to the utmost, that 5 planes, taken at random, divide space into 26 parts. Yet is can be rigidly proved that the right number is actually 26, and the proof is not even long or difficult.

Carrying out our plan, we check each step. Checking out step, we may rely on intuitive insight or on formal rules. Sometimes the intuition is ahead, sometimes the formal reasoning. It is an interesting and useful exercise to do it both ways. Can you see clearly that the step is correct? Yes, I can see it clearly and distinctly. Intuition is head; but could formal reasoning overtake it? Can you also PROVE that it is correct?

Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise. 

Condition is a principal part of a “problem to find.” A condition is called redundant if it contains superfluous parts. It is called contradictory if its parts are mutually opposed and inconsistent so that there is no object satisfying the condition.

Thus, if a condition is expressed by linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns it is usually just sufficient to determine the unknowns but may be, in exceptional cases, contradictory or insufficient.

Corollary is a theorem which we find easily in examining another theorem just found. The word is of Latin origin; a more literal translation would be “gratuity” or “tip”

Could you derive something useful from the data? We have before us unsolved problem, an open question. We have to find the connection between the data and the unknown. We may represent our unsolved problem as open space between the data and the unknown, as gap across which we have to construct a bridge. We can start constructing out bridge from either side, from the unknown or from the data.

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown.

Look at the data! Could you derive something useful from the data? This suggests starting the work from the data.

Decomposing and recombining are important operations of the mind.

You examine an object that touches your interest or challenges your curiosity: a house you intend to rent, unimportant but cryptic telegram, any object whose purpose and origin puzzle you, or any problem you intend to solve. You have an impression of the object as a whole but this impression, possibly, is not definite enough. A detail strikes you, and you focus your attention upon it. Then, you concentrate upon another detail; then, again, upon another. Various combinations of details may present themselves and after a while you again consider the object as a while but you see it now differently. You decompose the whole into its parts, and you recombine the parts into more or less different whole. 

If you go into details you may lose yourself in details. Too many or too minute particulars are a burden on the mind. They may prevent you from giving sufficient attention to the main point, or even from seeing the main point at all. Think of the man who cannot see the forest for the trees.

Of course, we do not wish to waste out time with unnecessary detail and we should reserve our effort for the essential. The difficulty is that we cannot say beforehand which details will turn out ultimately as necessary and which will not.

Therefore, let us first of all, understand the problem as a whole. Having understood the problem, we shall be in a better position to judge which particular points may be the most essential. Having examined one or two essential points we shall be in a better position to judge which further details might deserve closer examination. Let us go into detail and decompose the problem gradually, but no further than we need to.

Having understood the problem as a whole, its aim, its main point, we wish to go into detail. Where should we start? In almost all cases, it is reasonable to begin with the consideration of the principal parts of the problem which are the unknown, the data, and the condition. In almost all cases it is advisable to start the detailed examination of the problem with the questions: What is the unknown? What are the data? What is the condition? 

If we wish to examine further details, what should we do? Fairly often, it is advisable to examine each datum by itself, to separate the various parts of the condition, and to examine each part by itself.

We may find it necessary, especially if our problem is more difficult, to decompose the problem still further, and to examine still more remote details. Thus, it may be necessary to go back to the definition of a certain term, to introduce new elements involved by the definition, and to examine the elements so introduced. 

After having decomposed the problem, we try to recombine its elements in some new manner. Especially, we may try to recombine the elements of the problem into some new, more accessible problem which we could possibly use as an auxiliary problem.

There are, of course, unlimited possibilities of recombination. Difficult problems demand hidden, exceptional, original combinations, and the ingenuity of the problem-solver shows itself in the originality of the combination. There are, however, certain usual and relatively simple sorts of combinations, sufficient for simpler problems, which we should know throughly and try first, even if we may be obliged eventually to resort to less obvious means.

There is a formal classification in which the most usual and useful combinations are neatly placed. In constructing a new problem from the proposed problem, we may

	1.	keep the unknown and change the rest (the data and the condition); or
	2.	keep the data and change the rest (the unknown and the condition); or
	3.	change both the unknown and the data

Keeping the unknown and changing the data and the condition in order to transform the proposed problem is often useful. The suggestion LOOK AT THE UNKNOWN aims at problems with the same unknown. We may try to recollect a formerly solved problem of this kind: And try to think of a familiar problem having the same or a similar unknown. Failing to remember such a problem we may try to invent one: Could you think of other data appropriate to determine the unknown?

A new problem which is more closely related to the proposed problem has a better change of being useful. Therefore, keeping the unknown, we try to keep also some data and some part of the condition, and to change, as little as feasible, only one or two data and a small part of the condition. A good method is one in which we omit something without adding anything; we keep the unknown, keep only a part of the condition, drop the other part, but do not introduce any new clause or datum. 

Keeping the data, we may try to introduce some useful and more accessible new unknown. Such an unknown must be obtained from the original data and we have such an unknown in mind when we ask: COULD YOU DERIVE SOMETHING USEFUL FROM THE DATA?

Let us observe that two things are here desirable. First, the new unknown should be more accessible, the is more easily obtainable from the data than the original unknown. Second, the new unknown should be useful that is, it should be, when found, capable of rendering some definite service in the search of the original unknown. In short, the new unknown should be a sort of stepping stone. A stone in the middle of the creek is never to me than the other bank which I wish to arrive at and, when the stone is reached, it helps me on toward the other bank.

The new unknown should be both accessible and useful but, in practice, we must often content ourselves with less. If nothing better presents itself, it is not unreasonable to derive something from the data has some change of being useful; and it is also reasonable to try a new unknown which is closely connected with the original one, even if it does not seem particularly accessible from outset.

If our problem is to construct a circle, we have to find two things, its centre and its radius; our problem has two parts, we may say. In certain cases, one part is more accessible than the other and therefore, in any case, we may reasonably give a moment’s consideration to this possibility: could you love a part of the problem? Asking this, we weight the chances: Would it pay to concentrate just upon the center, or just upon the radius, and to choose one or the other as out new unknown? Questions of this sort are very often useful. In more complex or in more advanced problems, the decisive idea often consists in carving out some more accessible but essential part from the problem.

Changing both the unknown and the data we deviate more from our original course than in the foregoing cases. This, naturally, we do not like; we sense the danger of losing the original problem altogether. Yet we may be compelled to such extensive change if less radical changes have failed to produce something accessible and useful, and we may be tempted to recede so far from out original problem if the new problem has a good change of success. Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?

An interesting way of changing both the unknown and the data is interchanging the unknown with one of the data.

Example. In a crossword puzzle that allows puns and anagrams we find the following clue: “Forward and backward part of a machine (5 letters)”

What is unknown? A word.
What is the conditions? The word has 5 letters. It has something to do with some part of some machine. It should be, or course, and English word, and not too unusual one, let us hope.

Is the condition sufficient to determine the unknown? No. Or, rather, the condition may be sufficient but that part of the condition which is clear by now is a certainly insufficient. There are too many words satisfying it, as “lever,” or “screw,” or what not.

The condition is ambiguously expressed - on purpose, of course. If nothing can be found that could be plausibly described as a “forward part” of a machine and would be a “backward part” too, we may suspect that forward and backward reading might be meant. It may be a good idea to examine this interpretation of the clue.

Separate the various parts of the condition. There condition has two parts, one concerned with meaning of the word, the other with its spelling. The unknown word is required to be
	1.	a short word meaning some part of some machine;
	2.	a word with 5 letters which spelled backward give again a word of meaning some part of some machine.

If we keep only one part of the condition and drop the other part, the unknown is not completely determined. There are many words satisfying part (1) of the condition, we have a sort of locus. We may “describe” this locus (1), “follow” it to its “interaction” with locus (2). The natural procedure is to concentrate upon part (1) of the condition, to recollect words having the prescribed meaning and, when we have succeeded in recollecting some such word, to examine whether it has or has not the prescribed length and can or cannot be read backward. We may have to recollect several words before we run into the right one: level, screw, wheel, shaft, hinge, motor. Of course, “rotor”!

Definition of a term is a statement of its meaning in other terms which are supposed to be well known.

Technical terms in mathematics are of kinds. Some are accepted as primitive terms and are not defined. Others are considered as derived terms and are defined in due form; that is, their meaning is stated in primitive terms and in formerly defined derived terms. Thus, we do not give a formal definition of such primitive notions as point, straight line and plane. Yet we give formal definitions of such notions as “bisect of an angle” or “circle” or “parabola”.

Elimination of technical terms. In order to eliminate a technical term we must know its definition; but it is not enough to know the definition, we must use it.

Definitions and known theorems. If we know the name “parabola” and have some vague idea of the shape of the curve but do not know anything else about it, out knowledge is obviously insufficient to solve the problem proposed as example, or any other serious geometric problem about the parabola. What kind of knowledge is needed for such a purpose?

The science of geometry may be considered as consisting of axioms, definitions, and theorems. The parabola is not mentioned in the axioms which deal only with such primitive terms as point, straight line and so on. 

Going back to definition is important in inventing an argument but it is also important in checking it. 

Somebody presents an alleged new solution of Archimedes’ problem of finding the area of the surface of the sphere. If he has only a vague idea of the sphere, his solution will not be any good. He may have a clear idea of the sphere but if he fails to use this idea in his argument I cannot know that he has any idea at all, and his argument is no good. Therefore, listing to the argument, I am waiting for the moment when he is going to say something substantial about the sphere, to use its definition or some theorem about it. If such a moment never comes, the solution is no good.

We should check not only the argument of others but, of course, also our own arguments, in the same way. Have you taken into account all essential notions involved in the problem? How did you use this notion? Did you use its meaning, its definition? Did you use essential facts, known theorems about it?

Going back to definitions is an important operation of the mind. If we wish to understand why the definition of words are so important, we should realize first that words are important. We can hardly use our mind without using words, or signs, or symbols of some sort. Thus, words and signs have power. Primitive people believe that words and symbols have magic power. We may understand such belief but we should not share it. We should know that the power of a word does not reside in its sound, in the “voices flatus,” in the “hot air” produced by the speaker, but in the ideas of which the word reminds us and, ultimately, in the facts on which the ideas are based.

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 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.

Determination fluctuates with hope and hopelessness, with satisfaction and disappointment. It is easy to keep on going when we think that the solution is just around the corner; but it is hard to persevere when we do not see any way out of the difficulty. We are elated when our forecast comes true. We are depressed when the way we have followed with some confidence is suddenly blocked, and out determination wavers.

In scientific work, it is necessary to apportion wisely determination to outlook. You don’t take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in you whole personality if there is a great promise. If you purpose is set, you stick to it, but you don’t make it unnecessarily difficult for yourself. You do not despise little successes on the contrary, you seek them: If you cannot solve the proposed problem try to solve first some related problem.

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.

Diagnosis…

We are here particularly concerned with the student’s efficiency in solving problems. How can we characterize it? We may derive some profit from the distinction of the four phases of the solution. In fact, the behavior of the student in the various phases is quite characteristic.

Incomplete understanding of the problem, owning to lack of concentration, is perhaps the most widespread deficiency in solving problems. With respect to devising a plan and obtaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming. In carrying out the plan, the most frequent fault is carelessness, lack of patience in checking each step. Failure to checking the result at all is very frequent; the student is glad to get an answer, throws down his pencil, and is not shocked by the most unlikely results.

The teacher, having made a careful diagnosis of a fault of this kind, has some change to cure it by insisting on certain questions of the list.

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The questions we discussed aim at examining the completeness of our conception of the problem. Our conception is certainly incomplete if we fail to take into account any essential datum or condition or hypothesis. But it is also incomplete if we fail to realize the meaning of some essential term. Therefore, in order to examine our conception, we should also ask: Have you taken into account all essential notions involved in the problem?

The foregoing remarks, however, are subject to caution and certain limitations. In fact, their straightforward application is restricted to problems which are “perfectly stated” and “reasonable.”

A perfectly stated and reasonable “problem to find” must have all necessary data and not a single superfluous datum; also its condition must be just sufficient, neither contradictory nor redundant. In solving such a problem, we have to use, of course, all the data and whole condition.

The object of a “problem to prove” is a mathematical theorem. If the problem is perfectly stated and reasonable, each clause in the hypothesis of the theorem must be essential to the conclusion. In proving such a theorem we have to use, of course, each clause of the hypothesis. 

Mathematical problems proposed in traditional textbooks are supposed to be perfectly stated and reasonable. We should however not rely too much on this; when there is the slightest doubt, we should ask: IS IT POSSIBLE TO SATISFY THE CONDITION? Trying to answer this question, or a similar one, we may convince ourselves, at least to certain extent, that our problem is as good as it is supposed to be.

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?

…

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 mins at the mobilization of out formerly acquired knowledge. 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.

Examine your guess. You 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.

Many a guess has turned out to be wrong but nevertheless useful in leading to a better one.

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all. 

Don’t. Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise.

We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. 

The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically - he would especially hate critical examination, if he understood what that meant.

Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt?

Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination.

A non mathematical example. In a certain crossword puzzle we have to find a word with seven letters, and the clue is: “Do the walls again, back and forth.”
What is unknown? A word.
What are the data? The length of the word is given; it has seven letters.
What is the condition? It is stated in the clue. It has something to do with walls, yet it is still very hazy.

Thus, we have to reexamine the clue. As we do so, the last part may catch out attention: “… again, back and forth.” Could you solve a part of the problem? Here is chance to guess the beginning of the word. Since the repetition is so strongly emphasized, the word, quite possibly, might start with “re.” This is a pretty obvious guess. If we are tempted to believe it, we should realize what it means. The word required would look thus:

RE - - - - - 

Can you check the result? If another word of the puzzle crosses the one just considered in the first letter, we have an R to start that other word. It may be a good idea to switch to that other word and check the R. If we succeed in verifying that R or if, at least, we do not find any reason against it, we come back to our original word. We ask again: What is the condition? As we reexamine the clue, the very last part may catch out attention: “… back and forth.” Could this imply that the word we seek can be read not only forward but backward? This is a less obvious guess. 

At any rate, let us face this guess; let us realize what it means. The word would look as follows:

RE - - - ER

Moreover, the third letter should be the same at the fifth; it is very likely a consonant and the fourth or middle letter a vowel.

The reader can now easily guess the word by himself. If nothing else helps, he can try all the vowels, one after the other, for the letter in the middle.

Figures are not only the object of geometric problems but also an important help for all sorts of problems in which there is nothing geometric at the outset. Thus, we have two good reasons to consider the role of figures in solving problems.

	1.	If our problem is a problem of geometry, we have to consider a figure. This figure may be in our imagination, or it may be traced on paper. On certain occasions, it might be desirable to imagine the figure without drawing it; but if we have to examine various details, one detail after the other, it is desirable to draw a figure. If there are many details, we cannot imagine all of them simultaneously, but they are all together on the paper. A detail pictured in out imagination may be forgotten; but the detail traced on paper remains, and, when we com back to it, it remands us of our previous remarks, it saves us some of the trouble we have in recollecting our previous consideration.

	2.	………  The method of starting the examination of a problem of construction by drawing a sketch on which , supposedly, the condition is satisfied, goes back to the Greek geometers. It is hinted by the short, somewhat enigmatic phrase of Pappus: Assume what is required to be done as already done. The following recommendation is somewhat less terse but clearer: Draw a hypothetical figure which supposes the condition of the problem satisfied in all its parts.  This is a recommendation for problems of geometric construction but in fact there is no need to restrict us to any such particular kind of problem. We may extend the recommendation to all “problem to find” stating it in the following general form: Examine the hypothetical situation in which the condition of the problem is supposed to be fully satisfied.

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Generalization is passing from the consideration of one object to the consideration of a set containing that object; or passing from the consideration of a restricted set to that of a more comprehensive set containing the restricted one.

If, by some chance, we come across the sum
1 + 8 + 27 + 64 = 100

we may observe that it can be expressed in the curious form

1^3 + 2^3 + 3^3 + 4^3 = 10^2

Now, it is natural to ask ourselves: Does it often happen that a sum of successive cubes as

1^3 + 2^3 + 3^3 + … + n^3 

is a square? In asking this, we generalize. This generalization is a lucky one; it leads from one observation to a remarkable general law. May results were found by lucky generalization in mathematics, physics, and the natural sciences.

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Have you seen it before? It is possible that we have solved before the same problem that we have to do now, or that we have heard of it, or that we had very similar problem. These are possibilities which we should not fail to explore. We try to remember what happened. Have you seen it before? Or have you seen the same problem in a slightly different form? Even if the answer is negative such questions may start the mobilization of useful knowledge.

The question in the title of the present article is often used in a more general meaning. In order to obtain the solution, we have to extract relevant elements from out memory, we have to mobilize the pertinent parts of our dormant knowledge. We cannot know, of course, in advance which parts of our knowledge may be relevant; but there are certain possibilities which we should not fail to explore. Thus, any feature of the present problem that played a role in the solution of some other may play again a role. Therefore, if any feature of the present problem strikes us as possibly important, we try to recognize it. What is it? Is it familiar to you? Have you seen it before?

Here is a problem related to your and solved before. This is good news; a problem for which the solution is known and which is connected with our present problem, is certainly welcome. It is still more welcome if the connection is close and the solution simple. There is a good chance that such a problem will be useful in solving out present one.

The situation that we are discussing here is typical and important. In order to see it clearly let us compare it with the situation in which we find ourselves when we are working at an auxiliary problem. In both cases, our aim is to solve a certain problem A and we introduce and consider another problem B in the hope that we may derive some profit for the solution of the proposed problem A from the consideration of that other problem B. The difference is in out relation to B. Here, we succeeded in recollecting an old problem B of which we know the solution but we do not know yet how to use it. There, we succeed in inventing a new problem B; we know (or at least we suspect strongly) how to use B, but we do not know yet how to solve it. Our difficulty concerning B makes all the difference between the two situations. When this difficulty is overcome, we may use B in the same way in both cases; we may use the result or the method, and, if we are lucky, we may use both the result and the method. In the situation considered here, we know well the solution of B but we do not know yet how to use it. Therefore, we ask: Could you use it? Could you use its result? Could you use its method?

The intention of using a certain formerly solved problem influences out conception of the present problem. Trying to link up the two problems, the new and the old, we introduce into the new problem elements corresponding to certain important elements of the old problem. For example, out problem is to determine the sphere circumscribed about a given tetrahedron. This is a problem of solid geometry. We may remember that we have solve before the analogous problem of plane geometry of constructing the circle circumscribed about a give triangle. Then we recollect that in the old problem of plane geometry, we used the perpendicular bisectors of the sides of the triangle. It is reasonable to try to introduce something analogous into our present problem. Thus we may be led to introduce into out present problem, as corresponding auxiliary elements, the perpendicular bisecting planes of the edges of the tetrahedron. After this idea, we can easily work out the solution to the problem of solid geometry, following the analogous solution in plane geometry.

The foregoing example is typical. The consideration of formerly solved related problem leads us to the introduction of auxiliary elements, and the introduction of suitable auxiliary elements makes it possible for us to use the related problem to full advantage in solving out present problem. We aim at such an effect when, thinking about the possible use of formerly solved related problem, we ask: Should you introduce some auxiliary element in order to make its use possible?

Here is a theorem related to yours and proved before.

…

Heuristic, as an adjective, means “serving to discover.”

Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

Heuristic reasoning is often based on induction, or on analogy.
Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof.

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?

Induction and mathematical induction. Induction is process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. Mathematical induction is used in mathematics alone to prove theorems of a certain kind. It is rather unfortunate that the names are connected because there is very little logical connection between the two processes. There is, however, some practical connection; we often use both method together.

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Mathematics presented with rigour is a systematic deductive science but mathematics in the making is an experimental inductive science.

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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.

…

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.

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 out 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.

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Look at the unknown. This is old advice; the corresponding Latin saying is: “respice finem.” This is, look at the end. Remember your aim. Do not forget your goal. Think of what you are desiring to obtain. Do not lose sight of what is required. Keep in mind what you are working for. Look at the unknown. Look at the conclusion. The last two versions of “respice finem” are specifically adapted to mathematical problems, to “problems to find” and to “problems to prove” respectively.

Focusing our attention on our aim and concentrating out will on our purpose, we think of ways and means to attain it. What are the means to this end? How can you attain your aim? How can you obtain a result of this kind? What causes could produce such a result? Where have you seen such a result produced? What do people usually do to obtain such a result? And try to think of a familiar problem having the same or a similar unknown. And try to think of a familiar theorem having the same or similar conclusion. Again, the last two versions are specifically adapter to “problems to find” and to “problems to prove” respectively.

…

“Find the area of the surface of the sphere inscribed in a tetrahedron whose six edges are given.” If we know Archimedes’ result, we need not have Archimedes’ genius to solve the problem; it remains to express the radius of the inscribed sphere in terms of the six edges of the tetrahedron. This is not exactly easy but the difficulty cannot be compared with that of Archimedes’ problem. 

To know or not to know a formerly solved problem with the same unknown may make all the difference between an easy and a difficult problem.

…

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

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Modern heuristic endeavours to understand the process of solving problems, especially the mental operations typically useful in this process. It has various sources of information none of which should be neglected. A serious study of heuristic should take into account both the logical and the psychological background, it should not neglect what such older writers as Pappus, Descartes, Leibnitz, and Bolzano have to say about the subject, but it should least neglect unbiased experience. Experience in solving problems and experience in watching other people solving problems must be the basis on which heuristics is built. In this study, we should not neglect any sort of problem, and should find out common features in the way of handing all sorts of problems; we should aim at general features, independent of the subject matter of the problem. The study of heuristic has “practical” aims; a better understanding of the mental operations typically useful in solving problems could exert some good influence on teaching, especially on the teaching of mathematics.

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Trying to solve a problem, we consider different aspects of it in turn, we roll it over and over incessantly in our mind; VARIATION OF THE PROBLEM is essential to our work. We may vary the problem by DECOMPOSING AND RECOMBINING its elements, or by going back to the DEFINITION of certain of its terms, or we may use the great resources of GENERALIZATION, SPECIALIZATION, and ANALOGY. Variation of the problem may lead us to AUXILIARY ELEMENTS, or to the discovery of a more accessible AUXILIARY PROBLEM.

We have to distinguish carefully between two kinds of problems, PROBLEMS TO FIND, PROBLEMS TO PROVE. Our list is specially adapted to “problems to find.” We have to revise it and change some of its questions and suggestions in order to apply it to “problems to prove.”

In all sorts of problems, but especially in mathematical problems which are not too simple, suitable NOTATION and geometrical FIGURES are a great and often indispensable help.

The process of solving problems has many aspects but some of them are not considered at all in this book and others only very briefly. It is justified, I think, to exclude from a first short exposition points which could appear too subtle, or too technical, or too controversial.

Provisional, merely plausible HEURISTIC REASONING is important in discovering the solution, but you should not take it for a proof; you must guess, but also EXAMINE YOUR GUESS. 

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…

Speaking and thinking are closely connected, the use of words assists the mind. Certain philosophers and philologists went a little further and asserted that the use of words is indispensable to the use of reason.

…

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An important step in solving a problem is to choose the notation. It should be done carefully. The time we spend now on choosing the notation may be well repaid by the time we save later by avoiding hesitation and confusion. Moreover, choosing the notation carefully, we have to think sharply of the elements of the problem which must be denoted. Thus, choosing a suitable notation may contribute essentially to understanding the problem.

A good notation should be unambiguous, pregnant, easy to remember; it should avoid harmful second meanings, and take advantage of useful second meanings; the order and connection of signs should suggest the order and connection of things.

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…
Not only the most hopeless boys in the class but also quite intelligent students may have an aversion for algebra. There is always something arbitrary and artificial about notation; to learn a new notation is a burden for the memory. The intelligent student refuses to assume the burden if he does not see any compensation for it. The intelligent student is justified in his aversion for algebra if he is not given ample opportunity to convince himself by his own experience that the language of mathematical symbols assists the mind. To help him to such experience is an important task of the teacher, one of his most important tasks.

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…
“In analysis, we start from what is required, we take it for granted, and we draw consequences, till we reach a point that we can use as starting point in synthesis. For in analysis we assume what is required to be done as already done (what is sought as already found, what we have to prove as true). We inquire from what antecedent the desire result could be derived; then we inquire again what could be the antecedent of the antecedent, and so on, until passing from antecedent to antecedent, we come eventually upon something already known or admittedly true. This procedure we call analysis, or solution backwards, or regressive reasoning. 

“But in synthesis, reversing the progress, we start from the point which we reached last of all in the analysis, from the thing already known or admittedly true. We derive from it what preceded it in the analysis, and go on making derivations until, retracing our steps, we finally succeed in arriving at what is required. This procedure we call synthesis, or constructive solution, or progressive reasoning.

“Now analysis is of two kinds; the one is the analysis of the ‘problem to prove’ and aims at establishing true theorems; the other is the analysis of the ‘problems to find’ and aims at finding the unknown.

“If we have a ‘problem to prove’ we are required to prove or disprove a clearly stated theorem A. We do not know yet whether A is true or false; but we derive from A another theorem B, from B another C, and so on, until we come upon a last theorem L about which we have definite knowledge. If L is true, A will be also true, provided that all our derivation are convertible. From L we prove the theorem K which preceded L in the analysis and preceding in the same way, we retrace our steps; from C we prove B, from B we prove A, and so we attain our aim. If, however, L is false, we have proved A false.

“If we have a ‘problem to find’ we are required to find a certain unknown x satisfying a clearly stated condition. We do not know yet whether a thing satisfying such a condition is possible or not; but assuming that there is an x satisfying the condition imposed we derive from it another unknown y which has to satisfy a related condition; then we link y to still another unknown, and so on, until we come upon a last unknown z which we can find by some known method. If there is actual a z satisfying the condition imposed upon it, there will be also an x satisfying the original condition, provided that all our derivations are convertible. We first find z; then, knowing z, we find the unknown that preceded z in the analysis; proceeding in the same way, we retrace out steps, and finally, knowing y, we obtain x, and so we attain out aim. If, however, there is nothing that would satisfy the condition imposed upon z, the problem concerning x has no solution.”

We should not forget that the foregoing is not a literal translation but a free rendering, a paraphrase. Various differences between the original and the paraphrase deserve comment, for Pappus’s text is important in many ways.

…

The paraphrase has “mathematical problems” where the original means “geometrical problems.” This emphasizes that the procedures described by Pappus are by no means restricted to geometric problems; they are, in fact, not even restricted to mathematical problems. We have to illustrate this by examples since, in these matters, generality and independence from the nature of the subject are important. 

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…

Nonmathematical illustration. A primitive man wishes to cross a creek; but he cannot do so in the usual way because the water has risen overnight. Thus, the crossing becomes the object of a problem; “crossing the creek” is the x of this primitive problem. The man may recall that he has crossed some other creek by walking along a fallen tree. He looks around for a suitable fallen tree which becomes his new unknown, his y. He cannot find any suitable tree but there are plenty of trees standing along the creek; he wishes that one of them would fall. Could he make a tree fall across the creek? There is a great idea and there is a new unknown; by what means could he tilt the tree over the creek?

This train of ideas ought to be called analysis if we accept the terminology of Pappus. If the primitive man succeeds in finishing his analysis he may become the inventor of the bridge and of the axe. What will be the synthesis? Translation of ideas into actions. The finishing act of the synthesis is walking along a tree across the creek.

The same objects fill the analysis and the synthesis; they exercise the mind of the man in the analysis and his muscles in the synthesis; the analysis consists in thoughts, the synthesis in acts. There is another difference; the order is reversed. Walking across the creek is the first desire from which the analysis starts and it is the last act with which the synthesis ends.

The paraphrase hints a little more distinctly than the original the natural connection between analysis and synthesis. This connection is manifest after the foregoing examples. Analysis comes naturally first, synthesis afterwards; analysis is invention, synthesis, execution; analysis is devising a plan, synthesis carrying through the plan.

…

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…

Pedantry and mastery are attitudes toward rules.

To apply a rules to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally.

To apply a rule with natural ease, with judgement, noticing the cases where it fits, without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.

The questions and suggestions of our list may be helpful both to problem-solvers and to teachers. But, first, they must be understood, their proper use must be learned, and learned by trial and error, by failure and success, by experience in applying them. Second, their use should never become pedantic. You should ask no question, make no suggestion, indiscriminately, following some rigid habit. Be prepared for various questions and suggestions and use your judgement. You are doing a hard and exciting problem; the step you are going to try next should be prompted by an attentive and openminded consideration of the problem before you. You with to help a student; what you say to your student should proceed from a sympathetic understanding of his difficulties.

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And if you are inclined to be a pedant and must rely upon some rule learn this one: Always use you own brains first.

Practical problems are different in various respects from purely mathematical problems, yet the principal motives and procedures of the solution are essentially the same. Practical engineering problems usually involve mathematical problems. We will say a few words about the differences, analogies, and connections between these two sorts of problems.

An impressive practical problem in the construction of a dam across a river. We need no special knowledge to understand this problem. In almost prehistoric times, long before our modern age of scientific theories, men built dams of some sort in the valley of the Nile, and in other parts of the world, where the crops depended on irrigation.

Let us visualize the problem of constructing an important modern dam.

What is the unknown? Many unknowns are involved in a problem of this kind: the exact location of the dam, its geometric shape and dimensions, the materials used in its construction, and so on.

What is the condition? We cannot answer this question in one short sentence because there are many conditions. In so large a project it is necessary to satisfy many important economic needs and to hurt other needs as little as possible. The dam should provide electric power, supply water for irrigation or the use of certain communities, and also help to control floods. On the other hand, it should disturb as little as possible navigation, or economically important fish-life, or beautiful scenery; and so forth. And, of course, it should cost as little as possible and be constructed as quickly as possible.

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What are the data? The multitude of desirable data is tremendous. We need topographical data concerning the vicinity of the river and its tributaries; geological data important for the solidity of foundations, possible leakage, and available materials of construction; meteorological data about annual precipitation and the height of floods; economic data concerning the value of ground which will be flooded, cost of materials and labor; and so on.

Our example shows that unknowns, data, and conditions are more complex and less sharply defined in the practical problem than in a mathematical problem.

In order to solve a problem, we need a certain amount of previously acquired knowledge. The modern engineer has a highly specialized body of knowledge at his disposal, a scientific theory of the strength of materials, his own experience, and the mass of engineering experience stored in special technical literature. We cannot avail ourselves of such special knowledge here but we may try to imagine what was in the mind of an ancient Egyptian dam-builder.

He has seen, of course, various other, perhaps smaller, dams: banks of earth or masonry holding back the water. He has seen the flood, laden with all sorts of debris, pressing against the bank. He might have helped to repair the cracks and the erosion left by the flood. He might have seen a dam break, giving way under the impact of the flood. He has certainly heard stories about dams withstanding the test of centuries or causing catastrophe by an unexpected break. His mind may have pictured the pressure of the river against the surface of the sam and the train and stress in its interior. 

Yet the Egyptian dam-builder had no precise, quantitative, scientific concepts of fluid pressure or of strain and stress in a solid body. Such concepts form an essential 

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part of the intellectual equipment of a modern engineer. Yet the latter also uses much knowledge which has not yet quite reached a precise, scientific level; what he knows about erosion by flowing water, the transportation of silt, the plasticity and other not quite clearly circumscribed properties of certain materials, is knowledge of a rather empirical character.

Our example shows that the knowledge needed and the concepts used are more complex and less sharply defined in practical problems than in mathematical problems.

Unknowns, data, conditions, concepts, necessary preliminary knowledge, everything is more complex and less sharp in practical problems than in purely mathematical problems. This is an important difference, perhaps the main difference, and it certainly implies further differences; yet the fundamental motives and procedures of the solution appear to be the same for both sorts of problems.

There is a widespread opinion that practical problems need more experience than mathematical problems. This may be so. Yet, very likely, the difference lies in the nature of the knowledge needed and not in our attitude toward the problem. In solving a problem of one or the other kind, we have to rely on our experience with similar problems and we often ask the questions: Have you need the same problem in a slightly different form? Do you know a related problem?

In solving a mathematical problem, we start from very clear concepts which are fairly well ordered in our mind. In solving a practical problem, we are often obliged to start from rather hazy ideas; then, the clarification of the concepts may become an important part of the problem. Thus, medical science is in a better position to check infectious diseases today than it was in the times before Pasteur when the notion of infection itself was rather

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hazy. Have you taken into account all essential notions involved in the problem? This is a good question for all sorts of problems but its use varies widely with the nature of the intervening notions.

In a perfectly stated mathematical problem all data and all clauses of the condition are essential and must be taken into account. In practical problems we have a multitude of data and conditions; we take into account as many as we can but we are obliged to neglect some. Take the case of the designer of a large dam. He considers the public interest and important economic interests but he is bound to disregard certain petty claims and grievances. The data of his problem are, strictly speaking, inexhaustible. For instance, we would like to know a little more about the geologic nature of the ground on which the foundations must be laid, but eventually he must stop collecting geologic data although a certain margin of uncertainty unavoidably remains.

Did you use all the data? Did you use the whole condition? We cannot miss these questions when we deal with purely mathematical problems. In practical problems, however, we should put these questions in a modified form: Did you use all the data which could contribute appreciably to the solution? Did you use all the conditions which could influence appreciably the solution? We take stock of the available relevant information, we collect more information if necessary, but eventually we must stop collecting, we must draw the line somewhere, we cannot help neglecting something. “If you will sail without danger, you must never put to sea.” Quite often, there is a great surplus of data which have no appreciable influence on the final form of the solution.

The designers of the ancient Egyptian dams had to rely on the common-sense interpretation of their experience, 

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they had nothing else to rely on. The modern engineer cannot rely on common sense alone, especially if his project is of a new and daring design; he has to calculate the resistance of the projected dam, foresee quantitatively the strain and stress in its interior. For this purpose, he has to apply the theory of elasticity (which applies fairly well to constructions in concrete). To apply this theory, he needs a good deal of mathematics; the practical engineering problem leads to a mathematical problem.

This mathematical problem is too technical to be discussed here; all we can say about it is a general remark. In setting up and in solving mathematical problems derived from practical problems, we usually content ourselves with an approximations. We are bound to neglect some minor data and conditions of the practical problem. Therefore it is reasonable to allow some slight inaccuracy in the computations especially when we can gain in simplicity what we lose in accuracy.

….

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If you wish to solve a “problem to find” you must know, and know very exactly, its principal parts, the unknown, the data, and the condition. Our list contains many questions and suggestions concerned with these parts.

What is the unknown? What are the data? What is the condition?

Separate the various parts of the condition.

Find the connection between the data and the unknown.

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Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition?

If you wish to solve a “problem to prove” you must know, and know very exactly, its principal parts, the hypothesis, and the conclusion. There are useful questions and suggestions concerning these parts which correspond to those questions and suggestions of our list which are specially adapted to “problem to find.”

What is hypothesis? What is the conclusion? 
Separate the various parts of the hypothesis.
Find the connection between hypothesis and the conclusion.
Look at the conclusion! And try to think of a familiar theorem having the same or a similar conclusion.
Keep only a part of the hypothesis, drop the other part; is the conclusion still valid? Could you derive something useful from the hypothesis? Could you think of another hypothesis from which you could easily derive the conclusion? Could you change the hypothesis, or the conclusion? Could you change the hypothesis, or the conclusion, or both if necessary, so that the new hypothesis and the new conclusion are nearer to each other?
Did you use the whole hypothesis?

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Progress and achievement. Have you made any progress? What was the essential achievement? We may address questions of this kind to ourselves when we are solving a problem or to a student whose work we supervise. Thus, we are used to judge, more or less confidently, progress and achievement in concrete cases. The step from such concrete cases to a general description is not easy at all. Yet we have to undertake this step if we wish to make our study of heuristic somewhat complete and we must try to clarify what constitutes, in general, progress and achievement in solving problems.

In order to solve a problem, we must have some knowledge of the subject-matter and we must select and collect the relevant items of our existing but initially dormant knowledge. There is much more in our conception of the problem at the end than was in it at the outset; what has been added? What we have succeeded in extracting from our memory. In order to obtain the solution we have to recall various essential facts. We have to recollect formerly solved problems, known theorems, definitions, if our problem is mathematical. Extracting such relevant elements from our memory may be termed mobilization.

In order to solve a problem, however, it is not enough to recollect isolated facts, we must combine these facts, and their combination must be well adapted to the problem at hand. Thus, in solving a mathematical problem, we have to construct an argument connecting the materials recollected to a well adapted whole. This adapting and combining activity may be termed organization.

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In fact, mobilization and organization can never be really separated. Working at the problem with concentration, we recall only facts which are more or less connected with our purpose, and we have nothing to connect and organize but materials we have recollected and mobilized.

Mobilization and organization are but two aspects of the same complex process which has still many other aspects.

Another aspect of the progress of our work is that our mode of conception changes. Enriched with all the materials which we have recalled, adapted to it, and worked into it, out conception of the problem is much fuller at the end than it was at the outset. Desiring to proceed from our initial conception of the problem to a more adequate, better adapted conception, we try to various sanding and view the problem from different sides. We could make hardly any progress without VARIATION OF THE PROBLEM.

As we progress toward our final goal we see more and more of it, and the we see it better we judge that we are nearer to it. As our examination of the problem advances, we foresee more and more clearly what should be done for the solution and how it should be done. Solving a mathematical problem we may foresee, if we are lucky, that a certain known theorem might be used, that the consideration of a certain formerly solved problem might be helpful, that going back to the meaning of a certain technical term might be necessary. We do not foresee such things with certainty, only with a certain degree of plausibility. We shall attain complete certainty when we have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. Without considerations which are only plausible and provisional, we could never find the solution which is certain and final. We need HEURISTIC REASONING. 

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What is progress toward the solution? Advancing mobilization and organization of our knowledge, evolution of our conception of the problem, increasing prevision of the steps which will constitute the final argument. We may advance steadily, by small imperceptible steps, but now and then we advance abruptly, by leaps and bounds. A sudden advance toward the solution is called a BRIGHT IDEA, a good idea, a happy thought, a brain-wave (in German there is a more technical term, Einfall). What is bright idea? An abrupt and mementoes change of our outlook, a sudden reorganization of our mode of conceiving the problem, a just emerging confident prevision of the steps we have to take in order to attain the solution. 

The foregoing considerations provide the questions and suggestions of our list with the right sort of background.

Many of these questions and suggestions aim directly at the mobilization of our formerly acquired knowledge:
Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

There are typical situations in which we think that we have collected the right sort of material and we work for a better organization of what we have mobilized: Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

There are other typical situations in which we think that we have not yet collected enough material. We wonder what is missing: Did you use all the data? Did you use the whole condition? Have you taken into account all the essential notions involved in the problem?

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Some questions aim directly at the variation of the problem: Could you restate the problem? Could you restate it still differently? Many questions aim at the variation of the problem by specified means, as going back to the DEFINITION, using ANALOGY, GENERALIZATION, SPECIALIZATION, DECOMPOSING AND RECOMBINING.

Still other question suggest a trail to foresee the nature of the solution we are striving to obtain: Is it possible to satisfy the condition? If the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

The questions and suggestions of our list do not mention directly the bright idea; but, in fact, all are concerned with it. Understanding the problem we prepare for it, devising a plan we try to provoke it, having provoked it we carry it through, looking back at the course and the result of the solution we try to exploit it better. 

Puzzles. According to section 3, the questions and suggestion of our list are independent of the subject-matter and applicable to all kinds of problems. It is quite interesting to test this assertion on various puzzles. 

Take, for instance, the words

	DRY OXTAIN IN REAR.

The problem is to find an “anagram,” that is, a rear-rangement of the letters contained in the given words into one word. It is interesting to observe that, when we are solving this puzzle, several questions of our list are pertinent and even stimulating.

What is the unknown? A word.
What are the date? The four words DRY OXTAIL IN REAR.

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What is the condition? The desired word has fifteen letters, the letters contained in the four given words. It is probably a not too unusual English word.

Draw a figure. It is quite useful to mark out fifteen blank spaces:

	- - - - - - - - - - - - - - - - 

Could you restate the problem? We have to find a word containing, in some arrangement, the letters

	A A E I I O Y      D L N R R R T X

This is certainly an equivalent restatement of the problem. It may be an advantageous restatement. Separating the vowels from the consonants (this is important, the alphabetical order is not) we see another aspect of the problem. Thus, we see now that the desired word has seven syllables unless it has some diphthongs.

If you cannot solve the proposed problem, try to solve first some related problem. A related problem is to form words with some of the given letters. We can certainly form short words of this kind. Then we try to find longer and longer words. The more letters we use the nearer we may come to the desired word.

Could you solve a part of the problem? The desired word is so long that it must have distinct parts. It is, probably, a compound word, or it is derived from some other word by adding some usual ending. Which usual ending could it be?

	- - - - - - - - - - A T I O N
	- - - - - - - - - - - - - E L Y

Keep only a part of the condition and drop the other part. We may try to think of a long word with, possibly, as many as seven syllables and relatively few consonants, containing an X and a Y.

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The questions and suggestions of our list cannot work magic. They cannot give us the solution of all possible puzzles without ay effort on our part. If the reader wishes to find the word, he must keep on trying and thinking about it. What the questions and suggestions of the list can do is to “keep the ball rolling.” When, discouraged by lack of success, we are inclined to drop the problem, they may suggest to us a new trail, a new aspect, a new variation of the problem, a new stimulus; they may keep us thinking.

Reductio ad absurdum and indirect proof are different but related procedures.

Reduction ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity. “Reduction to an absurdity” is mathematical procedure but it has some resemblance to irony which is the favourite procedure of the satirist. Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity.

Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption. Thus, indirect proof has some resemblance to a politician’s trick of establishing a candidate by demolishing the reputation of his opponent. 

Both “reductio ad absurdum” and indirect proof are effective tools of discovery which present themselves naturally to an intent mind. Nevertheless, they are disliked by a few philosophers and many beginners, which is understandable; satirical people and tricky politicians do not appeal to everybody. 

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Reductio ad absurdum. Write numbers using each of the ten digits exactly once so that the sum of numbers is exactly 100. 

We may learn something by trying to solve this puzzle whose statement demands some elucidation.

What is the unknown? A set of numbers; and by numbers we mean here, of course, ordinary integers.
What is given? The number 100.
What is the condition? The condition has two parts. First, writing the desired set of numbers, we must use each of the ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, just once. Second, the sum of all numbers in the set must be 100.

Keep only a part of the condition, drop the other part. The first part alone is easy to satisfy. Take the set 19, 28, 37, 46, 50; each figure occurs just once. But of course, the second part of the condition is not satisfied; the sum of these numbers is 180, not 100. We could, however, do better. “Try, try again.” Yes,

19 + 28 + 30 + 7 + 6 + 5 + 4 = 99.

The first part of the condition is satisfied, and the second part is almost satisfied; we have 99 instead of 100. Of course, we can easily satisfy the second part if we drop the first:

19 + 28 + 31 + 7 + 6 + 5 + 4 = 100.

The first part is not satisfied: the figure 1 occurs twice, and 0 not at all; the other figures are all right. “Try, try again.”

After a few unsuccessful trials, however, we may be led to suspect that it is not possible to obtain 100 in the manner required. Eventually the problem arises: Prove that it is impossible to satisfy both parts of the proposed condition at the same time.

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Quite good students may find that this problem is above their heads. Yet the answer is easy enough is we have the right attitude. We have to examine the hypothetical situation in which both parts of the condition are satisfied.

We suspect that this situation cannot actually arise and our suspicion, based on the experience of our unsuccessful trails, has some foundation. Nevertheless, let us keep an open mind and let us face the situation in which hypothetically, supposedly, allegedly both parts of the condition are satisfied. Thus, let us imagine a set of numbers whose sum is 100. They must be numbers with one or two figures. There are ten figures, and these ten figures must be all different, since each of the figures, 0, 1, 2, … 9 should occur just once. Thus, the sum of all ten figures must be

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Some of these figures denote units and others tens. It takes a little sagacity to hit upon the idea that the sum of the figures denoting tens may be of some importance. In fact, let t stand for this sum. Then the sum of the remaining figures, denoting units, is 45 - t . Therefore, the sum of all numbers in the set must be 

10t + (45 - t) = 100

We have here an equation to determine t. It is of the first degree and gives

t = 55/9

Now, there is something that is definitely wrong. The value of t that we have found is not an integer and t should be, of course, an integer. Starting from the supposition that both parts of the proposed condition can be simultaneously satisfied, we have been let to a manifest absurdity. 

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How can we explain this? Our original supposition must be wrong; both parts of the condition cannot be satisfied at the same time. And so we have attained our goal, we have succeeded in proving that the two parts of the proposed condition are incompatible.

Our reasoning is a typical “reductio ad absurdum.”

Remarks. Let us look back at the foregoing reasoning and understand its general trend. 

We wuss ti drive that it is impossible to fulfill a certain condition, that is, that the situation in which all parts of the condition are simultaneously satisfied can never arise. But, if we have proved nothing yet, we have to face the possibility that the situation could arise. Only by facing squarely the hypothetical situation and examining it closely can we hope to perceive some definitely wrong point in it. And we must lay our hand upon some definitely wrong point if we wish to show conclusively that the situation is impossible. Hence we can see that the procedure that was successful in our example if reasonable in general: We have to examine the hypothetical situation in which all parts of the condition are satisfied, although such a situation appears extremely unlikely. 

The more experienced reader may see here another point. The main step of our procedure consisted in setting up an equation for t. Now, we could have arrived at the same equation without suspecting that something was wrong with the condition. If we wish to set up an equation, we have to express in mathematical language that all parts of the condition are satisfied, although we do not know yet whether it is actually possible to satisfy all these parts simultaneously. 

Out procedure is “open-minded.” We may hope to find the unknown satisfying the condition, or we may hope to show that the condition cannot be satisfied. 

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It matters little in one respect: the investigation, if it is well conducted, starts in both cases in the same way, examining the hypothetical situation in which the condition is fulfilled, and shows only in its later course which hope is justified. 

Indirect proof. The prime numbers, or primes, are the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … which cannot be resolved into smaller factors, although they are greater that 1. (The last clause excludes the number 1 which , obviously, cannot be resolved into smaller factors, but has a different nature and should not be counted as a prime.) The primes are the “ultimate elements” into which all integers (greater than 1) can be decomposed. For instance, 

630 = 2 * 3 * 3 * 5 * 7

us decomposed into a product of five primes.

Is the series of primes infinite or does it end somewhere? It is natural to suspect that the series of primes never ends. If it ended somewhere, all integers could be decomposed into a finite number of ultimate elements and the world would appear “too poor” in a manner of speaking. Thus arises the problem of proving the existence of an infinity of prime numbers. 

This problem is very different from elementary mathematical problems of the usual kind and appears at first inaccessible. Yet, as we wait, it is extremely unlikely that there should be a last prime, say P. Why is it so unlikely? 

Let us face squarely the unlikely situation in which hypothetically, supposedly, allegedly, there is a last prime P. There we could write down the complete series of primes 2, 3, 5, 7, 11, … P. Why is this so unlikely? What is wrong with it? Can we point out anything that is definitely wrong? Indeed, we can. We can construct the number

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Q = ( 2 * 3 * 5 * 7 * 11 …. P) + 1.

This number Q is greater than P and therefore, allegedly, Q cannot be a prime. Consequently, Q must be divisible by a prime. Now, all primes at our disposal are, supposedly, the numbers 2, 3, 5, … P but Q, divided by any of these numbers, leaves the rest 1; and so Q is not divisible by any of the primes mentioned which are, hypothetically, all the primes. Now, there is something that is definitely wrong; Q must be either a prime or it must be divisible by some prime. Starting from the supposition that there is a last prime P we have been led to a manifest absurdity. How can we explain this? Our original supposition must be wrong; there cannot be a last prime P. And so we have succeeded in proving that the series of prime numbers never ends.

We have established our theorem (that the series of primes never ends) by disproving its contradictory opposite ( that the series of primes ends somewhere ) which we have disproved by deducing from it a manifest absurdity. Thus we have combined indirect proof with “reductio ad absurdum”; this combination is also very typical.

Objections. The procedures which we are studying encountered considerable opposition. Many objections have been raised which are, possibly, only various forms of the same fundamental objection. We discuss here a “practical” form of the objection, which is on our level.

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To find a not obvious proof is a considerable intellectual achievement but to learn such a proof, or even to understand it thoroughly costs also a certain amount of mental effort. Naturally enough, we wish to retain some benefit from our effort, and, of course, what we retain in our memory should be true and correct and not false or absurd.

But it seems difficult to retain something true from “reductio ad absurdum.” The procedure starts from a false assumption and derives from it consequences which are equally, but perhaps more visibly, false till it reaches a last consequence which is manifestly false. If we do not wish to store falsehoods in our memory we should forget everything as quickly as possible which is, however, not feasible because all points must be remembered sharply and correctly during our study of the proof.

The objection to indirect proofs can be now stated very briefly. Listening to such a proof, we are obliged to focus our attention all the time upon a false assumption which we should forget and not upon the true theorem which we should retain.

If we wish to judge correctly of the merits of these objections, we should distinguish between two uses of “reductio ad absurdum,” as a tool of research and as a means of exposition, and make the same distinction concerning the indirect proof.

It must be confessed that “reductio ad absurdum” as a means of exposition is not a unmixed blessing. Such a “reductio,” especially if it is long, may become very painful indeed for the reader or listener. All the derivations which we examine in succession are correct but all the situations which we have to face are impossible. Even the verbal expression may become reties if it insists, as it should, on emphasizing that everything is based on a initial assumption; the words “hypothetically,” “supposedly,” “allegedly” myst secure incessantly, or some other device myst be applied continually. 

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We wish to reject and forget the situation as impossible but we have to retain and examine it as the basis for the next step, and this inner discord may become unbearable in the long run.

Yet it would be foolish to repudiate “reductio ad absurdum” as a tool of discovery. It may present itself naturally and bring a decision when all other means seem to be exhausted as the foregoing examples may show.

We need some experience to perceive that there no essential opposition between our two contentions. Experience shows that usually there is little difficulty in converting an indirect proof into a direct proof, or in rearranging a proof found by a long “reduction ad absurdum” into a more pleasant form from which the “reductio ad absurdum” may even completely disappear (or, after due preparation, it may be compressed into a few striking sentences).

In short, if we wish to make full use of our capacities, we should be familiar both with “reductio ad absurdum” and with indirect proof. When, however, we have succeeded in deriving a result by either of these methods we should not fail to look back at the solution and ask: Can you derive the result differently?

Let us illustrate by examples what we have said.

Rearranging a reductio ad absurdum. We look back at the reasoning presented under 1. The reductio ad absurdum started from a situation which, eventually, turned out to be impossible. Let us however carve out a part of the argument which is independent of the initial false assumption and contains positive information. Reconsidering what we have done, we may perceive that this much is doubtless true: If a set of numbers with one

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or two digits is written so that each of the ten figures occurs just once, then the sum of the set is of the form

10t + (45 - t) = 9 (t + 5).

Thus, this sum is divisible by 9. The proposed puzzle demands however that this sum should be 100. Is this possible? No, it is not, since 100 is not divisible by 9.

The “reductio ad absurdum” which let to the discovery of the argument vanished from our new presentation.

By the way, a reader acquainted with the procedure of “casting out nines” can see now the whole argument at a glance.

Converting an indirect proof. We look back at the reasoning presented under 3. Reconsidering carefully what we have done, we may find elements of the argument which are independent of any false assumption, yet the best clue comes from a reconsideration of the meaning of the original problem itself.

What do we mean by saying that the series of primes never ends? Evidently, just this: when we have ascertained any finite set of primes as 2, 3, 5, 7, 11, … P, where P is the last prime hitherto found, there is always one more prime. Thus, what must we do to prove the existence of infinity of primes? We have to point out a way of finding a prime different from all primes hitherto found. Thus, out “problem to prove” is in fact reduced to a “problem to find”: Being given the primes 2, 3, 5, … P, find a new prime N different from all the given primes.

Having restated our original problem in this new form, we have taken the main step. It is relatively easy now to see how to use the essential parts of out former argument for the new purpose. In fact, the number 

Q = ( 2 * 3 * 5 * 7 * 11 … P) + 1

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is certainly divisible by a prime. Let us take - this is the idea - any prime divisor of Q (for instance, the smallest one) for N. (Of course, if Q happens to be a prime then N = Q.) Obviously, Q divided by ay of the primes 2, 3, 5, … P leaves the reminder 1 and, therefore, none of these numbers can be N which is a divisor of Q. But that is all we need: N is a prime, and different from all hitherto found primes 2, 3, 5, 7, 11, … P.

This proof gives a definite procedure of prolonging again and again the series of primes, without limit. Nothing is indirect in it, no impossible situation needs to be considered. Yet, fundamentally, it is the same as out former indirect proof which we have succeeded in converting.

Routing problem …

In general, a problem is “routine problem” if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example. Setting a routine problem, the teacher thrusts under the nose of the student an immediate and decisive answer to the question: Do you know a related problem? Thus, the student needs nothing but a little care and patience in following a cut-and-dried precept, and he has no opportunity to use his judgement or his incentive faculties.

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Routine problems, even many routine problems, may be necessary in teaching mathematics but to make the student do no other kind is inexcusable. Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgement of the cook but mathematical recipes do not.

Rules of discovery. The first rule of discovery is to have brains and good luck. The second rules of discovery is to sit tight and wait till you get a bright idea.

It may be good to be reminded somewhat rudely that certain aspirations are hopeless. Infallible rules of discovery leading to the solution of all possible mathematical problems would be more desirable than the philosophers’ stone, vainly sought by the alchemists. Such rules would work magic, but there is no such thing as magic. To find unfailing rules applicable to all sorts of problems is an old philosophical dream; but this dream will never be more than a dream.

A reasonable sort of heuristic cannot aim at unfailing rules; but it may endeavor to study procedures (mental operations, moves, steps) which are typically useful in solving problems. Such procedures are practiced by every sane person sufficiently interested in his problem. They are hinted by certain stereotyped questions and suggestions which intelligent people put to themselves and intelligent teachers to their students. A collection of such questions and suggestions, stated with sufficient generality and neatly ordered, may be less desirable than the philosophers’ stone but can be provided. The list we study provides such a collection.

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Rules of style. The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.

Rules of teaching. The first rules of teaching is to know what you are supposed to teach. The second rules of teaching is to know a little more than what you are supposed to teach. 

First things come first. The author of this book does not think all rules of conduct for teachers are completely useless; otherwise, he would not have dared to write a while book about the conduct of teachers and students. Yet is should not be forgotten that teacher of mathematics should know some mathematics, and that a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself.

Separate the various parts of the condition. Our first duty is to understand the problem. Having understood the problem as a while, we go into detail. We consider its principal parts, the unknown, the data, the condition, each by itself. When we have these parts well in mind but no particularly helpful idea has yet occurred to us, we go into further detail. We consider the various data, each datum by itself. Having understood the condition as a while, we separate its various parts, and we consider each part by itself.

We see now the role the suggestion that we have to discuss here. It tends to provoke a step that we have to take when we are trying to see the problem distinctly and have to go into finer and finer detail. It is a step in DECOMPOSING AND RECOMBINING. 

Separate the various parts of the condition. Can you write them down? We often have opportunity to ask this question when we are SETTING UP EQUATIONS.

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Setting up equations is like translation from one language into another. this comparison, used by Newton in his Arithmetica Universalis, may help ti clarity the nature of certain difficulties often felt both by students and by teachers.

To setup equations means to express in mathematical symbols a condition that is stated in words; it is translation from ordinary language into the language of mathematical formulas. The difficulties which we may have in setting up equations are difficulties of translation. 

In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.

An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idioms, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the while meaning; before translating the sentence, we may have to rearrange it.

It is very the same in setting up equations. In easy cases, the verbal statement splits almost automatically into successive parts, each of which can be immediately written down in mathematical symbols. In more difficult cases, the condition has parts which cannot be immediately translated into mathematical symbols. If this is so, we myst pay less attention to the verbal statement, and concentrate more upon the meaning. 

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Before we start writing formulas, we may have to rearrange the condition, and we should keep and eye on the resources of mathematical notation while doing so.

In all cases, easy or difficult, we have to understand the condition, to separate the various parts of the condition, and to ask: Can you write them down? In easy cases, we succeed without hesitation in dividing the condition into parts that can be written down in mathematical symbols; in difficult cases, the appropriate division of the condition is less obvious.

The foregoing explanation should be read again after the study of the following examples.

Find two quantities whose sum is 78 and whose product is 1296.

We divide the page by a vertical line. On one side, we write the verbal statement split into appropriate parts. On the other size, we write algebraic signs, opposite to the corresponding part of the verbal statement. The original is on the left, the translation into symbols on the right.

Stating the problem

in English						|	in algebraic language
find two quantities			|	x, y
whose sum is 78 and		| 	x + y = 78
whose product is 1296		|	xy = 1296

In this case, the verbal statement splits almost automatically into successive parts, each of which can be immediately written down in mathematical symbols.

Find the breadth and the height of a right prism with square base, being given the volume, 63 cu. in., and the area of the surface, 192 sq. in.

What are the unknowns? The side of the base, say, x, and the altitude of the prism, say y.

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What are the data? The volume, 63, and the area, 102. 
What is the condition? The prism whose base is a square with side x and whose altitude is y must have the volume 63 and the area 102.

Separate the various parts of the condition. There are two parts, one concerned with the volume, the other with the area.

We can scarcely hesitate in dividing the whole condition just in these two parts; but we cannot write down these parts “immediately.”. We must know how to calculate the volume and the various parts of the area. Yet, if we know that much geometry, we can easily restate both parts of the condition so that the translation into equation is feasible. We write on the left hand side of the page an essentially rearranged and expanded statement of the problem, ready for translation into algebraic language.

Of a right prism with square base 			|
find the side of the base						|	x
and altitude										|	y
First. The volume is given.						| 	63
The area of the base which					| 
is a square with side x							|	x^2
and the altitude									|	y
determine the volume							|
which is their product.							|	x^2 * y = 63
Second. The area of the surface is given	|	102
The surface consists of two squares 		|
with side x										|	2*x^2
and of four rectangles, each					|
with base x and altitude y,						|	4*x*y
whose sum is the area							|	2*x^2 + 4 * x * y = 102.

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Being given the equation of a straight line and coordinates of a point, find the point which is symmetrical to the given point with respect to the given straight line.

This is a problem of plane analytical geometry.
What is the unknown? A point, with coordinates, say, p, q.
What is given? The equation of a straight line, say y = mx + n, and a point with coordinates, say, a, b.
What is the condition? The points (a, b) and (p, q) are symmetrical to each other with respect to the line y = mx + n.

We now reach the essential difficulty which is to divide the condition into parts each of which can be expressed in the language of analytic geometry. The nature of this difficulty must be well understood. A decomposition of the condition into parts may be logically unobjectionable and nevertheless useless. What we need here is a decomposition into parts which are fit for analytic expression. In order to find such a decomposition we must go back the definition of symmetry, but keep an eye on the resources of analytic geometry. What is meant by symmetry with respect to a straight line? What geometric relations can we express simply in analytic geometry? We concentrate upon the first question, but we should not forget the second. Thus, eventually, we may find the decomposition which we are going to state…

// page 177 decomposition statement

…

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Sign of progress. As Columbus and his companions sailed westward across an unknown ocean they were cheered whenever they saw birds. They regarded a bird as a favourable sign, indicating the nearness of land. But in this they were repeatedly disappointed. They watched for other signs too. They thought that floating seaweed or low banks of cloud might indicate land, but they were again disappointed. One day, however, the signs multiplied. On Thursday, the 11th of October, 1492, “they saw sandpipers, and a green reed near the ship. Those of the caravel Pinta saw a cane and a pole, and they took up another small pole which appeared to have been worked by iron; also another bit of cane, a land-plant, and a small board. The crew of the caravel Niña also saw signs of land, and a small branch covered with berries.Everyone breathed afresh and rejoiced at these signs.” And in fact the next day they sighted land, the first island of a New World.

Our undertaking may be important or unimportant, our problem of any kind - when we are working intensely, we watch eagerly for signs of progress as Columbus and his companions watched for signs of approaching land. We shall discuss a few examples in order to understand what can be reasonably regarded as a sign pf approaching the solution.

1. Examples. I have a chess problem. I have to mate the black king in, say, two moves. On the chessboard there is a white knight, quite a distance from the black king, that is apparently superfluous. What is it good for? I am obliged to leave this question unanswered at first. Yet after various trials, I hit upon a new move and observe that it would bring that apparently superfluous white knight into play. This observation gives me a new hope. I regard it as a favourable sign: that new move has some change to be the right one. Why?

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In a well-constructed chess problem there is no superfluous piece. Therefore, we have to take into account all 	chessmen on the board; we have to use all the data. The correct solution does certainly use all the pieces, even that apparently superfluous white knight. In this last respect, the new move that I contemplate agrees with the correct move that I am supposed to find. The new move looks line the correct move; it might be the correct move.

…

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Here is one more example. Some time ago, I wished to prove a theorem in solid geometry. Without much trouble I found a first remark that appeared to be pertinent; but then I got stuck. Something was missing to finish the proof. When I gave up that day I had a much clearer notion than at the outset how the proof should look, how the gap should be filled; but I was not able to fill it. The next day, after a good night’s rest, I looked again into the question and soon hit upon an analogous theorem in plane geometry. In a flash I was convinced that now I had got hold of the solution and I had, I think, good reason too be convinced. Why?

In fact, analogy is a great guide. The solution of a problem in solid geometry often depends on an analogous problem in plane geometry. Thus, in my case, there was a chance from the outset that the desired proof would use as a lemma some theorem of plane geometry the kind which actually came to my mind. “This theorem looks like the lemma I need; it might be the lemma I need” - such was my reasoning. 

If Columbus and his men had taken the trouble to reason explicitly, they would have reasoned in some similar way. They knew how the sea looks near the shore. They knew that, more often than on the open sea, there are birds in the air, coming from the land, and objects floating in the water, detached from the seashore. Many of the men must observed such things when from former voyages they had returned to their home port. The day before that memorable date on which they sighted the island of San Salvador, as the floating objects in the water because so frequent, they thought: “It looks as if we were approaching some land; we may be approaching from land” and “everyone breathed afresh and rejoiced at these signs.”

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2. Heuristic character of signs of progress. Let us insist upon a point which is perhaps already clear to everyone; but it is very important and, therefore, it should be completely clear. 

They type of reasoning illustrated by the foregoing example deserves to be noticed and taken into account seriously, although it yields only a plausible indication and not an unfailing certainty. Let us restate pedantically, at full length, in rather unnatural detail, one of these reasonings:

	If we are approaching land, we often see birds.
	Now we see birds.
	Therefore, probably, we are approaching land.

Without the word “probably” the conclusion would be a outright fallacy. In fact, Columbus and his companions saw birds many times but were disappointed later. Just once came the say on which they saw sandpipers followed by the day of discovery. 

With the word “probably” the conclusion is reasonable and natural but by no means a proof, a demonstrative conclusion; it is only an indication, a heuristic suggestion. It would be a great mistake to forget that such a conclusion is only probable, and to regard it as certain. But to disregard such conclusions entirely would be a still greater mistake. If you take a heuristic conclusion as certain, you may be fooled and disappointed; but if you neglect heuristic conclusions altogether you will make no progress at all. The most important signs of progress are heuristic. Should we trust them? Should we follow them? Follow, but your eyes open. Trust but look. And never renounce your judgment.

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3. Clearly expressible signs. We can look at the foregoing examples from another point of view.

In one of these examples, we regarded as a favourable sign that we succeeded in bringing into play a datum not used before (the white knight). We were quite right to so regard it. In fact, to solve a problem is, essentially, to find the connection between the data and the unknown. More over we should, at least in well-stated problems, use all the data, connect each of them with the unknown. Thus, brining one more datum into play is quite properly felt as progress, as a step forward.

In another example, we regarded as a favourable sign that an essential clause of the condition was appropriately taken into account by out formula. We were quite right to so regard it. In fact, we should use the whole condition. Thus, taking into account one more cause of condition is justly felt as progress, as a move in the right direction.

In still another example, we regarded as a favourable sign the emergence of a simpler analogous problem. This also is justified. Indeed, analogy is one of the main sources of invention. If other means fail, we should try to imagine an analogous problem. Therefore, if such a problem emerges spontaneously, by its own accord, we naturally feel elated; we feel that we are approaching the solution. 

…

Thus, understanding clearly the nature of the unknown means progress. Clearly disposing the various data so that we can easily recall any one also means progress. Visualizing vividly the condition as a whole, may mean an essential advance; and separating the condition into appropriate parts may be an important step forward. When we have found a figure that we can easily imagine, or a notation that we can easily retain, we can reasonably believe that we have made some progress. Recalling a problem related to ours and solved before may be a decisive move in the right direction.

…

4. Less clearly expressible signs. When we work intently, we feel keenly the pace of our progress: when it is rapid we are elated; when it is slow we are depressed. We feel such differences quite clearly without being able to point out any distinct sign. Moods, feelings, general aspects of the situation serve to indicate our progress. They are not easy to express. “It looks good to me,” or “It is not so good,” say the unsophisticated. More sophisticated people express themselves with some nuance: “This is well-balanced plan,” or “No, something is still lacking and that spoils the harmony.” 

[Page 184]

Yet behind primitive or vague expressions there is an unmistakable feeling which we follow with confidence and which leads us frequently in the right direction. If such feeling is very strong and emerges suddenly, we speak of inspiration. People usually cannot doubt their inspirations and are sometimes fooled by them. … Trust, but keep your eyes open.

Always follow your inspiration - with a grain of doubt.

	5.	How signs help. 

…

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Signs may guide our acts. Their absence may warn us of a blind alley and save us time and useless exertion; their presence may cause us to concentrate our effort upon the right spot.

Yet signs may also be deceptive. In once abandoned a certain path for lack of signs, but a man who came after me and followed that path a little farther made an important discovery - to my great annoyance and long-lasting regret. He not only had more perseverance than I did but also read correctly a certain sign which I had failed to notice. Again, I may follow a road cheerfully, encouraged by favorable signs, and run against an unsuspected and insurmountable obstacle.

Yes, signs may misguide us in any single case, but they guide us tight in the majority of them. A hunter may misinterpret now and then the traces of his game but he must be right on the average, otherwise he could not make a living by hunting.

It takes experience to interpret the signs correctly. … The expert knows more signs than the inexperienced, and he knows them better; his main advantage may consist in such knowledge. An expect hunter notices traces of game and appraises even their freshness or staleness where the inexperienced one is unable to see anything.

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The main advantage of the exceptionally talented may consist in a sort of extraordinary mental sensibility. With exquisite sensibility, he feels subtle signs of progress or notices their absence where the less talented are unable to perceive a difference.

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Specialization is passing from the consideration of a given set of objects to that of a smaller set, or of just one object, contained in the given set. Specialization is often useful in the solution of problems. 

[Page 196]

… 
In order to solve our original problem we have solved first another problem which we may call appropriately the auxiliary problem. This auxiliary problem is a special case of the original problem (the extreme special case in which one of the two ships is at rest). The original problem was proposed, the auxiliary problem invented in the course of the solution. The original problem looked hard, the solution of the auxiliary problem was immediate. The auxiliary problem was, as a special case, in fact much less ambitious than the original problem. How is it then possible that we were able to solve the original problem, on the basis of the auxiliary problem? Because in reducing the original problem to the auxiliary problem, we added a substantial supplementary remark (on relativity of motion).

We succeeded in solving out original problem thanks to two remarks. First, we invented an advantageous auxiliary problem. Second, we discovered an appropriate supplementary remark to pass from the auxiliary problem to the original problem. We solved the proposed problem in two steps as we might cross a creek in two steps provided we were lucky enough to discover an appropriate stone in the middle which could server as a momentary foothold.

To sum up, we used the less difficult, less ambitious, special, auxiliary problem as a stepping stone in solving the more difficult, more ambitious, general, original problem.

…

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Mathematics being a very abstract science should be presented very concretely.

Subconscious work. One evening I wished to discuss with a friend a certain author but I could not remember the author’s name. I was annoyed, because I remembered fairly well one of his stories. I remembered also some story about the author himself which I wanted to tell; I remembered, in fact, everything except the name. Repeatedly, I tried to recollect that name but all in vain. The next morning, as soon as I thought of the annoyance of the evening before, the name occurred to me without any effort.

The reader, very likely, remembers some similar experience of his own. And, if he is a passionate problem-solver, he has probably had some similar experience with problems. It often happens that you have no success at all with a problem; you work very hard yet without finding anything. But when you come back to the problem after a night’s rest, or a few days’ interruption, a bright idea appears and you solve the problem easily.

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The nature of the problem matters little; a forgotten word, a difficult word from a crossword-puzzle, the beginning of an annoying letter, or the solution of a mathematical problem may occur in this way.

Such happenings give the impression of subconscious work. The fact is that a problem, after prolonged absence, may return into consciousness essentially clarified, much nearer to its solution that it was when it dropped out of consciousness. Who clarified it, who brought it nearer to the solution? Obviously, oneself, working at it subconsciously. It is difficult to give any other answer; although psychologists have discovered the beginnings of another answer which may turn out some day to be more satisfactory.

Whatever may or may not be the merits of the theory of subconscious work, it is certain that there is a limit beyond which we should not force the conscious reflection. There are certain moments in which it is better to leave the problem alone for a while. “Take counsel of your pillow” is an old piece of advice. Allowing an interval of rest to the problem and to ourselves, we may obtain more tomorrow with less effort. “If today will not, tomorrow may” is another old saying. But it is desirable not to set aside a problem to which we with to come back later without the impression of some achievement; at least some little point should be settled, some aspect of the question somewhat elucidated when we quit working.

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 with.

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The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate. He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems. By these means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.

The intelligent problem-solver often asks himself questions similar to those contained in our list. He, perhaps discovered questions of this sort by himself; or, having heard such a question from somebody, he discovered its proper use by himself. He is possibly not conscious at all that he repeats the same stereotyped question again and again. Or the question is his particular pet; he knows that the question is part of his mental attitude appropriate in such and such a phase of the work, and he summons up the right attitude by asking the right question.

The intelligent problem-solver may find the questions and suggestions of out list useful. He may understand quite well the explanations and examples illustrating a certain question, he may suspect the proper use of the question; but he cannot attain real understanding unless he comes across the procedure that the questions tries to provoke in his own work and, by having experienced its usefulness, discovers the proper use of the question for himself.

The intelligent problem-solver should be prepared to ask all questions of the list but should ask non unless he is prompted to do so by careful consideration of the problem at hand and by his own unprejudiced judgement.

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In fact, he must recognize by himself whether the present situation is sufficiently similar or not to some other situation in which he saw the question successfully applied.

The intelligent problem-solver tries first of all to understand the problem as fully and as clearly as he can. Yet understanding alone is not enough; he must concentrate upon the problem, he must desire earnestly to obtain its solution. If he cannot summon up real desire for solving the problem he would do better to leave it alone. The one secret of real success it to throw your whole personality into your problem.

The intelligent reader of a mathematical book desires two things:
First, to see that the present step of the argument is correct.
Second, to see the purpose of the present step.

The intelligent listener to a mathematical lecture has the same wishes. If he cannot see that the present step of the argument is correct and even suspects that it is, possibly, incorrect, he may protest and ask a question. If he cannot see any purpose in the present step, nor suspect any reason for it, he usually cannot even formulate a clear objection, he does not protest, he is just dismayed and bored, and loses the thread of the argument.

The intelligent teacher and the intelligent author of textbooks should bear these points in mind. To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps in incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument, if he is not able to derive any suggestion from the presentation as to how he could find such an argument by himself.

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The questions and suggestions of our list may be useful to the author and to the teacher in emphasizing the purpose and the motives of his argument. Particularly useful in this respect in the question: DID WE USE ALL THE DATA? The author or the teacher may show by this question a good reason for considering the datum that has not been used heretofore. The reader or the listener can use the same question in order to understand the author’s or the teacher’s reason for considering such and such an element, and he may feel that, asking this question, he could have discovered this step of the argument by himself.

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Variation of the problem. 

An insect tries to escape through the windowpane, tries the same hopeless thing again and again, and does not try the next window which is open and through which it came into the room. A mouse may act more intelligently; caught in the trap, he tries to squeeze through between two bars, then between the next two bars, then between other bars; he varies his trials, he explores various possibilities. A man is able, or should be able, to vary his trials still more intelligently, to explore the various possibilities with more understanding, to learn by his errors and shortcomings. “Try, try again” is popular advice. It is good advice. The insect, the mouse, and the man follow it; but if one follows it with more success than the others it is because he varies his problem more intelligently.

At the end of our work, when we have obtained the solution, out conception of the problem will be fuller and more adequate than it was at the outset. Desiring to proceed from our initial conception of the problem to a more adequate, better adapted conception, we try various standpoints and we view the problem from different sides.

Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side. In order to find out which aspect is the right one, which side is accessible, we try various sides and aspects, we vary the problem.

Variation of the problem is essential. This fact can be explained in various ways. Thus, from a certain point of view, progress in solving the problem appears as mobilization and organization of formerly acquired knowledge. We have to extract from our memory and to work into the problem certain elements. Now, variation of the problem helps us to extract such elements. How?

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We remember things by a kind of “action by contact,” called “mental association”; what we have in our mind at present tends to recall what was in contact with it at some previous occasion. Varying the problem, we bring in new points, and so we create new contacts, new possibilities of contacting elements relevant to our problem.

We cannot hope to solve any worth-while problem without intense concentration. But we are easily tired by intense concentration of our attention upon the same point. In order to sleep the attention alive, the object on which it is directed must unceasingly change.

If our work progresses, there is something to do, there are new points to examine, our attention is occupied, our interest is alive. But if we fail to make progress, out attention falters, our interest fades, we get tired of the problem, out thoughts began to wander, and there is danger of losing the problem altogether. To escape from this danger we have to set ourselves a new question about the problem.

The new question unfolds untried possibilities of contact without previous knowledge, it revives out hope of making useful contacts. The new question reconquers our interest by varying the problem, by showing some new aspect of it.

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What we said a while ago about new questions which may reconquer out interest is important for the proper use of our list. 

A teacher may use the list to help his students. If the student progresses, he needs no help and the teacher should not ask him any questions, but allow him to work alone which is obviously better for his independence. But the teacher should, of course, try to find some suitable question or suggestion to help him when he gets stuck. Because then there is a danger that the student will get tired of his problem and drop it, or lose interest and make some stupid blunder out of sheer indifference.

We may use the list in solving our own problems. To use it properly we proceed as in the former case. When out progress is satisfactory, when new remarks emerge spontaneously, it would be simply stupid to hamper out spontaneous progress by extraneous questions. But when out progress is blocked, when nothing occurs to us, there is danger that we may get tired of our problem. 

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Then it is time to think of some general idea that could be helpful, of some question or suggestion of the list that might be suitable. And any question is welcome that has some chance of showing a new aspect of the problem; it may reconquer our interest, it may keep us working and thinking.

What is the unknown?

What is required? What do you want? What are you supposed to seek?
What are the data? What is given? What have you?
What is the condition? By what condition is the unknown linked to the data?

These questions may be used by the teacher to test the understanding of the problem; the student should be able to answer them clearly. Moreover, they direct the student’s attention to the principal parts of a “problem to find,” the unknown, the data, the condition. As the consideration of these parts may be necessary again and again, the questions may be often repeated in the later phases of the solution.

The questions are of the greatest importance for the problem-solver. He checks his own understanding of the problem, he focuses his attention on this or that principal part of the problem. The solution consists essentially in linking the unknown to the data. Therefore, the problem-solver has to focus those elements again and again, asking: What is the unknown? What are the data?

The problem may have many unknowns, or the condition may have various parts which must be considered separately, or it may be desirable to consider some datum by itself. Therefore, we may use various modifications of our questions, as: What are the unknowns? What is the first datum? What is the second datum? What are the various parts of the condition? What is the first clause of the condition?

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The principal parts of a “problem to prove” are the hypothesis and the conclusion, and the corresponding questions are: What is the hypothesis? What is the conclusion? We may need some variation of verbal expression or modification of these frequently useful questions as: What do you assume? What are the various parts of your assumption?

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Wisdom of proverbs. Solving problems is a fundamental human activity. In fact, the greater part of our conscious thinking is concerned with problems.

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When we do not indulge in mere musing or daydreaming, our thoughts are directed toward some end; we seek means, we seek to solve a problem.

Some people are more and others less successful in attaining their ends and solving their problems. Such differences are noticed, discussed, and commented upon, and certain proverbs seem to have preserved the quintessence of such comments. At any rate, there are a good many proverbs which characterize strikingly the typical procedures followed in solving problems, the points of common sense involved, the usual tricks, and the usual errors. There are many shrewd and some subtle remarks in proverbs but, obviously, there is no scientific system free of inconsistencies and obscurities in them. On the contrary, many a proverb can be matched with another proverb giving exactly opposite advice, and there is a great latitude of interpretation. It would be foolish to regard proverbs as an authoritative source of universally applicable wisdom but it would be a pity to disregard the graphic description of heuristic procedure provided by proverbs.

It could be an interesting task to collect and group proverbs about planning, seeking means, and choosing between lines of action, in short, proverbs about solving problems. Of the space needed for such a task only a small fraction is available here; the best we can do is to quote a few proverbs illustrating the main phases of the solution emphasized in our list, and discussed in sections 6 to 14 and elsewhere. The proverbs quoted will be printed in italics.

The very first thing we must do for our problem is to understand it: Who understands ill, answers ill. We must see clearly the end we have to attain: Think on the end before you begin. This is an old piece of advice; “respice finem” is the saying in Latin. Unfortunately, not everybody heeds such a good advice, and people often start speculating, talking, and even acting fussily without having properly understood the aim for which they should work.

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A fool looks to the beginning, a wise man regards the end. If the end is not clear in our mind, we may easily stray from the problem and drop it. A wise mean begins in the end, a fool ends in the beginning.

Yet it is not enough to understand the problem, we must also desire its solution. We have no chance to solve a difficult problem without a strong desire to solve it, but with such desire there is a chance. Where there is will there is a way.

Devising a plan, conceiving the idea of an appropriate action, is the main achievement in the solution of a problem.

A good idea is a price of good fortune, an inspiration, a gift of the goods, and we have to deserve it: Diligence is the mother of good luck. Perseverance kills the game. An oak is not felled at one stroke. If at first you don’t succeed, try, try again. It is not enough however to try repeatedly, we must try different means, vary our trials. Try all the keys in the bunch . Arrows are made of all sorts of wood. We must adapt our trials to the circumstances. As the wind blows you must set your sail. Cut your coat according to the cloth. We myst do as we may if we can’t do as we would. If we have failed, we must try something else. A wise man changes his mind, a fool never does. We should even be prepared from the outset for a possible failure of our scheme and have another one in reserve. Have two strings to your bow. We may, of course, overdo this sort of changing from one scheme to another and lose time. Then we may hear the ironical comment: Do and undo, the day is long enough. We are likely to blunder less if we do not lose sight of our aim. The end of fish is not angling but catching.

We work hard to extract something helpful from our memory, yet quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous.

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The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has has uses it better. A wise man will make more opportunities than he finds. A wise man will make tools of what comes to hand. A wise man turns chance into good fortune. Or, possibly the advantage of the expert is that he is continually on the lookout for opportunities. Have an eye to the main chance.

We should start carrying out our plan at the right moment, when it is ripe, but not before. We should not start rashly. Look before you leap. Try before you trust. A wise delay makes the read safe. On the other hand, we should not hesitate too long. If you will sail without danger you must never put to sea. Do the likeliest and hope the best. Use the means and God will give the blessing.

We must use our judgement to determine the right moment. And here is a timely warning that points out the most common fallacy, the most usual failure of our judgement: We soon believe what we desire.

Our plan gives usually but a general outline. We have to convince ourselves that the details fit into the outline, and so we have to examine carefully each detail, one after the other. Step after step the ladder is ascended. Little by little as the cat ate the flick. Do it by degrees.

In carrying out our plan we must be careful to arrange its steps in proper order, which is frequently just the reverse of the order of invention. What a fool does at last, a wise man does at first.

Looking back at the completed solution in an important and instructive phase of the work. He thinks not well that things not again. Second thoughts are the best.

Reexamining the solution, we may discover an additional confirmation of the result. Yet it must be pointed out to the beginner that such an additional confirmation is valuable, that two proofs are better than once. It is save riding at two anchors.

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We have by means exhausted the comments of proverbs on the solution of problems. Yet many other proverbs which could be quoted would scarcely furnish new themes, only variations on the themes already mentioned. Certain more systematic and more sophisticated aspects of the process of solution are hardly within the scope of the Wisdom of Proverbs.

In describing the more systematic aspects of the solution, the author tried now and then to imitate the peculiar run of proverbs, which is not easy. Here follow a few “synthetic” proverbs which describe somewhat more sophisticated attitudes.

The end suggest the means. 

Your five best friends are What, Why, Where, When and How. You ask What, you ask Why, you ask Where, you ask When and How - and nobody else when you need advice. 

Do not believe anything but doubt only what is worth doubting.

Looks around when you have got your first mushroom or made your first discovery; they grow in clusters.