How to Solve It: A New Aspect of Mathematical Method by G. Polya. Read online, or download in secure PDF or secure EPUB format. and solved before. Heuristic. Heuristic reasoning. If you cannot solve the proposed problem. Induction and mathematical induction. Inventor's paradox. Read "How to Solve It A New Aspect of Mathematical Method" by G. Polya available from Rakuten Kobo. Sign up today and get $5 off your first download.
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Where can I download free Art of Problem Solving textbooks (PDF)?. 82, Views Download >> How to Solve It: A New Aspect of Mathematical Method. k views What sites can I download free PDF and E-books?. Editorial Reviews. Review. Every prospective teacher should read it. In particular, graduate Advanced Search · site Store; ›; site eBooks; ›; Science & Math How to Solve It: A New Aspect of Mathematical Method by [Polya, G. In all my years as a student and teacher, I have never seen another that lives up to George Polya's title by teaching you how to go about solving problems.
The teacher can check this, up to a cer- tain extent; he asks the student to repeat the statement, and the student should be able to state the problem fluently. Example 7 data, the condition. Hence, the teacher can seldom afford to miss the questions: Whai are the data?
The student should consider the principal parts of the problem attentively, repeatedly, and from various sides. If there is a figure connected with the problem he should draw a figure and point out on it the unknown and the data. If it is necessary to give names to these objects he should introduce suitable notation; devoting some atten- tion to the appropriate choice of signs, he is obliged to consider the objects for which the signs have to be chosen.
There is another question which may be useful in this preparatory stage provided that we do not expect a definitive answer but just a provisional answer, a guess: In the exposition of Part II [p. Let us illustrate some of the points ex- plained in the foregoing section. We take the following simple problem: In order to discuss this problem profitably, the students must be familiar with the theorem of Pythagoras, and with some of its applications in plane geometry, but they may have very little systematic knowledge in solid geom- etry.
The teacher may rely here upon the student's un- sophisticated familiarity with spatial relations. The teacher can make the problem interesting by making it concrete.
The classroom is a rectangular paral- lelepiped whose dimensions could be measured, and can be estimated; the students have to find, to "measure indirectly," the diagonal of the classroom. The dialogue between the teacher and the students may start as follows: Which letter should de- note the unknown? I mean, is the condition sufficient to determine the unknown?
If we know a, b, c, we know the parallele- piped. If the parallelepiped is determined, the diagonal is determined.
Devising a plan. We have a plan when we know, or know at least in outline, which calculations, computa- tions, 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 prob- lem is to conceive the idea of a plan.
This idea may emerge gradually. Or, after apparently unsuccessful trials and a period of hesitation, it may occur suddenly, in a flash, as a "bright idea. The questions and suggestions we are going to discuss tend to provoke such an idea.
In order to be able to see the student's position, the teacher should think of his own experience, of his diffi- culties and successes in solving problems.
We know, of course, that 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 con- structing a house but we cannot construct a house with- out 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? The difficulty is that there are usually too many prob- lems which are somewhat related to our present problem, that is, have some point in common with it. How can we choose the one, or the few, which are really useful?
There is a suggestion that puts our finger on an essential com- mon point: If we succeed in recalling a formerly solved problem which is closely related to our present problem, we are lucky. We should try to deserve such luck; we may de- serve it by exploiting it.
Could you use it? If they do not work, we must look around for some other appropriate point of contact, and explore the vari- ous aspects of our problem; we have to vary, to transform, to modify the problem.
Some of the questions of our list hint specific means to vary the problem, as generalization, specialization, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into them now.
Variation of the problem may lead to some appropriate auxiliary problem: Trying to apply various known problems or theorems, considering various modifications, experimenting with various auxiliary problems, we may stray so far from our original problem that we are in danger of losing it alto- gether.
Yet there is a good question that may bring us back to it: We return to the example considered in section 8. As we left it, the students just succeeded in understanding the problem and showed some mild inter- est in it. They could now have some ideas of their own, some initiative. If the teacher, having watched sharply, cannot detect any sign of such initiative he has to resume carefully his dialogue with the students. He must be pre- pared to repeat with some modification the questions which the students do not answer.
He must be prepared to meet often with the disconcerting silence of the students which will be indicated by dots Do you know a problem hav- ing the same unknown? Example 11 "The diagonal of a parallelepiped. We have not had any problem yet about the diagonal of a parallelepiped. Did you never solve a problem whose un- known was the length of a line? For instance, to find a side of a right triangle.
Would you like to use it? Could you introduce some auxiliary element in order to make its use possible? Have you any triangle in your figure? Yet the teacher should be prepared for the case that even this fairly ex- plicit hint is insufficient to shake the torpor of the stu- dents; and so he should be prepared to use a whole gamut of more and more explicit hints.
Now, what will you do? You have now a triangle; but have you the unknown? And the other, I think, is not difficult to find. Yes, the other leg is the hypotenuse of another right triangle. Now I see that you have a plan. Carrying out the plan. To devise a plan, to con- ceive 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: To carry out the plan is much easier; what we need is mainly patience.
Example ourselves that the details fit into the outline, and so we have to examine the details one after the other, patiently, till everything is perfectly clear, and no obscure corner remains in which an error could be hidden. If the student has really conceived a plan, the teacher has now a relatively peaceful time.
The main danger is that the student forgets his plan. This may easily happen if the student received his plan from outside, and ac- cepted it on the authority of the teacher; but if he worked for it himself, even with some help, and conceived the final idea with satisfaction, he will not lose this idea easily.
Yet the teacher must insist that the student should check each step. We may convince ourselves of the correctness of a step in our reasoning either "intuitively" or "formally. The difference between "insight" and "formal proof" is clear enough in many important cases; we may leave further discussion to philosophers.
The main point is that the student should be honestly convinced of the correctness of each step. In certain cases, the teacher may emphasize the difference between "see- ing" and "proving": Can you see clearly that the step is correct?
But can you also prove that the step is correct? Let us resume our work at the point where we left it at the end of section The student, at last, has got the idea of the solution.
He sees the right triangle of which the unknown x is the hypotenuse and the given height c is one of the legs; the other leg is the diagonal of a face. The student must, possibly, be urged to introduce suitable notation. Thus, he may see more clearly the idea of the solution which is to introduce an auxiliary problem whose unknown is y.
Finally, working at one right tri- angle after the other, he may obtain see Fig. Thus, the teacher rna y ask: Even in the latter case, there is some danger that the answer to an incidental question may become the main difficulty for the majority of the students. Looking back. Even fairly good students, when they have obtained the solution of the problem and writ- ten down neatly the argument, shut their books and look for something else.
Doing so, they miss an important and instructive phase of the work. Looking Back date their knowledge and develop their ability to solve problems. A good teacher should understand and impress on his students the view that no problem whatever is com- pletely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution.
The student has now carried through his plan. He has written down the solution, checking each step. Thus, he should have good reasons to believe that his solution is correct. Nevertheless, errors are always possible, especially if the argument is long and involved. Hence, verifications are desirable. Especially, if there is some rapid and in- tuitive procedure to test either the result or the argument, it should not be overlooked.
How to Solve It
In order to convince ourselves of the presence or of 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 de- rive the result differently?
We prefer, of course, a short and intuitive argument to a long and heavy one: One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution. The students will find looking back at the solution really interesting if they have made an honest effort, and have the consciousness of having done well.
Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time. In section 12, the students finally ob- tained the solution: The teacher cannot expect a good answer to this question from inexperienced stu- dents.
The students, however, should acquire fairly early the experience that problems "in letters" have a great advantage over purely numerical problems; if the prob- lem is given "in letters" its result is accessible to several tests to which a problem "in numbers" is not susceptible at all. Our example, although fairly simple, is sufficient to show this. The teacher can ask several questions about the result which the students may readily answer with "Yes"; but an answer "No" would show a serious flaw in the result.
Do all the data a, b, c appear in your formula for the diagonal? Is the expression you obtained for the diagonal sym- metric in a, b, c? Does it remain unchanged when a, b, c are interchanged? Our problem is analogous to a problem of plane geometry: Is the result of our 'solid' problem anal- ogous to the result of the 'plane' problem?
Example parallelepiped becomes a parallelogram. Does your formula show this? If, in your formula, you substi- tute 12a, 12b, 12c for a, b, c respectively, the expression of the diagonal, owing to this substitution, should also be multiplied by Is that so?
These questions have several good effects. First, an in- telligent student cannot help being impressed by the fact that the formula passes so many tests. He was convinced before that the formula is correct because he derived it carefully. But now he is more convinced, and his gain in confidence comes from a different source; it is due to a sort of "experimental evidence.
The formula has therefore a better chance of being re- membered, the knowledge of the student is consolidated. Finally, these questions can be easily transferred to simi- lar problems. After some experience with similar prob- lems, an intelligent student may perceive the underlying general ideas: If he gets into the habit of directing his attention to such points, his ability to solve problems may definitely profit.
In the Classroom Can you check the argument? To recheck the argument step by step may be necessary in difficult and important cases. Usually, it is enough to pick out "touchy" points for rechecking. In our case, it may be advisable to discuss retrospectively the question which was less advisable to discuss as the solution was not yet attained: Can you prove that the triangle with sides x, y, c is a right tri- angle?
See the end of section Can you use the result or the method for some other problem? With a little encouragement, and after one or two examples, the students easily find applications which consist essentially in giving some concrete interpretation to the abstract mathematical elements of the problem. The teacher himself used such a concrete interpretation as he took the room in which the discussion takes place for the parallelepiped of the problem. A dull student may propose, as application, to calculate the diagonal of the cafeteria instead of the diagonal of the classroom.
If the students do not volunteer more imaginative remarks, the teacher himself may put a slightly different problem, for instance: Or they may use the method, introducing suitable right triangles the latter alternative is less obvious and somewhat more clumsy in the present case.
After this application, the teacher may discuss the con- figuration of the four diagonals of the parallelepiped, and the six pyramids of which the six faces are the bases, the center the common vertex, and the semidiagonals the lateral edges. Various Approaches back to his question: Now there is a better chance that the students may find some more interesting concrete interpretation, for instance, the following: To support the pole, we need four equal cables. The cables should start from the same point, 2 yards under the top of the pole, and end at the four corners of the top of the building.
How long is each cable? Various approaches. Let us still retain, for a while, the problem we considered in the foregoing sections 8, 10, 12, The main work, the discovery of the plan, was described in section Let us observe that the teacher could have proceeded differently. Starting from the same point as in section 10, he could have followed a somewhat different line, asking the following questions: Could you think of a simpler analogous prob- lem of plane geometry? What might be an analogous problem about a figure in the plane?
It should be concerned with -the diagonal-of-a rectangular-" "Parallelogram. Besides, if the students are so slow, the teacher should not take up the present problem about the paral- lelepiped without having discussed before, in order to prepare the students, the analogous problem about the parallelogram. Then, he can go on now as follows: Can you use it? It consists in conceiving the diagonal of the given parallelepiped as the diagonal of a suitable parallelogram which must be introduced into the figure as intersection of the parallelepiped with a plane passing through two opposite edges.
The idea is essentially the same as before section 10 but the ap- proach is different. In section 10, the contact with the available knowledge of the students was established through the unknown; a formerly solved problem was recollected because its unknown was the same as that of the proposed problem. In the present section analogy provides the contact with the idea of the solution.
The teacher's method of questioning shown in the foregoing sections 8, 10, 12, 14, 15 is essentially this: Begin with a general question or suggestion of our list, and, if necessary, come down gradually to more specific and concrete questions or suggestions till you reach one which elicits a response in the student's mind.
The Teacher's Method of Questioning 21 have to help the student exploit his idea, start again, if possible, from a general question or suggestion contained in the list, and return again to some more special one if necessary; and so on. Of course, our list is just a first list of this kind; it seems to be sufficient for the majority of simple cases, but there is no doubt that it could be perfected.
It is impor- tant, however, that the suggestions from which we start should be simple, natural, and general, and that their list should be short. The suggestions must be simple and natural because otherwise they cannot be unobtrusive. The suggestions must be general, applicable not only to the present problem but to problems of all sorts, if they are to help develop the ability of the student and not just a special technique.
The list must be short in order that the questions may be often repeated, unartificially, and under varying cir- cumstances; thus, there is a chance that they will be eventually assimilated by the student and will contribute to the development of a mental habit. It is necessary to come down gradually to specific sug- gestions, in order that the student may have as great a share of the work as possible. This method of questioning is not a rigid one; for- tunately so, because, in these matters, any rigid, mechani- cal, pedantical procedure is necessarily bad.
Our method admits a certain elasticity and variation, it admits various approaches section 15 , it can be and should be so applied that questions asked by the teacher could have occurred to the student himself. If a reader wishes to try the method here proposed in his class he should, of course, proceed with caution.
He should study carefully the example introduced in section 8, and the following examples in sections 18, 19, He should start with a few trials and find out gradually how he can manage the method, how the students take it, and how much time it takes. Good questions and bad questions. If the method of questioning formulated in the foregoing section is well understood it helps to judge, by comparison, the quality of certain suggestions which may be offered with the in- tention of helping the students.
Let us go back to the situation as it presented itself at the beginning of section 10 when the question was asked: Instead of this, with the best intention to help the students, the question may be offered: Could you apply the theorem of Pythagoras? The intention may be the best, but the question is about the worst. We must realize in what situation it was of- fered; then we shall see that there is a long sequence of objections against that sort of "help.
Thus the question fails to help where help is most needed. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive. And how could he, the stu- dent, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive. A Problem of Construction None of these objections can be raised against the pro- cedure described in section 10, or against that in sec- tion A problem of construction.
Inscribe a square in a given triangle. Two vertices of the square should be on the base of the triangle, the two other vertices of the square on the two other sides of the triangle, one on each. I am not so sure. If you cannot solve the proposed problem, try to solve first some related problem. Could you satisfy a part of the con- dition?
How many vertices are there? Keep only a part of the condition, drop the other part. What part of the condition is easy to satisfy? How far is the unknown now determined? How can it vary? A Problem to Prove it can vary; the same is true of its fourth corner. Draw more squares with three comers on the perimeter in the same way as the two squares already in the figure. Draw small squares and large squares. What seems to be the locus of the fourth corner? If the student is able to guess that the locus of the fourth corner is a straight line, he has got it.
A problem to prove. Two angles are in different planes but each side of one is parallel to the correspond- ing side of the other, and has also the same direction. Prove that such angles are equal. What we have to prove is a fundamental theorem of solid geometry. The problem may be proposed to stu- dents who are familiar with plane geometry and ac- quainted with those few facts of solid geometry which prepare the present theorem in Euclid's Elements.
The theorem that we have stated and are going to prove is the proposition 10 of Book XI of Euclid. Not only ques- tions and suggestions quoted from our list are printed in italics but also others which correspond to them as "problems to prove" correspond to "problems to find. Each side of one is parallel to the corresponding side of the other, and has also the same direction. Say it, please, using your nota- tion. And try to think of a familiar theorem having the same or a similar conclusion.
Now here is a theorem related to yours and proved before. A Problem to Prove triangles, about a pair of congruent triangles. Have you any triangles in your figure? But I could introduce some. Let me join B to C and B' to C'.
Then there are two triangles, 6, ABC 6. But what are these triangles good for? If you wish to prove this, what kind of tri- angles do you need? Now, what do you wish to prove? Look at the conclusion! You could have chosen a worse one. Thus, what is your aim? What is the hypothesis? Yes, of course, I must use that.
Is that all that you know about these lines? They are parallel and equal to each other. And so are AC and A'C'. How many parallelograms have you now in your figure? No, three. No, two.
A Rate Problem 29 which you can prove immediately that they are paral- lelograms. There is a third which seems to be a parallelo- gram; I hope I can prove that it is one. And then the proof will be finished! But after this last remark of his, there is no doubt.
This student is able to guess a mathematical result and to distinguish clearly between proof and guess. He knows also that guesses can be more or less plausible. Really, he did profit something from his mathematics classes; he has some real experience in solving problems, he can conceive and exploit a good idea.
Princeton Science Library
A rate problem. Water is flowing into a conical vessel at the rate r. The vessel has the shape of a right circular cone, with horizontal base, the vertex pointing downwards; the radius of the base is a, the altitude of the FIG. Find the rate at which the surface is rising when the depth of the water is y. The students are supposed to know the simplest rules of differentiation and the notion of "rate of change. The statement of the problem seems to sug- gest that you should disregard, provisionally, the numeri- cal values, work with the letters, express the unknown in terms of a, b, r, y and only finally, after having obtained the expression of the unknown in letters, substitute the numerical values.
I would follow this suggestion. Now, what is the unknown? Could you say it in other terms? But what is the rate of change? Go back to the definition. Now, is y a function? As we said before, we disregard the numerical value of y.
Can you imagine that y changes? How would you write the 'rate of change of y' in mathematical symbols? Thus, this is your unknown. You have to ex- press it in terms of a, b, r, y. By the way, one of these data is a 'rate. A Rate Problem "r is the rate at which water is flowing into the vessel. How would you write it in suitable notation?
Thus, you have to express 1e in terms of a, b, dV ,y. H ow Wl"II you d o It dt. If you do not see yet the con- nection between 1e and the data, try to bring in some simpler connection that could serve as a stepping stone.
For instance, are y and V independent of each other? When y increases, V must increase too. What is the connection? But I do not know yet the radius of the base. Call it something, say x. Now, what about x? Is it independent of y? When the depth of the water, y, increases the radius of the free surface, x, increases too. I would not miss profiting from it. Do not forget, you wished to know the connection between V and y.
This looks like a stepping stone, does it not? But you should not forget your goal. What to do?
Of course! Here it is. And what about the numerical values? 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 your- self with details for the moment. What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its pur- pose on your mind. The attention bestowed on the prob- lem 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 alto- gether. Isolate the principal parts of your problem. The hypothesis and the conclusion are the principal parts of a "problem to prove"; the unknown, the data, and the conditions are the principal parts of a "problem to find.
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 principal parts of your problem. Start when these principal parts are distinctly arranged and clearly con- ceived, thanks to your previous work, and when your memory seems responsive.
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 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 examine, try to perceive something useful in what you recognize.
What could I perceive? A helpful idea, perhaps a de- cisive 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. The situa- tion 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 the hazy ones, also for the supplementary ideas add- ing some precision to a hazy one, or attempting the cor- rection 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 your grasp of the main connection and you feel confident that you can supply the minor details that may be wanting. Make your grasp quite secure. Carry through in detail all the algebraic or geometric opera- tions which you have recognized previously as feasible.
Convince yourself of the correctness of each step by for- mal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distin- guish "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. A presentation of the solution each step of which is correct beyond doubt.
From the solution, complete and correct in each detail. 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 at 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. 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.
Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts. A rectangular parallelogram is analogous to a rec- tangular parallelepiped. In fact, the relations between the sides of the parallelogram are similar to those be- tween the faces of the parallelepiped: Each side of the parallelogram is parallel to just one other side, and is perpendicular to the remaining sides.
Each face of the parallelepiped is parallel to just one other face, and is perpendicular to the remaining faces. Let us agree to call a side a "bounding element" of the parallelogram and a face a "bounding element" of the parallelepiped. Then, we may contract the two fore- going statements into one that applies equally to both figures: Each bounding element is parallel to just one other bounding element and is perpendicular to the remaining bounding elements. Thus, we have expressed certain relations which are common to the two systems of objects we compared, sides of the rectangle and faces of the rectangular parallele- piped.
The analogy of these systems consists in this com- munity of relations. Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achieve- ments.
Analogy is used on very different levels. All sorts of analogy may play a role in the discovery of the solution and so we should not neglect any sort. In section 15, our original problem was concerned with the diagonal of a rectangular paral- lelepiped; the consideration of a simpler analogous prob- lem, concerned with the diagonal of a rectangle, led us to the solution of the original problem.
We are going to discuss one more case of the same sort. We have to solve the following problem: Find the center of gravity of a homogeneous tetra- hedron.
Without knowledge of the integral calculus, and with little knowledge of physics, this problem is not easy at all; it was a serious scientific problem in the days of Archimedes or Galileo. Thus, if we wish to solve it with as little preliminary knowledge as possible, we should look around for a simpler analogous problem. The corre- sponding problem in the plane occurs here naturally: Find the center of gravity of a homogeneous triangle.
But two questions may be easier to answer than just one question -provided that the two questions are intelligently con- nected.
To solve this problem, we have to know something about centers of gravity. The following principle is plausible and presents itself naturally. This principle yields all that we need in the case of the triangle. First, it implies that the center of gravity of the triangle lies in the plane of the triangle. Then, we may consider the triangle as consisting of fibers thin strips, "infinitely narrow'' parallelograms parallel to a certain side of the triangle the side AB in Fig.
The center of gravity of each fiber of any parallelogram is, obvi- ously, its midpoint, and all these midpoints lie on the line joining the vertex C opposite to the side AB to the midpoint M of AB see Fig. Thus, we are led to the conclusion that the center of gravity of the whole tri- angle lies on the same median.
Yet it must lie on the other two medians just as well, it must be the common point of intersection of all three medians. It is desirable to verify now by pure geometry, inde- pendently of any mechanical assumption, that the three medians meet in the same point.
We have now solved a problem analogous to our proposed problem and, having solved it, we have a model to follow. In solving the analogous problem which we use now as a model, we conceived the triangle ABC as consisting of fibers parallel to one of its sides, AB.
The midpoints of the fibers which constitute the tri- angle lie all on the same straight line, a median of the triangle, joining the midpoint M of the side AB to the opposite vertex C.
The midpoints of the fibers which con- stitute the tetrahedron lie all in the same plane, joining the midpoint M of the edge AB to the opposite edge CD see Fig. Therefore, these three medians must meet in one point which is precisely the center of gravity.
How to Solve It: A New Aspect of Mathematical Method
Therefore, these six median planes must meet in one point which is precisely the center of gravity. Thus, we have solved the problem of the center of gravity of the homogeneous tetrahedron. To complete our solution, it is desirable to verify now by pure geome- try, independently of mechanical considerations, that the six median planes mentioned pass through the same point.
When we had solved the problem of the center of grav- ity of the homogeneous triangle, we found it desirable to verify, in order to complete our solution, that the three medians of the triangle pass through the same point. This problem is analogous to the foregoing but visibly simpler. Again we may use, in solving the problem concerning the tetrahedron, the simpler analogous problem concern- ing the triangle which we may suppose here as solved. In fact, consider the three median planes, passing through the three edges DA, DB, DC issued from the vertex D; each passes also through the midpoint of the opposite edge the median plane through DC passes through M , see Fig.
Now, these three median planes intersect the plane of b. ABC in the three medians of this triangle. These three medians pass through the same point this is the r esult of the simpler analogous prob- lem and this point, just as D, is a common point of the three median planes. The straight line, joining the two common points, is common to all three median planes.
We proved that those 3 among the 6 median planes which pass through the vertex D have a common straight line. Connecting these facts suitably, we may prove that the 6 median planes have a common point. The 3 median planes passing through the sides of!: Now, by what we have just proved, through each line of intersection one more median plane must pass. Both under 5 and under 6 we used a simpler analo- gous problem, concerning the triangle, to solve a prob- lem about the tetrahedron.
Yet the two cases are different in an important respect. Under 5, we used the method of the simpler analogous problem whose solution we imi- tated point by point. Under 6, we used the result of the simpler analogous problem, and we did not care how this result had been obtained. Sometimes, we may be able to use both the method and the result of the simpler analogous problem.
Even our foregoing example shows this if we regard the considerations under 5 and 6 as different parts of the solution of the same problem. Our example is typical. In solving a proposed problem, we can often use the solution of a simpler analogous problem; we may be able to use its method, or its result, or both. Of course, in more difficult cases, complications may arise which are not yet shown by our example.
Especially, it can happen that the solution of the analo- gous problem cannot be immediately used for our orig- inal problem. Then, it may be worth while to reconsider the solution, to vary and to modify it till, after having tried various forms of the solution, we find eventually one that can be extended to our original problem.
It is desirable to foresee the result, or, at least, some features of the result, with some degree of plausibility. Such plausible forecasts are often based on analogy. Analogy 43 Thus, we may know that the center of gravity of a homogeneous triangle coincides with the center of gravity of its three vertices that is, of three material points with equal masses, placed in the vertices of the triangle.
Knowing this, we may conjecture that the center of gravity of a homogeneous tetrahedron coincides with the center of gravity of its four vertices.
This conjecture is an "inference by analogy. It would be foolish to regard the plausibility of such conjectures as certainty, but it would be just as foolish, or even more foolish, to disregard such plausible conjectures. Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning. The chemist, experimenting on animals in order to foresee the influence of his drugs on humans, draws conclusions by analogy.
But so did a small boy I knew. His pet dog had to be taken to the veterinary, and he inquired: An analogical conclusion from many parallel cases is stronger than one from fewer cases. Yet quality is still more important here than quantity. Clear-cut analogies weigh more heavily than vague similarities, systematically arranged instances count for more than random collec- tions of cases.
In the foregoing under 8 we put forward a con jec- ture about the center of gravity of the tetrahedron. We may strengthen the conjecture by examining one more analo- gous case, the case of a homogeneous rod that is, a straight line-segment of uniform density.
The analogy between segment triangle tetrahedron has many aspects. A segment is contained in a straight line, a triangle in a plane, a tetrahedron in space. Straight line-segments are the simplest one-dimensional bounded figures, triangles the simplest polygons, tetrahedrons the simplest polyhedrons.
The segment has 2 zero-dimensional bounding ele- ments 2 end-points and its interior is one-dimensional. The triangle has 3 zero-dimensional and 3 one-dimen- sional bounding elements 3 vertices, 3 sides and its interior is two-dimensional.
The tetrahedron has 4 zero-dimensional, 6 one-dimen- sional, and 4 two-dimensional bounding elements 4 vertices, 6 edges, 4 faces , and its interior is three-dimen- sional.
These numbers can be assembled into a table. The suc- cessive columns contain the numbers for the zero-, one-, two-, and three-dimensional elements, the successive rows the numbers for the segment, triangle, and tetrahedron: We found a remarkable regularity in segment, triangle, and tetrahedron.
If we have experienced that the objects we com- pare are closely connected, "inferences by analogy," as the following, may have a certain weight with us. Analogy 45 The center of gravity of a homogeneous rod coincides with the center of gravity of its 2 end-points.
The center of gravity of a homogeneous triangle coincides with the center of gravity of its 3 vertices. Should we not suspect that the center of gravity of a homogeneous tetrahedron coincides with the center of gravity of its 4 vertices? Again, the center of gravity of a homogeneous rod divides the distance between its end-points in the propor- tion I: The center of gravity of a triangle divides the distance between any vertex and the midpoint of the opposite side in the proportion 2: Should we not sus- pect that the center of gravity of a homogeneous tetra- hedron divides the distance between any vertex and the center of gravity of the opposite face in the proportion 3: It appears extremely unlikely that the conjectures sug- gested by these questions should be wrong, that such a beautiful regularity should be spoiled.
The feeling that harmonious simple order cannot be deceitful guides the discoverer both in the mathematical and in the other sciences, and is expressed by the Latin saying: This conjecture is an "inference by induction"; it illustrates that induction is naturally based on analogy. We finish the present section by considering briefly the most important cases in which analogy attains the precision of mathematical ideas.
I Two systems of mathematical objects, sayS and S', are so connected that certain relations between the ob- jects of S are governed by the same laws as those between the objects of S'. Auxiliary Elements This kind of analogy between S and S' is exemplified by what we have discussed under I; take as S the sides of a rectangle, asS' the faces of a rectangular parallelepiped.
II There is a one-one correspondence between the objects of the two systems S and S', preserving certain relations.
That is, if such a relation holds between the objects of one system, the same relation holds between the corresponding objects of the other system.
Such a connection between two systems is a very precise sort of analogy; it is called isomorphism or holohedral iso- morphism.
III There is a one-many correspondence between the objects of the two systems S and S' preserving certain relations. Such a connection which is important in vari- ous branches of advanced mathematical study, especially in the Theory of Groups, and need not be discussed here in detail is called merohedral isomorphism or homo- morphism; homoiomorphism would be, perhaps, a better term.
Merohedral isomorphism may be considered as another very precise sort of analogy. As our work progresses, we add new elements to those originally considered. An element that we introduce in the hope that it will further the solution is called an auxiliary element. There are various kinds of auxiliary elements.
Solv- ing a geometric problem, we may introduce new lines into our figure, auxiliary lines. Auxiliary Elements 47 2.
There are various reasons for introducing auxiliary elements. We are glad when we have succeeded in recol- lecting 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 tri- angles.
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 wish- ing to use it for our present one, we must often ask: Should we introduce some auxiliary element in order to make its use possible?
The example in section 10 is typical. Going back to definitions, we have another opportu- nity to introduce auxiliary elements. For instance, expli- cating the definition of a circle we should not only mention its center and its radius, but we should also introduce these geometric elements into our figure.
With- out 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 defini- tions are among the best reasons for introducing auxil- iary 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 should not introduce auxiliary elements wantonly.
How to Solve It
Construct a triangle, being given one angle, the altitude drawn from the vertex of the given angle, and the perimeter of the triangle. Let a denote the given angle, h the given altitude drawn from the vertex A of a and p the given perimeter.
We draw a figure in which we easily place a and h. Have we used all the data? Therefore we must intro- duce p. But how? We may attempt to introduce p in various ways. The attempts exhibited in Figs. If we try to make clear to ourselves why they appear so unsatis- factory, we may perceive that it is for lack of symmetry. Now, the sides band c play the same role; they are inter- changeable; our problem is symmetric with respect to b and c.
But b and c do not play the same role in our figures g, 10; placing the length p we treated b and c differently; the figures 9 and 10 spoil the natural sym- metry of the problem with respect to band c. We should place p so that it has the same relation to b as to c.
This consideration may be helpful in suggesting to place the length p as in Fig. II the triangle the segment CE of length b on one side and the segment BD of the length c on the other side so that p appears in Fig. In fact, it is not unreasonable to introduce elements into the problem which are particularly simple and familiar, as isosceles triangle. We have been quite lucky in introducing our auxiliary lines.
Examining the new figure we may discover that LEAD has a simple relation to the given angle a. In fact, we find using the isosceles triangles t: ABD and t: Trying this construction, we introduce an auxiliary problem which is much easier than the original problem. 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, intelli- gent students and readers are disappointed; they feel that they are cheated.
Mathematics is interesting in so far as it occupies our 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 as in the foregoing, under 3 or by carefully chosen questions and suggestions as in sections 10, 18, 19, 20 takes a lot of time and effort; but it may be worth while. The original problem is the end we wish to attain, the auxiliary problem a means by which we try to attain our end.
An insect tries to escape through the windowpane, tries the same again and again, and does not try the next window which is open and through which it came into the room. A man is able, or at least should be able, to act more intelligently. Human superiority consists in going around an obstacle that cannot be overcome directly, in devising a suitable auxiliary problem when the original problem appears insoluble. To devise an auxiliary prob- lem 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 problems intelligently. We obtain now a new problem: The new problem is an auxiliary problem; we intend to use it as a means of solving our original problem.
Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams.
Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going straight to the heart of the problem.
I'm not sure how I missed this all these years - might have come in handy 40 years ago. Excellent problem solving approaches. Filing away for reference in case someone needs it. I'm glad I saw a reference to it in The Organized Mind. Polya wrote this little book for teachers of elementary mathematics, but his clear thinking and writing make his explanation of problem solving useful in many fields.
Even better is his explanation of How to Solve It:Ideas are more or less complete. The reader may refer to the Dictionary for information about particular points.
Such tests take the form of careful experiments and measurements, and are combined with mathematical reasoning in the physical sciences. Let us still retain, for a while, the problem we considered in the foregoing sections 8, 10, 12, And try to think of a familiar problem having to consider auxiliary problems the same or a similar unknown. The teacher can ask several questions about the result which the students may readily answer with "Yes"; but an answer "No" would show a serious flaw in the result.
The worst may happen if the student embarks upon computations or construc- tions without having understood the problem. If a reader wishes to try the method here proposed in his class he should, of course, proceed with caution.