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[ G.Polya ] Variation of the problem an insect (as mentioned elsewhere) 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. 1. At the end of our work, when we have obtained the solution, our 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 . 2. 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 mobilisation and organisation 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? 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. (there is no space and no need to state more neatly the theory of association, or to discuss its limitation.) Varying the problem we bring in new points, and so we create new contacts, new possibilities of contacting elements relevant to our problem. 3. We cannot hope to solve any worthwhile problem without intense concentration. But we are easily tired by intense concentration of our attention upon the same point. In order to keep 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, our attention falters, our interest fades, we get tired of the problem, our thoughts begin 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 con- tact with our previous knowledge, it revives our hope of making useful contacts. The new question reconquers our interest varying the problem , by showing some new aspect of it. 4. Example Find the volume the frustum of a pyramid with square base, being given the side of the lower base , the side of the upper base b, and the altitude of the frustum h. The problem may be proposed to a class familiar with the formulas for the volume of prism and pyramid. If the students do not come forward with some idea of their own, the teacher may begin with varying the data of the problem. We start from a frustum with a.b what happens when b increases till it becomes equal to a? The frustum becomes a prism and the volume in question becomes a^2h what happens when b decreases till i becomes equal to o? The frustum becomes a pyramid and the volume in question becomes a^2h/3 This variation of the data contributes, first of all,. to the interest of the problem. Then, it may suggest using in some way or other, the results quoted about prism and pyramid. At any rate, we have found definite properties of the final result; the final formula must be such that it reduces to a^h for b=a and to a^2h/3 for b=o it is an advantage to foresee properties of the result we are try trying to obtain. Such properties may give valuable suggestions and, in any case, when we have found the final formula we shall be able to test it. We have thus, in advance, an answer to the question : CAN YOU CHECK THE RESULT? (See there,under 2.) 5. Example. Construct a trapezoid being given its four sides a,b,c,d. Let a be the lower base c the upper base; a and c are parallel but unequal, b and d are not parallel. If there is no other idea, we may begin with varying the data. We start from a trapezoid with a>c. What happens when c decreases till it becomes equal to o? The trapezoid degenerates into a triangle. Now a triangle is a familiar and simple figure, which we can construct from various data; there could be some advantage in introducing this triangle into the figure. We do so by drawing just one auxiliary line, a diagonal of the trapezoid (Fig. 21) examining the triangle we find however that it is scarcely useful; we know two sides, a and d, but we should have three data. Let us try something else. What happens when c in- creases till it becomes equal to a? The trapezoid becomes ______c_________ /..... _ / ..... _ b d / ..... _ / ..... a Fig 21 a parallelogram. Could we use it? A little examination (see Fig. 22) directs our attention to the triangle which we have added to the original trapezoid when drawing the parallelogram. This triangle is easily constructed; we know three data, its three sides, b,d, and a-c. ______c_________............... /..... _ b . d / ..... _ . a Fig 22 Varying the original problem (construction of the trapezoid) we have been led to a more accessible auxiliary problem (construction of the triangle). Using the result of the auxiliary problem we easily solve our original problem (we have to complete the parallelogram). Our example is typical. It is also typical that our first attempt failed. Looking back at it, we may see however that the first attempt was not so useless. There was some idea in it; in particular, it gave us an opportunity to think of the construction of a triangle as means to our end. In fact, we arrived at our second, successful trial by modifying our first, unsuccessful trial. We varied c; we first tried to decrease it, then to increase it. 6. As in t he foregoing example, we often have to try various modifications of the problem. We often have to try various modifications of the problem. We have to vary, to restate, to transform it again and again till we succeed eventually in finding something useful. We may learn by failure; there may be some good idea in an unsuccessful trial, and we may arrive at a more successful trial by modifying an unsuccessful one. What we attain after various trials is very often, as in the foregoing example, a more accessible auxiliary problem. 7. There are certain modes of varying the problem which are typically useful, as going back to the DEFINITION DECOMPOSING AND RECOMBINING introducing AUXILIARY ELEMENTS , GENERALISATION , SPECIALISATION , and the use of ANALOGY 8. What we said a while ago (under 3) about new questions which may reconquer our interest is important of 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 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 proceeded as in the former case. When our progress is satisfactory, when new remarks emerge spontaneously, it would be simply stupid to hamper our spontaneous progress by extraneous questions. But when our progress is blocked, when nothing occurs to us, there is danger that we may get tired of our problem. Then it is time to think of some general idea that could be help- ful, 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.