HOW TO SOLVE IT:Specialization

[G.Polya]

Specialisation 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. Specialisation is often useful in the solution of problems. 1. Example. In a triangle, let r be the radius of the inscribed circle, R the radius of the circumscribed circle and H the longest altitude. Then r + R <= H We have to prove (or disprove) this theorem 9; we have a "problem to prove." The proposed theorem is of an unusual sort. We can scarcely remember any theorem about triangles with a similar conclusion. If nothing else occurs to us, we may test some special case of this unfamiliar assertion. Now, the best known special triangle is the equilateral triangle for which r = H / 3 R = 2H / 3 so that, in this case, the assertion is correct. If no other idea presents itself, we may test the more extended special case of isosceles triangles. The form of an isosceles triangle varies with the angle at the vertex and there are two extreme (or limiting) cases, the one in which the angle at the vertex becomes 0 degrees, and the other in which it becomes 180 degrees. In the first extreme case the base of the isosceles triangle vanishes and visibly r = 0 R = H/2 thus, the assertion is verified. In the second limiting case, however, all three heights vanish and r = 0 R = infty H = 0 The assertion is not verified. We have proved that the proposed theorem is false, and so we have solved our problem. By the way, it is clear that the assertion is also false for very flat isosceles triangles whose angle at the vertex is nearly 180 degrees so that we may "officially" disregard the extreme cases whose consideration may appear as not quite "orthodox." 2. "L'exception confirme la r'egle." "The exception proves the rule." We must take this widely known saying as a joke, laughing at the laxity of a certain sort of logic. If we take matters seriously, one exception is enough, of course, to refute irrefragably any would-be rule or gen- eral statement. The most usual and, in some respects, the best method to refute such a statement consists precisely in exhibiting an object that does not comply with it; such an object is called a counter-example by certain writers. The allegedly general statement is concerned with a certain set of objects ; in order to refute the statement we specialise,we pick out from the set an object that does not comply with it. The foregoing example (under I) shows how it is done. We may examine at least first any sim- ple special case, that is, any object chosen more or less at random which we can easily test. If the test shows that the case is not in accordance with the general statement, the statement is refuted and our task finished. If, how- ever, the object examined complies with the statement we may possibly derive some hint from its examination. We may receive the impression that the statement could be true, after all, and some suggestion in which direction we should seek the proof. Or we may receive, as in our example under 1, some suggestion in which direction we should seek the counter example, that is which other special cases should we test. We may modify the case we have just examined, vary it, investigate some more ex- tended special case, look around for extreme cases, as exemplified under 1. Extreme cases are particularly instructive. If a general statement is supposed to apply to all mammals it must apply even to such an unusual mammal as the whale. Let us not forget this extreme case of the whale. Exam- ining it, we may refute the general statement; there is a good chance for that, since such extreme cases are apt to be overlooked by the inventors of generalisations. If, however, we find that the general statement is verified even in the extreme case, the inductive evidence derived from this verification will be strong, just because the prospect of refutation was strong. Thus, we are tempted to reshape the saying from which we started: "Prospective exceptions test the rule." 3. Example. Given the speeds of two ships and their positions at a certain moment; each ship steers a recti- linear course with constant speed. Find the distance of the two ships when they are nearest to each other. What is the unknown? The shortest distance between two moving bodies. The bodies have to be considered as material points. What are the data? The initial positions of the moving material points, and the speed of each. These speeds are constant in amount and direction. B * / |/_ Q P <---------* A Fig. 19. What is the condition? The distance has to be ascer- tained when it is the shortest, that is, at the moment when the two moving points (ships) are nearest to each other. Draw a figure. Introduce suitable notation. In fig 19, the points A and B mark the given initial positions of the two ships. The directed line-segments (vectors) AP and BQ represent the given speeds so that the first ship proceeds along the straight line through the points A and P, and covers the distance AP in unit time. The second ship travels similarly along the straight line BQ. What is the unknown? The shortest distance of the two ships, the one travelling along AP and the other along BQ. It is clear by now what we should find; yet, if we wish to use only elementary means, we may be still in the dark how we should find it. The problem is not too easy and its difficulty has some peculiar nuance which we may try to express by saying that "there is too much variety." The initial positions, A and B, and the speeds, AP and BQ can be given in various ways; in fact, the four points A, B,P,Q may be chosen arbitrarily. Now, whatever the data may be, the required solution must apply and we do not see yet how to fit the same solution to all these pos- sibilities. Out of such feeling of "too much variety" this question and answer may eventually emerge: Could you imagine a more accessible related problem? A more special problem? Of course, there is the extreme special case in which one of the speeds vanishes. Yes, the ship in B may lay at anchor, Q may coincide with B. The shortest distance form the ship at rest to the moving ship is the perpendicular to the straight line along which the latter moves. 4. If the foregoing idea emerges with the premonition that there is more ahead and with the feeling that that extreme special case (which could appear as too simple to be relevant) has some role to play-then it is a bright idea indeed. Here is a problem related to yours, that specialised problem you just solved. Could you use it? Could you use its result? Should you introduce some auxiliary element in order to make its use possible? It should be used, but how? How could the result of the case in which B is at rest be used in the case in which B is moving? Rest is a Special case of motion. And motion is relative-and, therefor, whatever the given velocity of B may be I can consider B as being at rest! Here is the idea more clearly; If I impart to the whole system, consisting of both ships, the same uniform motion, the relative positions do not change, the relative distances remain the same, and so does especially the shortest relative distance of the two ships required by the problem. Now, I can impart a motion that reduces the speed of one of the ships to zero, _ /| / B * /. |/_ . * ___ . S --- ___ _________ __ \-____ . /\ /\ --___ / P <------------* A Fig. 20 and so reduces the general case of the problem to the special case just solved. Let me add a velocity, opposite to BQ but of the same amount, both to BQ and to AP. This is the auxiliary element that makes the use of the special result possible. See Fig.20 for the construction of the shortest distance, BS. 5. The foregoing solution (under 3,4) has a logical pattern that deserves to be analyzed and remembered. In order to solve our original problem (under 3, first lines) we have solved first another problem which we may call appropriately the auxiliary problem (under 3 last lines). 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 int he 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 pos- sible 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 our original problem thanks t to two remarks. First, we invented an advantageous aux- iliary problem. Second we discovered an appropriate supplementary remark to pass from the auxiliary prob- lem 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 stepping stone in the middle which could serve 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. 6. Specialisation has many other uses which we can- not discuss here. It may be just mentioned that it can be useful in testing the solution( CAN YOU CHECK THE RESULT?,2). A somewhat primitive kind of specialisation is often useful to the teacher. It consists in giving some concrete interpretation to the abstract mathematical elements of the problem. For instance, if there is a rectangular parallelepiped in the problem, the teacher may take the class- room in which he talks as example (section 8). In solid analytic geometry, a corner of the classroom may serve as the origin of coordinates, the floor and two walls as coordinate planes, two horizontal edges of the room and one vertical edge as coordinate axes. Explaining the notion of a surface of revolution, the teacher draws a curve with chalk on the door and opens it slowly. These are certainly simple tricks but nothing should be omitted that has some chance to bring home the mathematics to the students: Mathematics being a very abstract science should be presented very concretely.