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Can you use the result? To find the solution of a problem by our own means is a discovery. If the problem is 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 possibility opened up by the new result, we should try to use it against the procedure used. Exploit your success! Can you used the result, or the method, for some other problem? 1. We can easily imagine new problems if we are some what familiar with the principal means of varying a problem as GENERALISATION , SPECIALISATION ANALOGY , DECOMPOSING AND RECOMBINING . We start from a proposed problem, we derived 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. 2. We are going to illustrate some of the foregoing points by the same example that we discussed in section 8, 10, 12, 14, 15. Thus we start from the following problem: given the three dimensions (length, breadth, and height) of a rectangular parallelepiped, find the diagonal. If we know the solution of this problem, we can easily solve any of the following problems (of which the first two were almost stated in section 14). Given the three dimensions of a rectangular parallelepiped, find the radius of the circumscribed sphere The base of a pyramid is a rectangle of which the centre is the foot of the altitude of the pyramid. Given the altitude of the pyramid and the sides of its base, find lateral edges. Given the rectangular coordinates (x1,y1,z1), (x2,y2,z2) of two points in space, find the distance of these points. We solve these problems easily because they are scarcely different form the original problem whose solution we know. In each case, we add some new motion to our original problem, as circumscribed sphere, pyramid, rectangle coordinates. These notions are easily added and easily eliminated, and, having got rid of them, we fall back upon our original problem. The foregoing problems have a certain interest be cause the notions that we introduced intro the original problem are interesting. The last problem, that about the distance of two points given by their coordinates, is even a important problem because rectangular coordinates are important. 3. Here is another problem which we can easily solve if we know the solution of our original problem: Given the length, the breadth, and the diagonal of a rectangular parallelepiped, find the height. In fact, the solution of our original problem consists essentially in establishing a relation among four quantities, the three dimensions of the parallelepiped and its diagonal. If any three of these four quantities are given, we can calculate the fourth from the relation. Thus we can solve the new problem. We have here a pattern to derive easily solvable new problems from a problem we have solved; we regard the original unknown and given and one of the original data as unknown. The relation connecting the unknown and the data is the same in both problems, the old and the new. Having found this relation in one, we can use it also in the other. this pattern of deriving new problems by interchanging the roles is very different form the pattern followed under 2. 4. Let us now derive some new problems by others means. A natural generalization of our original problem is the following: Find the diagonal of a parallelepiped being given the three edges issued from an end-point of the diagonal, and the three angles between these three edges. By Specialization we obtain the following problem: Find the diagonal of a cube with given edge. We ay be led to an inexhaustible variety of problems by analogy Here are a few derived from those considered under 2. Find the diagonal of a regular octahedron with given edge. Find the radius of the circumscribed sphere of a regular tetrahedron with given edge. Given the geographical coordinates, latitude and longitude, of two points on the earth's surfaces, which we regard as a sphere find their spherical distance. These problems are interesting but only the one obtained specialisation can be solved immediately on the basis of the solution of the original problem. 5. We may derive new problems from a proposed one by considering certain of its elements as variable. A special case of a problem mentioned under 2 is to find the radius of a sphere circumscribed about a cube whose edge is given. Let us regard the cube, and he common- centre of the cube and sphere as fixed, but let us vary the radius of the sphere. If this radius is small, the sphere is contained in the cube. As the radius increases, the sphere expands(as a rubber balloon in the process of being inflated). At a certain moment, the sphere touches the faces of the cube, a little later, its edges; still later the sphere pusses though the vertices. Which values does the radius assume at these three critical moments.? 6. The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself . The teacher may show the derivation of new problems from one just solved and, doing so , provoke the curiosity of the students. The teacher may also leave some part of the invention to the students; For instance, he may tell about the expanding sphere we just discussed(under 5) and ask: "What would you try to calculate? Which value of the radius is particularly interesting?"