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[ G.Polya ] Did you use all the data? Owing to the progressive mobilisation of our knowledge, there will be much more in our conception of the problem at the end than was in it at the outset( PROGRESS AND ACHIEVEMENT ,1). But how is it now? Have we got what we need? Is our conception adequate? Did you use all the data? Did you use the whole condition? The corresponding question concerning "problems to prove" is : Did you use the whole hypothesis? 1. For an illustration, let us go back to the "parallelepiped problem" stated in section 8 ( and followed up in sections 10,12,14, 15) . It may happen that a student runs into the idea of calculating the diagonal of a face, s sqrt(a2+b2), but then he gets stuck. The teacher may help him by asking: Did you use all the data? The student can scarcely fail to observe that the expression sqrt(a2+b2) does not contain the third datum c. Therefor, he should try to bring c into play. Thus, he has a good chance to observe the decisive right triangle whose legs are sqrt(a2+b2) and c, and who's hypotenuse is the desired diagonal of the parallelepiped. (For another illustration see AUXILIARY ELEMENTS ,3.) The questions we discuss here are very important. Their use in constructing the solution is clearly shown by the foregoing example. They may help us to find the weak spot in our conception of the problem. They may point out a missing element. When we know that a certain element is still missing, we naturally try to bring it into play. Thus we have a clue, we have a definite line of inquiry to follow, and have a good chance to meet with the decisive idea. 2. The questions we discussed are helpful not only in constructing an argument but also in checking it. In order to be more concrete, let us assume that we have to check the proof of a theorem whose hypothesis consists of three parts, all three essential to the truth of the theorem. That is, if we discard any part of the hypothesis, the theorem ceases to be true. Therefor, if the proof neglects to use any part of they hypothesis, the proof must be wrong. Does the poof use the whole hypothesis? Does it use the first part of the hypothesis? Where does it use the first part of the hypothesis? Where does it use the second part? Where the third? Answering to all these questions we check the proof. This sort of checking is effective, instructive and almost most necessary though understanding if the argument is long and heavy as THE INTELLIGENT READER should know. 3. 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. Therefor, in order to examine our conception, we should also ask : Have you taken into account all essential notions involved in the problem>? See DEFINITION,7 . 4. The foregoing remarks, however, are subject to caution and certain limitations. In fact, their straight for ward 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 the whole condition. The object of a "problem to prove" is a mathematical theorem. If the problem is perfectly stated and the reason able, 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 text books 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 a certain extent, that our problem is as good as it is supposed to be. The question stated in the title of the present article and allied questions may and should be asked without modification only when we know that the problem before us is reasonable and perfectly stated or when, at least, we have no reason to suspect the contrary. 5. There are some non-mathematical problems which may be, in a certain sense, "perfectly stated." For in stance, good chess problems are supposed to have but one solution and no superfluous piece on the chess board, etc. PRACTICAL PROBLEMS however are usually far from being perfectly stated and require a thorough reconsideration of the questions discussed in the he present article.