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[ G.Polya ] Problems to find, problems to prove We draw a parallel between these two kinds of problems. 1. The aim of a "problem to find" is to find a certain object, the unknown of the problem. The unknown is also called "quaesitum", or the thing sought, or the thing required. "Problems to find" may be theoretical or practical, abstract or concrete, serious problems or mere puzzles. We may seek all sorts of unknowns; we may try to find, to obtain, to acquire, to produce, or to construct all imaginable kinds of objects. In the problem of the mystery story the unknown is a murderer. In a chess problem the unknown is a move of the chess men. In certain riddles the unknown is a word. In certain elementary problems of algebra the unknown is a number. In a problem of geometric construction the unknown is a figure. 2. The aim of a "problem to prove" is to show conclusively that a certain clearly stated assertion is true, or else to show that it is false. We have to answer the question: Is this assertion true or false? And we have to answer conclusively, either by proving the assertion true, or by proving it false. A witness that the defendant stayed at home a certain night. The judge has to find out whether this assertion is true or not and, moreover, he has to give as good grounds pas possible for his finding. Thus, the judge has a "problem to prove." Another"problem to prove" is to "prove the theorem of Pythagoras." We do not say: "Prove or disprove the theorem of Pythagoras". It would be better in some respects to include in the statement of the problem the possibility of disproving but we may neglect it because we know that the chances for disprove disproving the theorem of Pythagoras are rather slight. 3. The principal parts of a problem to find" are the unknown the data and the condition If we have to construct a triangle with sides a,b,c, the unknown is a triangle, the data are the three lengths a b,c, and the triangle is required to satisfy the condition that its sides have the given lengths a,b,c. If we have to construct a triangle whose altitudes are a,b,c, the unknown- known is an object of the same category as before, the data are the same, but the condition linking the unknown to the data is different. 4. If a "problem to prove" is a mathematical problem of the usual kind, its principal parts are the hypothesis and the conclusion of the theorem which has to be proved or disproved. "If the four sides of a quadrilateral are equal, then the two diagonals are perpendicular to each other." The second part starting with "then" is the conclusion, the first part starting with "if" is the hypothesis. [Not all mathematical theorems can be split naturally into hypothesis and conclusion. Thus, it is scarcely possible to split so the theorem:"There are an infinity of prime numbers." 5. If you wish to solve a "problem to find" you must know, and know very exactly, its principal parts, the unknown, the data, and the condition. Our list contains many questions and suggestions concerned with these parts. What is the unknown? What are the data? what is the condition? Separate the various parts of the condition. Find the connection between the data and the unknown Look at the unknown! And try to think of a familiar problem having the same or a similar unknown keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown, or the data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? 6. If you wish to solve a problem to prove you must know, and know very exactly its principal parts, the hypothesis, and the conclusion. There are useful questions and suggestions concerning these parts which correspond respond to those questions suggestions of our list which are specially adapted to problems to find. What is the hypothesis? what is the conclusion? Separate the various parts of the hypothesis Find the connection between the hypothesis and the conclusion Look at the conclusion! And try to think of a familiar theorem having the same or a similar conclusion. Keep only a part of the hypothesis, drop the other part; is the conclusion still valid? Could you derive some- thing useful for the hypothesis? Gould you think of another hypothesis from which you could easily derive the conclusion? Gould you change the hypothesis, or the conclusion, or both if necessary, so that the new hypothesis and the new conclusion are nearer to each other? Did you use the whole hypothesis? 7. "Problems to find" are more important in elementary mathematics, "problems to prove" more important in advanced mathematics. In the present book, "problems to find" are more emphasised then the other kind. but the author hopes to reestablish the balance in a fuller treatment of the subject.