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State Evaluation and Hill Climbing. Working Backward. Relations Between Problems. Topics in Mathematical Representation. Problems from Mathematics Science and Engineering. Problem Theory. How to Solve Mathematical Problems Dover Books on Mathematics Series Dover books on mathematical and logical puzzles, cryptography and word recreations Dover books on mathematical and word recreations. Also, from clue 3, we know that E is not married to I and, from clue 7, that B is not married to G. Otherwise, Mr. We obtain Figure 1. But now we seem to be stuck. It appears that we have not made full use of clue 7; but how can we use it?

Since, in Figure 1. Let us first marry Georgette off to Mr. Then, from clue 7, Mr. Dunstan owns the pony that is named after Mr. But we know that Mr. We thus reach a contradiction, and so this marriage does not work, and perhaps never could have:. Note that when an assumption logically leads to a contradiction that is, something is shown to be true that has previously been seen to be false, or vice verso , then the assumption must be incorrect.

The item assumed to be true must therefore be false. Using clue 7 again, Mr.

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Elmsby owns the pony named after Mr. But Mr. It follows that:. As a final step in the solution, regardless of which approach was used, the original problem should be checked to verify that the conclusions reached are consistent with the given facts. In our two approaches to the solution of Problem 1. There are other useful visual aids. We would then place a check in, say, the F-row, J-column if Francine is the name of the horse that belongs to the man who is married to Jasmine.

This figure tells us that Mr. Alloway is married to Francine and owns Georgette, Mr. Bennington does not own Inez, and Mr. Dunstan owns Francine but we do not yet know to whom he is married. In general, it is difficult, if not impossible, to know in advance exactly what type of visual aid will be most helpful. Sometimes, none is needed; other times, more than one is necessary. You might start out with one aid and realize that another is better.

The best guideline seems to be to try to use the chart or other aid that will incorporate as much of the given information as possible, in the least complicated way possible. However, if you wish to present your solution to others, exhibiting a completed chart is unconvincing. They will want an explanation of the method of solution. You can do this by numbering the steps or clues and using these numbers for reference or as explanation of an entry, as we did in Figure 1.

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This is a good idea even when you do not intend to present your solution to anyone else, as it makes it easier for you to retrace your steps in the event that you find a mistake or want to check your work. An illustration of the technique is coming up soon. It is also important to be critical of your own work before you present a solution to someone else.

If you are not fully convinced by your argument, it is not likely that anyone else will be convinced. As a first step, it is important to understand exactly what information is given and what is to be found. It is also important to decide what information is relevant. Sometimes, restating the problem will make these tasks easier. Often, it is useful either to make a list of the given information or to use a chart, diagram, or other visual aid.

After all the given information is listed or charted, add to it all the information that you can logically deduce from what is given. Continue to add to the list deductively, until either the desired information is obtained or an impasse is reached—that is, no further conclusions seem possible but the desired information has not yet been obtained.

Before you decide that an impasse has been reached, it is often a good idea to run through all the given clues once again, bearing in mind what has been found so far. Sometimes this leads to additional conclusions that had been overlooked. In the event of an impasse, choose some aspect of the problem for which only a finite preferably small number of nonoverlapping alternatives are possible.

Arbitrarily assume any one of these alternatives to be true; add it temporarily to the list or chart of facts that are known; and see what conclusions you can logically draw. If a contradiction is reached, then conclude that the alternative assumed is, in fact, not correct and eliminate it as a possible alternative. Also, add the fact that the alternative is not correct to the list of things that are known. This fact may now be used in further arguments.

What happens, though, if an assumption does not lead to a contradiction? It may, for example, lead to an answer. In this case, you cannot yet conclude that this is the complete solution to the problem. Each alternative assumption must also be investigated: Every possible case must be considered. If all these other assumptions lead either to contradictions or to the same answer found originally, then you can conclude that this answer is the only solution to the problem.

However, if some alternatives lead to different answers, it follows that either the problem has more than one solution—and this should be noted—or that the problem is not well posed. If the wording of the problem indicates that there should be a unique solution and if, in fact, there can be more than one solution, then the problem is not well posed.

If a problem does have more than one solution, the truly inquisitive mind would not be satisfied until all solutions are at least categorized, if not worked out in detail. Since you must always return eventually to the point at which you make an assumption—either because the assumption leads to a contradiction or to consider other alternatives—it is a good idea to enter that assumption in your visual aid with a pencil, preferably using a different color from the one you started with.

Use the same pencil to note down all conclusions that are based on the particular assumption. Then, when you do return to the point of making the assumption, you will easily be able to eliminate all information based on it, leaving only what was known before the assumption was made. It is possible that after you make an assumption, you still obtain neither an answer nor a contradiction—you still face an impasse. In this case, you may have to make a secondary assumption.

We will consider this possibility in more detail in the solution of Problem 1. Each solution you obtain should be shown to satisfy completely the conditions of the original problem. Sometimes an answer appears when you use part but not all of the given information. It is possible that this answer contradicts some of the other conditions. This is the reason a final check is important, especially if more than one answer has been obtained. A diagrammatic aid helpful in the solution of this problem is just a simple picture: the dyer is seated somewhere.

Call the chair in which he is seated chair 1 and number the other chairs in the clockwise direction.


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By clue 2, Mr. Hosier is in chair 4 Figure 1.

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How can we use the other information? We can chart the numbers of the seats that could possibly be occupied by each of the people in clues 3, 4, and 5 Figure 1. There are only two possibilities left for Mr. Dyer: seat 2 or seat 3. Case I Assume he is in seat 3, which means that the glover is in seat 4. In this case, we can eliminate some more seating possibilities Figure 1.

We have not yet arrived at a solution or a contradiction. However, at this point, we see that there are only two possibilities for Mr. Baker: He is in seat 2 or in seat 5. We consider these as subcases, that is, as alternative secondary assumptions. Then, the baker is in seat 5, and Mr. Farmer must be in seat 1 by Figure 1.

Baker is assumed to be in seat 2 , with the farmer in seat 2. This gives us Figure 1. Does this complete the solution of the problem? Not yet. Our conclusion has been based on two assumptions. We must still investigate the alternatives. We begin with the alternative to our secondary assumption. Case Ib Assume that Mr. Baker is in seat 5.

Remember that we are still working under the primary assumption that Mr. Dyer is in seat 3. Then, the baker must be in seat 3, and the farmer must therefore be in seat 2, with Mr. Farmer in seat 1. This gives the seating arrangement shown in Figure 1. Note that although the seating arrangements in Figures 1. Farmer is the dyer. We are still not yet satisfied. We must return to consider the alternative to our original assumption.

It follows that the glover is in seat 3 see Figure 1. Again we can use this information to eliminate some of the possibilities from Figure 1. Specifically, Mr. Baker must be in seat 1, with the baker in seat 4 see Figure 1. Since the dyer is also in seat 1, this seems to give us a different answer—that Mr. Baker is the dyer. However, we must follow through to make sure that this situation is consistent with the given information. This completes the solution of the problem. We have found that two of the possible seating arrangements Figures 1.

In both, the dyer is Mr. Farmer; and so that is the answer. Note that we could also conclude that the glover is Mr. Hosier, since that is the case in both diagrams; but we cannot identify the occupations of the other three gentlemen. In the solution of Problem 1. However, there are many problems in which this technique is unnecessary—the solution may be deduced directly from the given information, without analysis by cases. On the other hand, in a problem in which we have to assume something, even after we make that assumption we are not always able to proceed directly to a solution; we may still have an impasse.

We may then have to consider subcases and make a secondary assumption, possibly followed by further assumptions. Whenever we do so, however, we must remember to consider each possible alternative to every assumption we make. This procedure, of considering cases and subcases, is sometimes referred to as case analysis. To help keep track of these assumptions or cases and their alternatives, a diagram can be helpful. For example, in Problem 1.

Hosier in seat 4, we saw in Figure 1. Dyer, three possibilities 1, 2, and 5 for the seat occupied by Mr. Baker, and three possibilities 1, 2, and 3 for the seat occupied by Mr. We can indicate this diagrammatically, as in Figure 1. This figure is a tree diagram, so called because it branches out as a tree does.

The points at which the branching takes place are called nodes or branch points.

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Each line segment connecting two nodes is called a branch, and a sequence of branches connecting the starting point of the tree to a terminal point is called a path. Each path indicates a case to be considered; the whole tree indicates all possible cases. The tree in Figure 1. Dyer, Mr. Baker, and Mr. Farmer could be seated. We read the cases in the tree in the following manner:. Case 1 Mr. Dyer is in seat 2, Mr. Baker in seat 1, and Mr. This case cannot actually occur. Case 2 Mr. Farmer in seat 2.

This case cannot occur either. Case 4 Mr. Baker in seat 2 again an impossibility , and Mr. Obviously, many of the branches in the tree do not represent alternatives that can actually occur. Cases in which two of the three people occupy the same seat are eliminated immediately. In the remaining cases, the locations of Mr. Farmer determine the locations of the glover, the baker, and the farmer respectively.

In five of the seven cases, we again find two different people occupying the same seat. For each path leading to a contradiction, the people occupying the same seat are indicated at the right.


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The two cases that remain correspond to the two solutions that we found when we originally solved the problem. In both, Mr.

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An alternative tree diagram that more closely describes the procedure we actually followed in solving Problem 1. Many times we are interested in knowing the number of cases that could possibly arise in a problem, that is, how many paths there would be in a tree diagram.

Note that this number is the same as the number of terminal branches. In the original tree diagram Figure 1. Each of these splits into three branches, each of which in turn splits into three others. The answer lies in the Multiplication Principle. The principle says that, if a tree has m primary branches, each of which splits into n secondary branches, then there are mn different paths through the tree Figure 1. Atlantic 3. This classic undergraduate text by an eminent educator acquaints students with the fundamental concepts and methods of mathematics.

In addition to introducing many noteworthy historical figures from the eighteenth through the mid-twentieth centuries, the book examines the axiomatic method, set theory, infinite sets, the linear continuum and the real number system, and groups. Additional topics include the Frege-Russell thesis, intuitionism, formal systems, mathematical logic, and the cultural setting of mathematics.

Students and teachers will find that this elegant treatment covers a vast amount of material in a single reasonably concise and readable volume. Each chapter concludes with a set of problems and a list of suggested readings.