As stated in Chapter 2, problems contain information concerning givens, actions, and goals. The first and most basic step in problem solving is to represent this information in either symbolic or diagram matic form. Symbolic form refers to the expression of information in words, letters, numbers, mathematical symbols, symbolic logic nota tion, and so on. Diagrammatic form refers to the expression of in formation by a collection of points, lines, angles, figures, directed lines (vectors), matrices, plots of functions, graphs, and the like. Often the same information should be represented using a variety of symbolic or diagrammatic notations. In fact, diagrammatic representation is generally labeled ; for example, points, lines, and cells in a matrix have symbols attached to them in the diagram. Of course, problems are stated originally in some form, often relying heavily upon verbal lan guage. The first step in solving such a problem is to translate from the representation given explicitly or implicitly in the original state ment of the problem to a more adequate representation.
This chapter is concerned with selected topics in the mathematical or precise representation of information in problems. Although precise representation of the information in a problem is the first step to take in trying to solve a problem, I deferred discussing this important topic to this late chapter of the book for two reasons.
First, although some general statements can be made about the representation of information in a large variety of problems, most of the principles of representation are specific to particular problem areas. Effective representation for problems from some area of mathematics, science, or engineering depends upon knowing centuries of conceptual development in the relevant areas of mathematics, science, and en gineering. I doubt that mankind will ever develop a general method for determining what are the useful concepts to define in any particular area. Certainly, no such general principles of how to define good con cepts are presented in this book. The best I can do is to present those types of concepts and the principles for representing them that have proved the most useful in a wide variety of areas of formal problem solving. This is what is done in the present chapter, without any claim to completeness (which would be preposterous) and with only minimal claim to logical organization of the concepts and the principles of mathematical representation.
Second, although some of the principles of mathematical representa tion are reasonably simple and can be communicated to even the most minimally prepared student, some of the principles discussed in the latter half of this chapter are concerned with concepts from various areas of mathematics with which some readers will be unfamiliar. I hope that these readers will profit from the sections on sets, relations, operations, mappings, functions, and real-valued functions of a real variable. However, it seemed wisest to put this material near the end of the book so as not to discourage readers with less mathematical sophistication.
The material in the latter portion of this chapter is really a brief, simple discussion of selected mathematical topics, largely modern algebra and combinatorial mathematics. This material is primaril y intended for students who have some background in these topics i n college, high school, o r grade school new math courses. For such stu dents, these sections are intended as review of the relation of certain mathematical concepts to the general methods of problem solving dis cussed in this book. For students with no background in set theory, modern algebra, and combinatorial mathematics, these sections may be rather hard going and require considerable study. Such students should consult regular mathematics books concerned with these topics,
rather than try to master the material on the basis of the rather brief discussion presented here.
The primary basis for selecting the mathematical concepts discussed in this chapter is their applicabil ity to the puzzle-type problems characteristic of recreational mathematics, which constitute the pri mary example problems in this book. A large subclass of all recreational mathematics problems consists of "insight" problems, where a major difficulty may be to recognize the important concepts for representing the information in the problem.
REPRESENTATION ON PAPER OR IN THE HEAD
This section has a simple message : use pencil and paper extensively when you are trying to solve problems. Of course, the primary repre sentation of information is in your head , but virtually all problems can be solved faster by representing some of the information on paper (or a blackboard or other writing surface) than they can without a written graphic aid. Written representation of information is useful for both verbal symbolic information and visual diagrammatic information. To try to solve problems without using pencil and paper is to subject yourself to an unnecessary handicap. Although an occasional problem may be solved faster purely "in the head," the vast majority of all problems will be more quickly solved by representing information on paper at an earl y stage in working on the problem. No one can say for sure why this is so, but there are at least four plausible reasons. First, writing down the components of a problem focuses your atten tion on the need to give names (symbols, diagrammatic representation) to each of the important concepts in the problem.
Second , it automatically draws your attention to the information stated in the problem as you attempt to represent that information on paper.
Third , as you derive inferences or get to intermediate stages in the solution of the problem, writing aids your memory for these inferences or intermediate stages at later stages in the solution of the problem. After working on a problem for some time, it is easy to forget some of the given information or inferences you drew from the given informa tion, and some of this information may be helpful later. H aving this information written down allows you to use rapid visual scanning to jog your memory for prior concepts and facts that might usefully be combined with the concepts and facts to which you are currently paying attention.
especially difficult to retain as a visual image purely in the mind . Such information is very efficiently represented by means of a table written on paper. For an example of the importance of constructing tables to represent information, see the Smith, Jones, Robinson problem in Chapter 7. Similar conclusions apply to graphs and other figures, which may be difficult to accurately imagine and remember purely mentally, without graphic aids.
Whatever the reason, experience indicates that pencil and paper representation of information is very useful in problem solving. So do not be lazy. Always have pencil and paper ready when you start to work on problems, and make extensive use of them through all stages of problem solving.
DIAGRAMMATIC REPRESENTATION
When a problem in some way involves spatial concepts - points, lines, angles, directions, vectors, surfaces or plane figures, solids, contiguity, connectedness, inside, outside, around - diagrammatic representation may be an extremely useful aid to symbolic representation, whether verbal, logical , or algebraic. Even when the problem does not seem to involve any spatial concepts, it sometimes happens that you can form an analogy between the concepts in the problem and spatial concepts, so that you could draw a diagram that might be of some aid in solving the problem. For example, overlapping circles might be used to repre sent overlapping sets, points to represent elements of a set, and sets of arrows to represent mappings from one set to another.
Verbal symbolic representation is probably somewhat more im portant than visual diagrammatic representation in problem solving and in abstract thinking in general. The communication of the givens, operations, and goals of a problem is largely in verbal symbolic terms. Even when we employ diagrams in the solution of problems, they are usually labeled ; that is, symbol s are attached to the points, lines, and angles. For example, in solving for the lengths of lines or the magni tudes of the angles between lines in geometric figures, we invariably make extensive use of symbols attached to various points, line seg ments, or angles in the diagram (see Fig. 1 0- 1 ).
Furthermore, all the spatial information represented by a diagram like Fig. 1 0- 1 can be represented symbolically without having to employ diagrammatic representation. For example, the spatial informa tion represented in Fig. 1 0- 1 can be represented symbolically as fol lows: lines a, b, and h meet at common vertex B, lines a and d meet at vertex A , lines d, h, and c meet at vertex D, lines b and c meet at