The problem presented a Request for information (What is ASKED FOR); the solver made a Response (How would I FIND OUT?), and produced a Result.
A physicist might refer to this process as "the hydrogen atom of problem solving." Just as understanding the hydrogen atom provided the basis for developing the analysis of more complex atomic and molecular systems, the problem-solving model illustrated here forms the building block to develop solutions of more complex problems.
These three principles form a reliable approach to problem solving. They also unify the process of problem solving with other communication processes. For example, in expository writing an essay begins with a thesis sentence which informs the reader of the intent, or goal, of the essay.
The thesis sentence that establishes what the essay is about is analogous to the starting equation in a problem solution. An essay is developed in paragraphs. Typically, a paragraph is introduced by a topic sentence. This structure corresponds to the successive indentations in the solution structure above.
In both writing and public speaking the concept of "opening, body and close" provides the rules which are fundamental to effective communication. (Fowler 1983) Problem solving entails precise communication involving the process of "opening, body, and close."
In the apples puzzle solution, the first equation forms the opening. The indented equations are "paragraphs" of the body of the solution. The answer is the close.
In this simple puzzle individual paragraphs have not been closed; that is, the conclusions have not been shown for each paragraph. This can readily be done when desired. One example of an indented structure that has been closed is shown below.
Sometimes one needs the additional information that can be obtained by closing an intermediate indented step.
Major topics, subheadings, and sub-subheads are all quite evident when you look at outlines, flow charts, hierarchical computer files, bulleted memos, instruction manuals, the table of contents of a book and many other written communications. The indented structure shown here is the same as that used so successfully in computer programming, where indented and nested structures organize the parts and subparts of a computer program.
The reason this structural form has been so widely adopted is its proven effectiveness as a logical approach to assembling and using knowledge. It is a basic factor in critical thinking. It is actually quite surprising that the ideas are missing in published solutions to examples and problems in typical introductory texts in mathematics, physics and other subjects.
The same structure organizes a problem solution quite effectively. Each indent level is a subproblem of the previous outdented level. Just as in any outline, there may be several subordinate indent levels and there may be several items of the same indent level under any superior level.
The solution will always return to the primary outdented (superior) level when all of the subordinate levels have been satisfied.
The three problem solving principles and the indented presentation of the solution provide the control needed for reliable problem solving.
H. Ramsey Fowler, The Little, Brown Handbook, Second Edition, Little, Brown and Company, 1983. Return to text.
