In starting the solution of the puzzle, we went directly to what was ASKED FOR, ignoring the various statements that provided given material and constraints. Ignoring the given material is so important to reliable problem solving that it is necessary to make it a part of the first principle, which is restated as follows:
Unreliable problem solving often can be traced to trying to build a solution out of the given material. As the solution above shows, equations ASK FOR given material when it is needed. Given material must be ignored until the equations DEMAND that it be provided.
In most problem solving situations, knowledge of subject matter is needed to respond to what is ASKED FOR. Then the question, "How would I FIND OUT?" requires a law, definition or principle which provides the instructions for determining the answer to the question asked.
Since the first two questions, "What is ASKED FOR" and "How would I FIND OUT?" are applied recursively as the problem solution evolves, several laws, definitions and principles will generally be needed.
When course content is involved, the ideas are frequently represented by symbols (letters) that represent physical or mathematical quantities. Such symbols usually have specific meaning within the context of the course involved.
In learning any new subject, few things are more important than immediately identifying the meanings of symbols, definitions and laws pertaining to that subject, even though they are not yet fully understood. Those meanings are essential to answering, "How would I FIND OUT?"
When the necessary quantity or expression is identified, one must know what it means. It is not just a symbol written on a piece of paper or on a computer screen. The question, What is it? is therefore added to the second principle, giving:
Asking "What is it?" is a means of focusing on the specific meaning of a symbol and the ideas it represents. A clear answer to "What is it?" forces attention on the concepts being studied, not on the given material.
Working with the components of an equation in systematic left-right, top-down fashion, until all the unknowns are identified, may overcome the usual desire to start putting in the given material.
If you are tempted, don't do it! Some of the given material may never be needed and handling it may simply add unnecessary complications. So we add to the third principle:
Now consider a second puzzle, which may appear frivolous but actually is a good illustration of using the three problem-solving principles without any "scientific" knowledge being required.
A rope over the top of a fence has the same length on each side and weighs one-third of a pound per foot. On one end hangs a monkey holding a banana, and on the other end a weight equal to the weight of the monkey. The banana weighs two ounces per inch. The length of the rope in feet is the same as the age of the monkey, and the weight of the monkey in ounces is as much as the age of the monkey's mother.
The combined ages of the monkey and its mother are 30 years. One-half the weight of the monkey plus the weight of the banana is one fourth the sum of the weights of the rope and the weight.
The monkey's mother is one-half as old as the monkey will be when it is three times as old as its mother was when she was one half as old as the monkey will be when it is as old as its mother will be when she is four times as old as the monkey was when it was twice as old as its mother was when she was one-third as old as the monkey was when it was as old as its mother was when she was three times as old as the monkey was when it was one-fourth as old as it is now.
How long is the banana?
Applying the first problem solving principle, we establish that what is ASKED FOR is the length of the banana. This TELLS us exactly what to look for in applying the second problem solving principle. How to FIND OUT is in the sentence, "The banana weighs two ounces per inch." This provides a starting equation which contains the instructions for solving the puzzle.
We now know that the length of the banana is equal to the total weight of the banana divided by its weight per inch, or
The simplicity of the starting equation is very important. Indeed, among the several starting points a particular problem may have, the simplest true statement that can be made is the starting equation of choice. The beginning point is sometimes so simple that it may be difficult to write it down or even be consciously aware of it!
To provide a clearer picture of the beginning steps of the solution to this puzzle, the steps are now shown without the intervening words:
A schematic of the first five steps of the problem looks like this:
Continuing in this way, you can show that the length of the banana is five and three-fourths inches. To see one complete solution click here.
This puzzle teaches us a number of things about problem solving. Perhaps the most striking lesson is that problems solve themselves.
As shown in the schematic diagram, the first equation is the answer to the question ASKED. The second equation responds to what is ASKED FOR by the first equation. The third equation responds to what is ASKED FOR by the second equation. The responses of the indented group produce an optional intermediate result.
In working this puzzle, "How would I FIND OUT?" is answered by the puzzle (or "problem statement") each time. In a subject matter course, the answers to "How would I FIND OUT?" are normally available from the laws, definitions and principles of that specific subject.
Each successive step is DEMANDED by the previous equation. You begin to feel that
And indeed they do. They really TELL us what to do next. It is not hard to learn to read their messages.
A problem from Algebra, Scott, Foresman, 1984. Return to text.
