Problem theory

Problem theory is the science of specifying a problem. It is a subject and there are books upon the subject (see citations).
A problem must state initial state and conditions (as example only: an equation, initial values, rules) and final state and conditions. A question on the other hand may only allude to a final state. Identifying initial and final state is the first task.
If a problem does not do that, or contains questionable material: the solver receiving the malformed questioned must stop, find who wrote the question, and insure what these are. Many times when mathematitians receive questions in the real world the question (or problem) is wrong and in asking to uncovering a correct question: a satisfying solution is found during the process of asking.
The second task is to specify the difference between the initial and final states. Moving from the initial to final state may mean many things.
One typical book problem type: the initial state is an equation, and the final state is a different equation. This means solving is defined by transforming one equation to another without making either untrue (by the zero property of algebra*) and without loosing solutions. (often books say which transformation is desired)
Another typical book problem type: the initial state is an equation and some states given, the final state is numeric, solving means using the zero property of algebra to find as a solution numbers that satisfy both the initial and final states and conditions. It is essential the that initial equations are in standard form and have the zero property*. It is essential that the initial values be substituted immediately because algebraic variables in standard form are by definition immediate substituted replacements of variables.
Often forgotten in the first task: is solution counting. For example, if the initial state is an equation and has 4 possible solutions and the final state is an equation: it must have the same 4 possible solutions.
I skip entirely a discussion of rules that interact with intermediate solving steps and whether they effect the final state (which must be satisfied despite any such).
I very much suggest at least an introduction to these Problem Theory books. Many math texts give ad-hoc advice on "solving word problems" (problems in some books are confusing because they are poorly written, problem theory shows one how to show this). After studying Problem Theory from a book on the subject there will be absolutely no mystery involved; only clear tasks to move the problem to a next step toward completion.
Last but not least the cost, reason, and payment for solving a problem is important. See Operations Research
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* the zero property is essentially algebra and important to test. Consider "y=x" and observe 0 is not seen, with simple assumptions. One correct equation is "y-x=0". If y-x is not always zero for all y and x, "y=x" is not an equation and algebra methods cannot be used to determine or solve anything. (think: what if x and y are not equatable?)
(citation source of the overview above: Master of Science Mr. Anwari at NVCC Loudoun)
(Please help this article with book citations and more book material. Books that have an introduction to Problem Theory are difficult to find.)

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