Classes of Problems

There are two classes of problems: those that are considered well defined and others that are considered ill defined.Well-defined problems are those problems whose goals, path to solution, and obstacles to solution are clear based on the information given. For example, the problem of how to calculate the price of a sale item is well defined. You see the original price on the tag, calculate the discount percentage, and subtract this amount from the original price. The solution is a straightforward calculation. In contrast, ill-defined problems are characterized by their lack of a clear path to solution. Such problems often lack a clear problem statement as well, making the task of problem definition and problem representation quite challenging. For example, the problem of how to find a life partner is an ill-defined problem. How do you define “life partner”? What traits should that individual have? Where do you look to find such a person?

Only after considerable work has been done to formulate the problem can an ill-defined problem become tractable. Even at this stage, however, the path to solution may remain fuzzy. Multiple revisions of the problem representation may be necessary in order to find a path to a solution. In contrast to well-defined problems, ill-defined problems can lead to more than one “correct” solution.

The solution process for well-defined problems has been studied extensively, often using algorithms to describe how each step of a problem is solved (e.g., Newell & Simon, 1972). A well-defined problem can

be broken down into a series of smaller problems. The problem may then be solved using a set of recursive operations or algorithms. In contrast, algorithms cannot be used to solve ill-defined problems precisely because the problem cannot be easily defined as a set of smaller components. Before a path to solution is found, ill-defined problems often require a radical change in representation. For example, consider the following problem:

You have a jug full of lemonade and a jug full of iced tea. You simultaneously empty both jugs into one large vat, yet the lemonade remains separate from the iced tea. How could this happen? At first, this puzzle is difficult. You imagine two pitchers of refreshing drinks being poured into a common vessel and wonder how they could not mix. (It is safe to assume that the lemonade and iced tea have similar densities). However, if you change your mental representation of the lemonade and iced tea, you see that frozen drinks could be easily poured into the same vat without mixing. Though the problem itself does not specify the state of the drinks, most people assume that they are liquid, as is usually the case. But this constraint is simply an assumption. Of course, this puzzle is a fairly trivial one. But in life, we often make unwarranted assumptions in our everyday problem solving. Such assumptions can interfere with our ability to discover a novel solution to an ordinary problem.

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