How to Solve Problems, by Wayne Wickelgren, presents several strategies for solving problems. The first steps that Wickelgren suggests are similar to the early phases of problem solving that Thinking Mathematically describes. Wickelgren advocates working towards a precise understanding of the question before trying to answer it. In order to achieve this understanding one must identify thegivens, operations and goal of the question. The givens are the information, materials and rules presented in the problem. An operation is an action that the problem solver is allowed to perform on the givens. The goal of the question is the desired result of solving the problem. One may also need to draw inferences from the information in the problem.
In order to achieve a solution, which Wickelgren defines as a sequence of operations the produces the goal expression, one should begin with random trial and error. Systematic trial and error will help to clarify patterns. Next, the problem solver should attempt classificatory trial and error, which requires organizing the actions discovered in earlier trial and error into categories. This strategy is very similar to specialization, as discussed in Thinking Mathematically.
In contrast to Thinking Mathematically, Wickelgren does not organize his problem solving strategies into phases. Instead, he presents several, and the problem solver is left to decide which path to take. The first of these strategies is called "state evaluation and hill climbing". State evaluation is evaluating possible operations to help determine paths to the goal expression. Hill climbing is systematically choosing one of these paths. Another strategy that Wickelgren presents is creating "sub-goals", or breaking the problem into simpler problems.
Wickelgren introduces two strategies that require operations to be performed on the goal, as well as the givens. "Contradiction" is a method of problem solving in which one proves that the goal could not possibly be obtained from the givens. " Working backwards" is an effective strategy for problems that have a uniquely defined goal and for which several givens must be used to derive the goal. By working backwards from the single goal, the problem solver may be directed to the relevant givens.
Similarly to Thinking Mathematically and Polya, Wickelgren emphasizes the usefulness of earlier problems. The relationship of one problem to an earlier problem can be described as analogous, similar (having some common elements), a special case of the earlier problem or a generalization of the earlier problem. Recognizing the relationship between the two problems can be a powerful problem solving strategy.
Essay author: Bronwyn Collins