B.5.QD:1 - Problem frame

Use this when a result, counterexample or newly available construction changes what you could usefully investigate, but the next question is still unclear. A failed conjecture may reveal a missing condition. A successful calculation may supply an operation that makes a previously impractical question approachable. The work is to turn that change into a question you can begin to answer.

First useful move. Recover the step at which the earlier reasoning failed, or the operation that has become available. Ask what it now lets you distinguish, construct, explain or change. For example, a triangle refutes the claim that every connected graph can be divided into two groups with every edge crossing between groups. It also suggests asking for a construction that returns either the division or an obstruction.

The result is a question with an understood answer form, a useful first attempt and a reason to pursue or retain it. The answer may support further theory, a different investigation or an action in the world. Developing a concept or a reusable operation can be that contribution even before its eventual applications are known.

Use an already adequate question directly. If the question is clear and the missing contribution is a way to answer it, C.39 supplies that search. This pattern is useful when the result sought or the question’s conditions themselves need to be developed.