One approach: try to prove it doesn't have a solution. Would solving the problem entail things that obviously aren't true? Or does it entail things that probably aren't true, forcing you to dig farther about situations where they might be? Attempting to prove impossibility inverts the problem, and lets you act as critic on an argument that you want to be proved false.

This is a common approach to mathematical problems, but it's helped me break down more practical situations too.