There are a number of ways you can approach any problem. A lot depends on the area you are talking about but you haven't provided anything to narrow the scope.

There are a number of ways you can approach any problem. A lot depends on the area you are talking about but you haven't provided anything to narrow the scope. For example, Logic Mathematical Inference and a few other areas have formal methods which you can use to identify properties which can then be used towards or against proving something. Computation for example has System's Properties that are required (such as determinism, time invariance, and step), and certain problems are derived from that which dictate some of the fundamental limits of computation. When you know what the limits are, you know when you start pushing up against impossible problems. Sometimes its useful to think of impossible problems as possible, and try to build a bridge, but these often require significant background knowledge and experience to avoid circular thinking of self reference.