what would Polya do?

It turns out there is a whole genre of problem solving frameworks.  Many represent "business paperback" level thinking and clearly wouldn't be much help doing anything difficult.

How to Solve It (1945) is a famous little volume by mathematician George Pólya describing general methods of problem solving.  I particularly like his heuristic approach:

Heuristic	Informal Description	Formal analogue
Analogy	Can you find a problem analogous to your problem and solve that?	Map
Generalization	Can you find a problem more general than your problem?	Generalization
Induction	Can you solve your problem by deriving a generalization from some examples?	Induction
Variation of the Problem	Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?	Search
Auxiliary Problem	Can you find a subproblem or side problem whose solution will help you solve your problem?	Subgoal
Here is a problem related to yours and solved before	Can you find a problem related to yours that has already been solved and use that to solve your problem?	Pattern recognition Pattern matching Reduction
Specialization	Can you find a problem more specialized?	Specialization
Decomposing and Recombining	Can you decompose the problem and "recombine its elements in some new manner"?	Divide and conquer
Working backward	Can you start with the goal and work backwards to something you already know?	Backward chaining
Draw a Figure	Can you draw a picture of the problem?	Diagrammatic Reasoning [28]
Auxiliary Elements	Can you add some new element to your problem to get closer to a solution?	Extension[disambiguation needed]

I like this book so much I have a page of it stuck to my office wall.  Every once in a while I look at it when stuck, and sometimes it inspires a new perspective.