I'd wager most programmers do not particularly solve complex problems. Instead, their applications are relatively simple and leverage existing solutions (libraries, frameworks, etc) to solve the complex problems. For most, programming is simply the application of existing solutions to a specific domain. The meat and potatoes on the other hand are those solving real technical problems.
Hmmm, that's very much what I gather is the philosophy of of MIT's new "CS1" (http://ocw.mit.edu/courses/electrical-engineering-and-comput...) which is very different from its groundbreaking 6.001 back in the early '80s. It's gluing together Python libraries to get robots to do difficult things (e.g. differential equations are required, at least in the early version of the course).
