I may have the hammer-solution methodology, but I have a lot of hammers. This generally keeps me from obsessing on one particular solution, whether that be a detriment or advantage (and there are advantages to the obsessive solutioner; if nothing else, he gets good at implimenting that solution).

Now, as for what to do with the two programmers above, is get the first one solving the P=NP problem, and while he's doing that, put a bullet into the back of his head. With that as a warning, teach the other guy to include more frequent error trapping as a second-pass implimentation refactoring, and you'll end up with one trained programmer with the right mix of habits. No one that believes that software can be complete can be allowed to live; its bad for the gene-pool. Mainly because I'd rather manage five programmers who believe that code should be in on time than ten of which five aren't hitting the implimentation deadlines. Retasking two of the five on refactoring for error trapping as the final set of deadlines nears assures that I'll get efficient, clean error-trapping on time and thus is a net win.

(This is why I have so much fun in management courses. Of course, I've been refered to as Torquemada, too.)