I find value in reading and doing exercises in books about areas of mathematics that I didn't get to explore while in school. I try to do each exercise in each chapter with only the material contained in each book, but I sometimes can't think of a solution in a reasonable amount of time without using outside resources. In short, here's my main question:
When is it appropriate to use outside resources to complete book exercises during self-study?
As background, I am a software developer with a bachelors degree in math.
In terms of technology, I consider using outside resources to solve technical problems as a short-cut. It is more efficient to use a proven statistical library to do computations on a data set, but that won't necessarily teach me what the results actually mean. When I decide to work through a book, I work under the assumption that the book is self-contained, except where it says otherwise in the book itself. I realize that outside resources can accelerate my understanding, but I fear that I may miss the subtle points the exercise intends to emphasize by doing so.
In other words, how can someone ensure that eir learning of a new subject isn't harmed by turning to Google for help?
Not every mathematics book is self-contained, and in particular, many include exercises or problems that require either a good deal of cleverness or knowledge of techniques that are not clearly explained in the book.
On the other hand, the point of the exercises is to gain technical strength and conceptual grasp by attempting to solve them on your own. After trying enough that you are convinced that nothing you are ever likely to come up with on your own, then that is the time to look to outside resources for the critical hint or solution technique. This site is a good resource for those hints or solutions.
Outside material can be helpful in other ways as well.
For example, if you are working your way through Munkries Topology, then the book "Counterexamples in Topology" only provides direct answers to a smattering of exercises, but it does provide examples that help you understand, for example, the sense in which topological spaces obeying different separation axioms are different.