Nice writeup, certainly helps in organizing your thoughts.

It reminded me of the Feynman method a bit though :)

1. Write down the problem.

2. Think real hard.

3. Write down the solution.

More seriously, the classic book by Gy. Polya "How to solve it" is also very useful:

http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d... (non-affiliate)

http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d... (affiliate)

May I suggest a couple of books?

How to Solve It by George Pólya's 1945 http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d...

and

The Art of Doing Science and Engineering Learning to Learn by Richard W.Hamming http://www.amazon.com/gp/product/9056995014/ref=olp_product_...

Great books, and they will introduce you to learn the mathematical thinking, so you can deduce the patterns yourself. Great for math and hack I think.

Another great book is "How to Solve It" by Polya (http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d...) as it gives a heuristic approach to solving problems.