What inspired you to work on... whatever you work on?

Grad students often wonder how people get ideas of things to work on. The usual advice I give is (1) go to talks, (2) read papers, (3) talk to people, (4) follow through on all of the above. That's fine advice as far as it goes; however, I ask my readers to give examples, and I do the same.

Here are examples of how I found things to work on.

1. In ninth grade when I first heard that the fifth degree equation was not solvable I thought I really want to learn why that is true. I went to college and majored in math to find out. I arranged my courses to take Abstract Algebra as soon as possible and did find out and was happy. I could have dropped out right then; however, I heard that there were problems that could not be solved at all so I decided to stick around and learn about those.
2. As an undergrad at SUNY Stonybrook, in a combinatorics class taught by Joel Spencer, he said Infinite combinatorics is easier than finite combinatorics because all of those messy constants go away. This later inspired me to look at Bounded Queries in Recursion Theory. The reasoning was that P vs NP is hard, but maybe an infinite version of it would be easier. And in recursion theory you don't have time or space so I looked at number-of-queries as a complexity measure. I got out several papers and a book in this area; however, I never did solve P vs NP. Oh well. (Michael Sipser had far more mature thoughts along the lines of trying to solve an infinitary version of P vs NP. Some of that lead to Furst-Saxe-Sipser paper on parity not in constant depth poly number of gates.)
3. In grad school I read up on oracles that made P=NP and that made P ≠ NP. I wondered about other classes. Nobody had looked at oracles for NP vs DTIME(2^n). So I worked on that.
4. In grad school I saw Anil Nerode give a a GREAT talk on Recursive Mathematics. I was particularly impressed that we could take some of the objections of the constructivists and the Intuitionists and turn them into well defined math problems. Rather than say This proof is nonconstructive hence its BAD we now say This proof is nonconstructive, can I prove that it has to be so? How nonconstructive is it? Can some variant of it be constructive? This inspired me to work on recursive combinatorics. Right after the talk I marched to the Library and read two papers on Recursive Graph Theory.
5. In grad school I saw the Chandra-Furst-Lipton paper on Multiparty Communication Complexity at STOC. It really intrigued me that there was a matching upper and lower bound that was based on a combinatorial quantity that was not known. This talk got me interested in BOTH Ramsey Theory and Communication complexity!
6. When I got to UMCP in 1985 Carl Smith was working on Inductive Inference. At the time the field did not use much recursion theory beyond the recursion theorem and its variants. Carl knew the field, I knew recursion theory , so we collaborated.
7. I saw Dan Ullman give a talk on Richman games which inspired me to use Games as a way to motivate math and a good topic for high school and college projects.
8. At STOC 2000 I saw a talk on PIR's. I was intrigued by the model and read up on the area.
9. I reviewed Communication Complexity by Kushilevitz and Nisan which inspired me to work in that area.
10. In 2002 (or so) Jacob Fox (now a professor at MIT working on Combinatorics) emailed me asking about applications of Ramsey Theory. This inspired me to make a website of applications of Ramsey Theory. This forced me to learn some new ones. (I still maintain the website but its getting harder to do so since Ramsey Theory has gotten to be a standard tool that people use without much commentary.)
11. I read about the Polynomial Van der Warden Theorem, and that it had an elementary proof, on the web. I immediately tracked it down and read it.
12. Because I maintain the Applications of Ramsey Theory website Jon Katz emailed me a paper that applied Ramsey Theory to Proving that Programs Terminate. At RATLOCC I had heard about the reverse mathematics of the transitive Ramsey Theorem. I wrote an expository article on this application of Ramsey Theory but was also able to add some commentary from what I learned at RATLOCC (and by being in email contact with both the Applications people and the transitive Ramsey People.) NOTE- Once people know that you are interested in XXX they will send you articles on XXX to help you further the interest.
13. (A non-math example) There was an episode of Babylon 5 with the title A tragedy of telepaths. The authors intent was that this term be like A cast of hawks or A pretension of professors or A wedge of swans. I became fascinated with the question: are these really phrases? They ONLY appear in lists of such words. The phrase a tragedy of telepaths isn't even used in the episode! This inspired me to study the (ill defined) question When is a phrase a phrase? Now whenever I hear a new (to me) word or phrase that intrigues me, I type it into a file (which in LaTeX in alphabetical order), do a Google Search on it, and type in some commentary on its usage. I DO NOT look at word lists- this is a purely personal project.
14. In Kindergarten the teacher had a PhD in combinatorics (the job market was tough). Once when we were being bad she made us sit at our desks and Color the numbers {1,...,9} with RED and BLUE such that there is no three numbers equally spaced that are the same color. I was not able to do this (in fact, it can't be done) but I found the problem very interesting. Some other kid did manage to do it by using RED and BLUE on the same number to get PURPLE. The teacher asked that kid to try to do it for {1,...,27}. He cried. (I didn't remember any of this until much later when I learned VDW's theorem.)
15. Help me on my phrase list: Is there a term for a Math Phd in Combinatorics who is teaching Kindergarten and inspires one of her students to study Ramsey Theory?