I enjoy using mathematical puzzles as a teaching tool. To that end, I have started a collection of my favorite puzzles (including the one above) on what I'm calling the Math Puzzle Wiki. It is intended primarily for fellow math teachers as a resource for in class activities and extra credit problems but it might also be interesting to puzzle enthusiasts. You are welcome to add puzzle or improve ones already there.
As an example of how math puzzles can become math research, you might be interested in a recent math talk on counting liars and truth-tellers I presented at the Joint Mathematics Meeting in January 2012. A paper on this topic was recently published in the College Mathematics Journal.