First of all: nice post. Personally I’ve never considered these puzzles (code golf, project euler, etc.) as tools to make me a better programmer. They are in the same league with sudoku, puzzles, etc. for me. They make me think, do some brain workout. It’s primarily fun, and it never hurts to think a bit more, or a bit outside of the daily routine. I guess it’s not just me to state that these ‘artificial’, primarily mathematcial problems don’t really arise in real-world daily coding tasks.
Apart from that, they can make you a better programmer, enforcing some stricter, more methodological thinking. They also can help in discovering tweaks and tips of a certain language while you’re rewriting your code for better performance. Most of the easier problems are only about this. A simple solution almost instantly comes to your mind, but it’s usually the slowest one So you start to think, start to look for possible bottlenecks (both algorithmical and language-specific).
I have to agree with the point that rookies should not start with such problems. They should do many little “simple” everyday tasks, and after reaching a certain level, they might consider spending some of their free time with such mind games.