Jason Rosenhouse has just put online the first draft of the first chapter of his upcoming book on the Monty Hall Problem! This is a famous probability problem that is notorious for tripping up a lot of smart people (even mathematicians) and stirring quite a bit of controversy. I explained the puzzle myself some months ago, but Rosenhouse does it much better.
I thought the history of the problem was especially interesting. I've known about the problem itself for a long time, but I never knew about the story about Marilyn vos Savant answering the problem in Parade. It's rather interesting to see how a simple math problem causes such a furor. Rosenhouse suggests that people don't recognize it as math, and therefore forget to give their socially programmed "Oh, I'm no good at math." response.
Other puzzles that tend to provoke a lot of arguments? There's the problem where you fill a container (usually an "urn") with an infinite number of balls. There's also one where you're asked if 3.99999... is equal to 4. There's also a number of problems where you're asked to resolve a paradox of some sort. Oh, and plenty of other probability problems!