"An induction argument can be used to show that once the guest has made her announcement all individuals with red hats will eventually leave the table. This contradicts one's feeling that nothing should continue to happen; after all, apparently the guest has not provided any new information."
--Mathematical Fallacies, Flaws, and Flimflam
After I thought and thought about this for way too long, the answer just popped into my head and it was so freaking simple I was annoyed at myself. Induction is kind of cool.

