Moments ago, @dandersod tweeted about an interesting, but non-intuitive probability problem. There are plenty in this genre such as the Monty Hall Problem and many in other areas like the uncountability of irrationals, area/perimeter relationships (@CmonMattTHINK), etc.
I think these problems are like lifting the wolverine's lip to show the teeth. Some kids will be like, "Wow! Those are sharp! So cool! I wonder what that wolverine would do to a rabbit..." Others will just be, "AHHH! Run away! Don't ever make me go near that thing ever again!"
These kinds of problems really separate those who end up loving math from those who do not. I present one or a dozen of these throughout the school year to my various math classes. Some of the kids LOVE it and try to figure out why it is and will read up on it when they go home and have all sorts of questions and really work to pay attention to the "proof." Others will just throw their hands up and say, "See! This is why I hate math!"
Basically, it polarizes the classroom. Those who enjoy thinking this way will get excited and encouraged to find out more. Those who already hate it will have fodder for their fears.
So, I guess what I'm asking here is: Is it a good idea to put forth these "paradoxical" problems in class?