One of my favourite ones is the unexpected tiger (although it has a number of different names.)

A guy shows you a certain number of doors, let's say five, and behind one of the doors they have put a tiger, saying that "the door which the tiger is behind will be a total surprise."

Suppose that you open the doors in a given sequence, and that the first four open with no tiger. The fifth door then obviously has the tiger, therefore it wouldn't be a surprise. Hence the tiger is definitely NOT behind the fifth door.

Now you have four doors left. If you open those in a particular order then after you've opened three doors, you know the tiger can't be behind the fourth, because it wouldn't be a surprise.

And so on down to one door, where because you've already ruled out the last four doors, it obviously can't be behind the first one as it wouldn't be a surprise.

Therefore there is no tiger.