It's impossible for the Barber to exist. That's why it's a paradox.
My favourite paradox is the Unexpected Test. It goes like this:
On Monday, a teacher announces that there will be a test some time this week. He will not tell his students on which day the test will occur, so they will not be able to just spend the night before studying. The test will be sprung upon them in a surprise attack one morning, between Tuesday and Friday, inclusive.
One student of sharp intellect thinks she's figured something out. She tells her classmate that the test couldn't possibly be on Friday, because if the test hadn't happened up till then, everyone would know as soon as it was clear the test wouldn't happen Thursday morning. Everyone could spend all Thurday afternoon and night studying for the test they knew must be tomorrow. So that rules out Friday, the students agreed. After some contemplation, the other student declares that the unexpected test cannot happen at all. His friend asks his to explain. He says that since they have ruled out Friday, Thurday is the last possible day on which the test can occur. But actually they can rule out that possibility the same way they just did with Friday. This line of thought works the same with Wednesday and Tuesday. The two students decide to keep their discovery to themselves and don't bother studying at all.
Then BAM! the test happens on Wednesday morning. They didn't expect it.
So where was the flaw in the students' reasoning?
The
book I read this in says the problem is just insoluble. But that's not true. I figured it out. Several times actually, I keep forgetting >.>