TITLE: Examples of non-constructive results
QUESTION [11 upvotes]: I'm giving a talk on constructive mathematics, and I'd like some snappy examples of weird things that happen in non-constructive math.  
For example, it would be great if there were some theorem claiming $\neg \forall x. \neg P(x)$, where no $x$ satisfying $P(x)$ were known or knowable.
But other examples are welcome as well.

REPLY [2 votes]: Brouwer's fixed point theorem in 2 dimensions is equivalent the fact that the game of Hex has a winning strategy but no one knows what that strategy is.