TITLE: Best Fake Proofs? (A M.SE April Fools Day collection)
QUESTION [237 upvotes]: In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen. 
I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's very easy to draw the diagram wrong and have two lines intersect in the wrong place (or intersect when they shouldn't). If someone could find and link this, I would appreciate it very much.

REPLY [6 votes]: We shall prove that $x=e^{\pi/2} \approx 4.8$ is a solution to $x^x=x$.
First, we compute $i^i = e^{i \log i} = e^{i(\pi/2i)} = e^{- \pi/2}$. Next we compute $i^{i^i} = e^{-pi/2 i} = -i$. Then $x=i^{i^{i^i}}=(-i)^i=e^{i\log (-i)}=e^{i(-\pi/2 i)}=e^{\pi/2}$. 
Finally, $x^i=i^{i^{i^{i^i}}}=e^{i \pi/2}=i$.
Raise each side of the previous equation to the $i^{i^i}$th power. We get
$$ x^{i^{i^{i^i}}}={i^{i^{i^i}}}$$
or
$$x^x=x$$
where $x=e^{\pi/2}$.