What is your favorite mathematical proof?

Proving all the axioms of probability theory using game theory instead of measure theory.

A couple of years back I wrote up a short proof of the law of large numbers using game theory. http://www.marksaroufim.com/2015/02/14/probability-without-m...

All the ideas are inspired by this book by Shafer and Vovk https://www.amazon.com/Game-Theoretic-Foundations-Probabilit...

The proof of the Theorem on friends and strangers [0] from Ramsey Theory, which is a special case of Ramsey's theorem [1]. I like it because it is a fun proof to show people to demonstrate a few different proof techniques while remaining very simple. You can draw it out on a napkin and even people who don't usually feel that they are mathematically inclined can follow along.

Another favorite of mine is Cantor's diagonal argument for proving the existence of uncountable sets [2].

[0] https://en.wikipedia.org/wiki/Theorem_on_friends_and_strange...

[1] https://en.wikipedia.org/wiki/Ramsey%27s_theorem#2-colour_ca...

[2] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Throw a uniform random dart at the interior of the unit circle. What's its mean distance from the origin?

Instead of integrating, approximate the circle with a regular n-gon and use the centroids of the n isosceles triangles connecting the polygon's vertices to the origin.

The proof that 1 is equal to 2. It has a fatal flaw, but it is fun to show people. In my experience, people who are active University students will figure it out. Others go OMG WTF!

Oh, several:

* Banach-Tarski

* Existence of transcendentals;

* Two-colourable <=> no odd cycles;

* Graph 3-colouring is NP-Complete;

* Wilson's Theorem;

... so many more, depending on my mood.

Showing that e^ix = cos x + i sin x