Proving all the axioms of probability theory using game theory instead of measure theory.<p>A couple of years back I wrote up a short proof of the law of large numbers using game theory. <a href="http://www.marksaroufim.com/2015/02/14/probability-without-measure.html" rel="nofollow">http://www.marksaroufim.com/2015/02/14/probability-without-m...</a><p>All the ideas are inspired by this book by Shafer and Vovk <a href="https://www.amazon.com/Game-Theoretic-Foundations-Probability-Statistics/dp/0470903058/ref=pd_sbs_14_1/131-3331144-0774962?_encoding=UTF8&pd_rd_i=0470903058&pd_rd_r=e456f6ff-380f-476a-b76d-f7568027c793&pd_rd_w=D2GaI&pd_rd_wg=DJ4GT&pf_rd_p=52b7592c-2dc9-4ac6-84d4-4bda6360045e&pf_rd_r=TWKZ5ATFJKT785CHB6HR&psc=1&refRID=TWKZ5ATFJKT785CHB6HR" rel="nofollow">https://www.amazon.com/Game-Theoretic-Foundations-Probabilit...</a>
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.<p>Another favorite of mine is Cantor's diagonal argument for proving the existence of uncountable sets [2].<p>[0] <a href="https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers#Sketch_of_a_proof" rel="nofollow">https://en.wikipedia.org/wiki/Theorem_on_friends_and_strange...</a><p>[1] <a href="https://en.wikipedia.org/wiki/Ramsey%27s_theorem#2-colour_case" rel="nofollow">https://en.wikipedia.org/wiki/Ramsey%27s_theorem#2-colour_ca...</a><p>[2] <a href="https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument" rel="nofollow">https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument</a>
Throw a uniform random dart at the interior of the unit circle. What's its mean distance from the origin?<p>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:<p>* Banach-Tarski<p>* Existence of transcendentals;<p>* Two-colourable <=> no odd cycles;<p>* Graph 3-colouring is NP-Complete;<p>* Wilson's Theorem;<p>... so many more, depending on my mood.