Ask HN: What is your favorite mathematical proof? - anant90
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formalsystem
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...](http://www.marksaroufim.com/2015/02/14/probability-without-
measure.html)

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

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eindiran
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...](https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers#Sketch_of_a_proof)

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

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

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SamReidHughes
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.

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sloaken
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!

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ColinWright
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.

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

