Mind-blowing mathematics experiments We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very counter-intuitive but can be explained using mathematics? I mean things such as the Monty Hall problem. I know it is not exactly an experiment or phenomenon (you could say a thought-experiment), but things along the line of this (so with connections to real life). I have stumbled across this interesting question, and this are the type of phenomena I have in mind. This question however only discusses differential geometry. 

There's the famous Buffon Needle problem: if you drop a needle of length $a$ at random onto a floor made of parallel wooden strips of identical width $l \gt a$, what is the probability that the needle will lie across a line separating two strips? The answer involves $\pi$, and you can actually use this process to experimentally approximate $\pi$.
One that definitely blew my mind when I first heard about it was Khinchin's constant: if you take just about any real number (excluding the rationals and a few others) and write down its (unique) continued fraction expansion,
$$x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + ...}}} = [a_0;a_1,a_2,a_3,...]$$
and then form the geometric mean of the first n coefficients,
$$\left(\,\prod_{k=1}^n a_k\right)^{1/n}$$
you can ask what happens if you do it again with larger and larger n, i.e. in the limit as $n \rightarrow \infty$. The intuitive guess is that this depends on the $a_k$ (meaning, on the $x$ you picked.) Surprisingly, it turns out that this isn't true: basically, no matter what $x$ you pick (i.e. almost always), this infinite product will converge to the same value, which is called Khinchin's constant.
Weirdest of all, while it's really easy to check this numerically (just pick a real number at random and start computing), and while there are several different proofs that this happens almost always, it hasn't been proven for any one particular case: for example, $\pi, \gamma$ and even Khinchin's constant itself all seem to have this property, but no one has a proof.
