The first mathematician orders a beer.
The second mathematician orders two beers.
The third mathematician orders three beers.
"Oh, I see where this is going" says the bartender and pays a quarter to the first mathematician. "This should cover your check, then."
[0]: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
-1/12 beers * $3 / beer = -$0.25
negative -> pay them for taking the beer
The -1/12 result is pretty widely known, but not particularly true. It's called "analytic continuation." There's a nonrigorous proof that's accessible at a highschool level - see [here](https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...)
Delightfully, this comes up in string theory, in computing the dimensionality of spacetime.
The catch is that this isn't valid in what we consider standard mathematics, and you can find many discussions of this online, but this one is fairly short and straightforward: http://curiouscheetah.com/BlogMath/infinity-and-string-theor...
The first mathematician orders a beer.
The second mathematician orders two beers.
The third mathematician orders three beers.
"Oh, I see where this is going" says the bartender and pays a quarter to the first mathematician. "This should cover your check, then."