If you hang around people who like to play with maths, eventually they'll start playing tricks with it. And by tricks, we mean they'll use math to do something seemingly impossible with it that makes absolutely no sense. Or, they'll do something that seems to make perfect sense, but produces results that can't possibly be true.
These are mathematical illusions, and in the following hour-long video, Gresham College Professor of Geometry Sarah Hart works through several mathematical proofs that give impossible results before explaining where many of them have gone wrong (at the 48-minute mark). Starting with a highly unlikely proof that reveals the number 1 is equal to 0....
The trick to using math to produce impossible results is to muck up the mathematical logic you're using while you're doing it, which can happen either on purpose or, as happens more often than many like to admit, without realizing it.
And as Professor Hart shows, "once you have 'proved' one false claim, you can prove absolutely any statement at all". Even if it's impossible.