Start with a single paper square:
Fold from upper right to lower left (colors added to both sides of the paper for clarity):
Follow up with one more fold:
Voila, a proof that the square root of 2 is irrational.
(Mind you, there is some reasoning involved, but what’s the fun in giving that away? Start with “if the square root of 2 is rational, then there is some isosceles right triangle where the sides are the smallest possible integers.”)