Cut a solid ball into pieces, reassemble them, and get two identical balls
A famous 1924 theorem says a solid ball can be cut into a finite number of pieces and reassembled, no stretching, no added material, into two solid balls, each exactly the same size as the original. It sounds like it breaks conservation of volume outright. How can cutting and rearranging pieces possibly double the ball?
Reveal the answer
It's true in pure mathematics, though not in the physical world. Stefan Banach and Alfred Tarski proved it using the axiom of choice: the 'pieces' are so infinitely jagged that they have no well-defined volume at all, so ordinary conservation of volume simply doesn't apply to them. No physical knife could ever cut such a piece. Published by Banach and Tarski in 1924, building on Felix Hausdorff's 1914 paradoxical sphere decomposition.