The Banach–Tarski Theorem
Stefan Banach and Alfred Tarski · 1924
The puzzle
A solid ball can be partitioned into five pieces and reassembled, by rotation and translation alone, into two balls each the size of the original.
Note
The pieces are not solids but pathological non-measurable point sets. They exist only because the axiom of choice lets you select uncountably many points without a rule. The theorem isn’t against intuition; it is about what kinds of sets can exist if you accept choice. Some set theorists take it as evidence that choice should be rejected or weakened; most accept it as the price of a powerful axiom and note that the construction does not survive a measure restriction. You cannot perform the trick on real apples; the cuts are not the kind of cuts that physical matter admits. But the apples-are-different argument has not satisfied anyone who expected mathematical sets to behave decently. The theorem stands as a permanent footnote to the foundations.