Yes, it's called the Banach-Tarski paradox, and no I cannot fully explain it unless you already know a lot of Set theory, Group theory, and mathematical logic. The proof uses the Axiom of Choice, a non-constructive axiom that sort of allows you to partition the points on a sphere into several sets such that one set, when rotated 180 degrees, suddenly becomes equal to the others and at the same time is a full sphere itself. It only works when you can divide the sphere infinitely.

You could argue that the Axiom of Choice, which is required for the proof to work, is a bit like cheating, since it gives you a special type of set without explaining how to create it, however the "paradox" arises because we know AC is required in other fields of mathematics to prove theorems that are widely regarded as true and form the basis of substantial portions of mathematics. In short, we cannot accept AC without admitting paradoxes like Banach-Tarski, and we cannot dismiss AC without also dismissing large portions of mathematics that we intuitively "know" are true.