A. Sentence B is true.

B. Sentence A is false.


If sentence A is true, then B is false, and if B is false then A must also be false. But if sentence A is false, then B must be true, and if B is true, then A must be true. In this case neither sentence talks about itself, but taken together they keep changing the value of the other, so that eventually, we are unable to say whether either sentence is true or false.

That's an old one.