A single sentence that can't be true and can't be false
"This statement is false" — call it true, and it's lying; call it false, and it's telling the truth. First posed by Eubulides of Miletus in the 4th century BC, the liar paradox drove two and a half millennia of work on truth and self-reference, from medieval "insolubilia" to Tarski's 20th-century hierarchy of languages.