Entscheidungsproblem
The Entscheidungsproblem (German, "decision problem") is a famous problem of mathematics. David Hilbert formulated the problem in 1928: Is there an algorithm that will take a formal language, and a logical statement in that language, and that will output "True" or "False", depending on the truth value of the statement? The algorithm does not tell how it reaches the answer, nor prove it, as long as the answer is always correct.