It seems all of the answers assume "You cannot assume there is only one correct answer." However in "...answer in this list is the correct answer..." to my mind the word "the" requires only one correct answer. If it was worded '...a correct answer...' or '...one of the correct answers...' ok sure, there can be more than one. But as it stands there can only be one correct answer so the "correct" answers so far, are based on a false premise.
Regardless of the interpretation it's not a false premise. The uniqueness of the answer is just an assumption that we don't need.
Suppose the question was instead: "What is the unique real number x for which x^3 = 8?" We can guess and verify that x = 2 is a solution. And if we're allowed to assume that this equation has a unique solution, then that's enough. But if we work a little harder, then we don't actually need that assumption: we can prove that x = 2 is the unique solution.
Similarly, in the original problem we could guess and verify that FFFFTF is a solution. And if we're allowed to assume that the solution is unique, then we're done. But again, if we work a little harder, then we don't need that assumption: we can prove that FFFFTF really is the only solution.
