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.
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.