In mathematical logic, this is known as the “decision problem.” Many logical systems are plagued with intractable decision problems. For instance, given a pile of dominos of various sizes, we have no tractable way to decide if we can arrange them to fill a square of a given size. But once an arrangement is proposed, it takes no time at all to verify whether it constitutes a solution.
There may be multiple paths or none at all. Really this is an NP-hard problem.
