There are two verses in the Bible that are contradictory like that. "Do not answer a fool according to his folly, or you yourself will be just like him" is one, the other is "Answer a fool according to his folly, or he will be wise in his own eyes." The two verses are right next to each other (Proverbs 26:4-5), so the juxtaposition must have been intentional. My guess is that putting them together was intended to imply that some problems have no good answer.