Do not answer a fool according to his folly, or you will be like him yourself. Answer a fool according to his folly, or he will be wise in his own eyes (Proverbs 26:4-5).
Some skeptics like to talk about "all those contradictions" in the Bible. And while most of these apparent contradictions can be explained with a bit of research into geography, history, cultural values, and logic -- it's pretty hard to get around the obvious and presumably intentional self-contradiction of Proverbs 26:4-5. "Do not answer a fool according to his folly... Answer a fool according to his folly." So which is it? Am I supposed to answer the fool or not? And why would King Solomon so blatantly contradict himself within two consecutive verses?!?
My interpretation of these contrasting back-to-back statements is that they tell us not so much about how to manage fools as much as they tell us if we should manage fools at all. In effect, these two statements right on top of each other express the time-honored message that, when you're dealing with fools, you're "damned if you do; damned if you don't." Really, the best option of all is to avoid fools all together! There can be no lessons learned -- no moral victories -- when a fool is doing his thing. If you try to correct him, you'll find yourself spinning your wheels, going nowhere, and using your own time and words foolishly. But if you try to ignore him, you'll implicitly reinforce the fool's way of thinking. Either way, you're in trouble.
But what are we supposed to do if we just cannot avoid interacting with a fool? Are we supposed to answer him according to his folly or not? To me, it seems like the parting comment is perhaps the most applicable. The one which would have us ultimately do our best to confront the folly and answer it as best we can. This isn't necessarily because of the chronological arrangement (although I do think it's interesting that they are positioned like that). To me, the greatest indication that an answer is ultimately better than no answer -- hopeless though either one may be -- is found seven verses later, in Proverbs 26:12. There it says, "Do you see a man wise in his own eyes? There is more hope for a fool than for him." And given everything negative that the Proverbs have to say about foolish people, a situation in which someone becomes wise in his own eyes appears to be a very serious situation indeed. So if the only way to keep a fool from being wise in his own eyes is to answer the fool... then I suppose we need to answer the fool.
Don't expect it to go well. Don't expect yourself to get anywhere (except becoming a bit more foolish yourself, in the process of trying to correct his folly). But if your back is against the wall and you don't have any other options, it seems that you've got to go the route that will ensure that no one will become wise his own eyes.