> “I don’t know yet enough about the problem to give you even a rough estimate. If you’d like, I can take a day to dig into it and then report back.”

This is my default. Even when I am fairly confident I know what the code does, I want to double check. And it isn't just so I don't short change myself on time. If I tell them X days and deliver on 0.X days several times, they'll start thinking I overestimate everything and cut me off at the knees. It pays to get as accurate as possible.