I usually got a bit of help from the math teacher when I was doing the really hard problems. She would give me some hints. But the last one one the page-- I figured that one out all by myself!

She did not tell me where she was from. But based on her accent, the way she dressed, and the way she styled her hair, I figured out that she was from Brooklyn.

There are a lot of options given by LEO for "to figure out" but I do not know the best way to say this idionatically for the contexts given in the examples above.