I once read a paperback novel by an author that I can't seem to remember that revolved around a female mathematician. She ends up nearly be killed by her sister by being placed in a kind of learning machine and manage to barely retain her sanity by simulating a solar system in her head for a few days. At some point, she wins a ship in a game of cards and it turns out that the ship has a mirrored ceiling that she sells off to fund... something. She ends up becoming a pilot and is well known for some kind of treatise that she wrote on astrogation.
I recall that the book started on a planet that had a strong clan system and that she was a member of a small clan on the verge of collapse due to financial difficulties.
I read this much while standing around waiting for someone some ten years ago but can't for the life of me remember what it was called or who it was by. I'd estimate that it was published around the late 90's.
Edit: I'd asked a couple different librarians and spent hours trying to come up with search queries and never even got close and you guys nail it within a day.
Answer
I think it's one of the books in Sharon Lee & Steve Miller's Liaden Universe — Scout's Progress.
From the blurb:
All of her life, Aelliana Caylon has lived by the rules of her overbearing brother, the head of the Caylon family. Though she is a brilliant mathematician, he has convinced her that she has no worth beyond what value she might have in an arranged marriage.
Then, on a dare, she plays a game of chance—and wins a starship. It is her way to escape her home, her planet, her drab life—if she can qualify as a pilot.
I think the learning program trap is near the end of the novel, you can see the preview on Baen Scout's Progress
Also wikipedia has entry about the entire series, the Liaden universe.
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