Please help!

I'm trying to find this reincarnation manhwa that I read a while ago. It was your typical fantasy reincarnation, where the fl transmigrates into the body of the villainess. She has purple hair and, ofc, has an awful relationship with her family. Her dad doesn't really like her (I think she died giving birth to fl) but the fl finds a box with letters from her mom and bribes her dad with them. She also finds a less complicated mathmatical solution to making magic circles and everyone is impressed. oh, and from what I saw when reading it, the original fl of the story she transmigrated into is actually a bitch or smth? idk, it didn't have too many chapter when I read it.

thank you!!!