1. As the film opens, the protagonist Catherine (Gwyneth Paltrow) is sitting on a sofa well after midnight, watching television as rain spatters on the roof and windows. What day is it? Independence Day Her birthday Easter Sunday Thanksgiving Day

2. A young man -- an aspiring professor of mathematics named Hal Dobbs (Jake Gyllenhaal) -- is in the house, looking through Catherine's father's writings. As he leaves for the night, he invites Catherine to watch him play a gig with his band, and she becomes angry. What does she accuse him of stealing? her sister's photo albums her mother's engagement ring her father's notebooks her wine

3. On the morning of their father's funeral, Catherine's older sister Claire (Hope Davis) arrives in Chicago. She is clearly worried about Catherine. Why? She fears that Catherine has inherited their father's madness as well as his talent. She dislikes Catherine's boyfriend. She is concerned that Catherine's grades will drop. She doesn't think that Catherine eats enough.

4. At the funeral, Catherine makes an unplanned and hostile speech. She begins by asking the mourners where they were during her father's illness, describes his dementia in graphic detail, and ends with the wrenching line "I'm glad he's dead." Why is she so much more resentful than Claire? Because Catherine's childhood was much more difficult, as we learn through flashbacks. Because Catherine has always blamed everyone else for everything. Because it was Catherine who dropped out of college to take care of their father. Because Claire has a much sunnier disposition, and always sees the best in everyone.

5. Claire throws a party at the family home on the night after the funeral. Catherine hates the idea of a party, and retreats to a quiet corner to talk with whom? Claire Professor Bhandari, an old teacher of hers An unnamed theoretical physicist Hal Dobbs

6. The next morning, Catherine gives her mathematician friend the key she wears around her neck, and directs him to a drawer in her father's desk where he finds a proof. She claims she wrote it, but Claire thinks their father wrote it. What is the evidence that their father wrote the proof? It's written in a notebook just like the ones he used, he had a better education, and the handwriting looks just like his. It's written in a notebook just like the ones he used, and every page is dated (a habit of his). The notebook is signed with his name. The handwriting looks just like his, and the doodles in the margins are in his style.

7. What is the evidence that Catherine wrote the proof? At the end of the notebook, there is a note from her father congratulating her on her achievement. She had a better education, he had not produced anything else that was lucid during his illness, and he would never have doodled bunny rabbits in the margins. The notebook is signed with her name. Her handwriting looks like her father's, she had access to the same books and materials that he did, and he had not produced anything else that was lucid during his illness.

8. Hal spends several days with a department's worth of mathematicians, working through and verifying the proof. In the meantime, Claire has extended her stay in Chicago so that she can complete what task? Locating an excellent Chicago mental institution to house Catherine. Completing the registry for her upcoming wedding. Visiting with old friends from high school. Packing up the family house so that she can sell it.

9. Why does Hal conclude that Catherine must have written the proof? He finds another notebook with Catherine's father's unsuccessful attempt at the proof. He notices that some of the i's are dotted with hearts, which is out of character for Catherine's father. The pages aren't dated, in contrast with Catherine's father's undisputed notebooks, and the proof is too "hip" to have been written by the aging mathematician. There are several notes partway through the proof that must have been written by Catherine, like grocery lists and lists of college courses.

10. How does "Proof" end? Catherine sits with Claire on an airplane to New York, brooding over the notebook with her proof. The notebook with the proof lies at the base of Catherine's father's tombstone. A drop of rain falls. Hal works in his office late at night, writing up the proof for publication under Catherine's name. Catherine sits with Hal on the University of Chicago campus, explaining her proof to him line by line.