Continuity mistake: In one of the first scenes where Nash is being introduced to everyone around the punchbowl, the trees around them have CGI leaves added on since it was shot in January and had to look like early fall. In shots that show the trees' shadows on building walls, all we see is branches. Shouldn't there be leaves?
Continuity mistake: When the student approaches Prof. Nash in the Princeton library, he tells Prof. Nash he has been working on a mathematical theory. When the student hands Prof. Nash what appears to be a binder he does so by holding the binder with his left hand, palm down and fingers on top of the binder. In the immediately subsequent shot of both Prof. Nash and the student, Prof. Nash takes the binder from the student who is now holding the binder palm up, fingers under the binder.
Continuity mistake: The second time Nash sees his old roommate's niece he drops his briefcase to hug her. A couple of shots later as he is walking away, his briefcase is back in his hands, without him having picked it up. Later in the film we see him from another man's point of view, and he is kneeling down, stroking the face of a girl who isn't there. He moves like he's really interacting with the imaginary people, so it can't be that he never really dropped his case.
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