Continuity mistake: When they are crossing the bridge to go back into Boston, after leaving the bar in Cambridge, a Red Line subway train is shown and then in the next shot there are several buildings in the background as they cross the bridge. These buildings are in Boston, but this shot has them driving away from the buildings, which means they would be re-entering Cambridge.
Continuity mistake: In the scene "When Did You Know?", when Will tells Sean that he read his book last night, an aerial shot looking down at them shows Will with his foot up on the table. But two shots later Will has his foot up on the chair instead. The chair in the first shot would also be too far away for Will to reach from his own chair, so that would have been moved in the later shot for him to reach.
Continuity mistake: In the scene where they are all gathered in the bar together Ben is telling the story about his uncle. Throughout the entire scene, Ben has nothing in his hands but as he comes to the end of the story there is a camera shot over his right shoulder showing him holding a cigarette. However, when the next shot moves to in front of him, the cigarette has gone again.
Revealing mistake: When Matt Damon and Minnie Driver are sitting outdoors at Au Bon Pain in Harvard Square, Matt is telling her how he is able to "see" mathematical equations, just like Beethoven was able to "see" the notes on a piano. At one point during the conversation a girl wearing a red backpack (possibly a Harvard student) walks by in the background, stops and looks at the camera, smiles, and then continues on her way.
Continuity mistake: At the end of the film during Will and Sean's last meeting, there is a shot showing both Will and Sean and we see Will with his hands in his lap. It then cuts to a shot over Sean's shoulder and we see Will, but now his hands are raised and his elbows are on the arms of the chair.
Factual error: Neither of the two problems that Will completed on the hallway chalkboard were actually proofs. With a "proof," you start with a specific conclusion and show the work necessary to verify mathematically that the conclusion is correct. With both problems on the hallway chalkboard, however, the objectives were simply to calculate and arrive at the answers without knowing in advance what the answers were.
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Deliberate mistake: Skylar's leaving is edited together with footage of Will sitting in a park watching planes take off - including, presumably, her plane. The sun is directly overhead in his scenes, while hers were filmed near sunset. (Most obvious when you see the very long shadows during the same airport footage which is also used in the deleted scene "Pudge Fisk Card").
Other mistake: When Gerald and Sean go to have lunch Sean asks his friend to bring them 2 sandwiches, and yet doesn't specify what type of sandwiches, nor does his friend ask Gerald what type he would like. He probably knows what type of sandwich Sean likes but he's meeting Gerald for the first time, so obviously he would have no idea what type of sandwich Gerald would want.