When in court for his assault on a police officer, Will begins to quote Henry Ward Beecher's "Prayers from the Plymouth Pulpit" and dates it to 1887. It was actually published in 1867. Henry Ward Beecher died in 1887.
When Will and Skylar are at the outdoor cafe near Harvard Square talking about the complicity of Organic Chemistry, her front shots show her hair on her shoulders, and the rear shots show her hair in the middle of her back.
When Will burns his proof of a problem and Gerald puts it out, the amount of burn damage changes from almost completely destroyed (when Gerald first extinguishes the flames) to minimally burnt (near the end of the scene), and various other degrees in between.
When Sean is packing his suitcase at the end of the movie, the window to his left has the curtains closed. But in the very next shot, the curtains are open as the camera is looking at Sean's left side.
The handwriting and layout of the first math problem on the hallway chalkboard changes between the scene when Will first sees it, and the scene during the alumni reunion when Lambeau and the students are looking at the solution that Will had (anonymously) put up on the board.
When Chuckie is having the conversation with Will about how he wishes one day Will wouldn't be home when he comes to pick him up, he finishes the scene with a sip of beer from a can. The can is in his right hand as he takes the sip, but in the next shot as he lowers the can from his lips, it has miraculously jumped to his left hand.
In an early scene Will states that he works as a janitor at MIT because it was part of his probation work placement program. In a later scene Lambeau questions Will accusingly about why he would ever seek out a job at MIT in the first place. This may be a misunderstanding between the characters to demonstrate Lambeau's ignorance towards Will, however it also leaves a lot of open ended questions about whether or not this was a deliberate path that Will chose or a circumstance of fate.
The position of the mop handle when Professor Lambeau catches Will solving the formula. From Will's perspective the handle is always leaning against the wall; from the professor's perspective it is in the air.
When Will and Sean are discussing Will's record and how they were both physically abused as kids, Sean's cardigan sweater goes from hanging near his shoulders to being pulled up with his shirt collar tucked underneath and back again.
At the end after Will leaves and Chuckie goes to pick him up and realizes he's not there, while he's walking back to the car he tells the other guys that Will's not there. However, it can be seen from his lip movements that this is not what he actually says.
In the novelty shop, as Skylar delivers the line "in the end, my brain is going to be worth $250,000", she picks up the "magic wand" with her right hand. Immediately after, when the camera shows her from the front, it is in her left hand.
Near the ending during the scene where Chuckie is at Will's house to pick him up, as Morgan gets out of the back seat of the car to change and sit in Will's old seat, while he's climbing into the passenger seat at the bottom left hand corner of the screen you can see a crew member with a red hat on watching from the sidelines.
When Professor Lambeau is interrupted at the alumni cocktail reception, the alumni wearing the red jackets are all too young to be wearing the he "Artillery" Red jacket, which is the MIT 50-year alumnus jacket.
The U.S. Army general who sits next to will during his interview at the NSA has several things wrong with his uniform; he's missing the thick black band that generals wear on their lower sleeve, he has some non-US decoration under his name badge that shouldn't be there, he wears Infantry branch insignia (crossed rifles) on his lapels which a general shouldn't wear and his cap has a braided 'chin strap' (above the visor/peak) which is wrong for any US army cap.
After the verbal confrontation in the bar, the gang leaves and is passing by the restaurant where the blond college boy, Clark, is now sitting with his friends, Will slaps a paper with the Skylar's phone number (555-1294) against the glass, saying "I got her number. How do you like them apples?" In the United States of America, there is no 555 telephone exchange.
A central plot point of the film is that Will is not yet 21 years old. That means that in Massachusetts, Will would not be old enough to purchase alcohol on his own or to enter a bar. He may have a fake ID, but this detail is not addressed in the film because in the one scene where they are entering the Harvard bar where Will first meets Skylar, the doorman is a friend of Chuckie's who let's them in without carding them. It is also not explained how Will entered the pub in the later scene where Skylar tells the joke about Paddy's blow job and they are all sitting around drinking pints of beer. One explanation is that Will is a local boy from a rough neighborhood and he knows enough people that they let him into the bar, however, Will's underage drinking is never fully addressed in the film.
When Will is seen driving to Stanford during the closing credits, the sign on a bridge over the highway says "Prospect St., Stockbridge", which points to him being on Rt. 90 west (MassPike). This means Will was taking an 18-mile/1-hour detour through upstate NY and Cleveland, Ohio, as the faster way to go (according to Google Maps) is via Rt. 84 south through Connecticut.
However, prior to Google Maps (and indeed even today), most people driving from Boston to San Francisco take the direct route of I-90 to Chicago and then I-80 from there. In spite of Google Maps, due to higher speed limits, Will's route is actually the faster route.
While at the race track, Skylar, while rooting on her dog, mentions that it is winning the race. However, the camera cuts to the race and the "2" dog overtakes the leader and wins. She then states how happy she is that she won.
Earlier in the film, Chuckie expresses his wish to come to Will's door, to find that he has left without notice. This happens at the end, however, the group has already given Will his car, eliminating the need for Chuckie to come pick him up.
When Sean tells Will about how his wife used to fart all the time, even in her sleep, everybody is laughing. Even the cameraman. The camera starts bobbing up and down at two very funny moments during that dialogue.
In the scene where Skylar and Will are sitting at a table with the chess players behind them, the white noise in the background of the shot lessens for Will's line "and I can't play the piano" and the intensifies again afterwards. If you listen closely, you can tell that the one line is a voice over.
At the end of the movie when Chuckie goes to pick up Will, he's wearing short sleeves and shorts. It's probably August or September and the trees still have their leaves. However, it's cold enough that you can see Chuckie's breath several times.