When Miles shows Ben the source code of the robot, Ben looks at them and immediately says this is very good. Actually, even a highly experienced programmer could not understand a source code just by looking at it for 2 seconds, especially the code of a really complex system such as this robot.
When Ben is in the casino playing for the first time, the count is +18. We then see a face card (worth 10) appear and Ben wins. Mickey then asks him what the count is. He says +18, but it's really +17 because a face card has a value of -1.
When Ben is riding his bike on the bridge, a young guy with a lady are passing in the back stage. After this, when a yellow stripped bus passed behind Ben, that young guy and lady disappeared (it seems, they jumped into water) then another lady in white jacket appears who was absent before in the back stage.
When Ben meets with Mickey and the group for the first time, he explains the basic rules of 21 and deals himself a face-up card, then the camera cuts to Ben then back to Mickey and now Mickey's card is face down.
Later in the movie Jill tells Ben the Hard Rock has comped her a suite and asks him if he would like to see it. In the next scene they are in a suite over looking the Bellagio fountains. The Hard Rock is not on the Strip and does not have that view to the Bellagio fountains. The suite is more likely to be in the Planet Hollywood (the old Aladdin) or even Paris.
There are multiple errors in Vegas hotel locations - seeming to be in one hotel when the shot from the window makes it clear they are elsewhere, particularly when visiting a suite said to be at the Hard Rock but clearly the view is from a tall Strip hotel. The Hard Rock Hotel is neither.
In the montage scene showing the "extravagant" lifestyle of the group, the film cuts to scenes at the blackjack table, inside of clubs, etc. It's supposed to happen in or around the Planet Hollywood Casino/Hotel, but many of the locations shown are from Red Rock Casino game room and the club "Cherry". Red Rock is about 12.5 miles northwest of the Strip, surrounded by houses and deserts.
In the beginning of the film, Ben receives a promotion in his job at the menswear store, and a raise to $8 an hour. In Massachusetts in 2008 the minimum wage was (and still is) $8.25 for retail workers.
In the shot of Ben counting cards on the first arrival in Vegas, Ben's tray-table is down. If the plane is as close to arriving as the view from the window shows, tray-tables must be in the "up and locked" position.
The method mentioned as the Newton method is in fact called the Newton-Raphson method because Raphson's contribution is well established (i.e. not something that would be impressive for a student to know) and in fact, the way Raphson approached it was simpler and is the one featured in books today despite the fact it bears both names. It's also well known that Newton developed his method first even though he didn't publish it for quite some time after Raphson published as was Newton's modus operandi. The Newton-Raphson method is also quite basic and is covered in most high school curricula around the world ergo not something that merits a lengthy conversation in class, particularly in MIT.
The two-siblings question posed by Rosa in his classroom is a popular thought problem, which often appears in the first chapter of introductory level probability books or in newspaper puzzle pages. It most certainly would not have had such a stymieing effect on a class of MIT mathematics students.
The notion you require an exceptional mind to count cards is faulty. Any person of average intelligence can gain proficiency in even the very sophisticated systems with sufficient practice however the method used in the film, "Hi-Lo" counting is the most basic system that can be learned very quickly by anyone.
The Monty Hall problem is well known even to lay people and so it is unlikely to stupefy any student of mathematics especially one of Ben's caliber so why would Professor Rosa expect to catch him out with it?
The other members of the team repeatedly make mistakes in tallying the count or even in terms of basic strategy, which they should have definitely mastered, usually so that Ben may show off his "exceptional mind" however since they all play integral roles, they could not function as a team if this were the case with or without Ben as the spotters need to keep the count ready to signal the big player and the big player needs to keep the count so that they can leave when the count is no longer favourable.
In two separate places in the film, the player is hollering for the dealer to draw a "monkey" (a face card or a ten) when, in fact, a monkey would have given the dealer a winning hand. The first instance is when an Asian woman introduces Ben to the slang. The more egregious of the two instances is when Ben (who, by this point, has established his big player credentials) calls out for a monkey twice when either of those times would have given the dealer a winning hand.
All the cards in the casino scenes have rounded corners opposite the index numbers, identifying them as used cards that have since been removed from use on the casino floor and altered so they can never be sneaked back into play.
The goof item below may give away important plot points.
A key clue that makes Cole Williams suspect Ben of being a card counter is that Ben did not double down on 11 when the count was "cool". Basic blackjack strategy dictates always doubling down on 11 (because the odds of making 21 are very high - even casual players know this). The only situation where it would make sense not to double down is if the count was very negative (lots of low cards in the deck). But as the "big player" in the MIT system, Ben would have left the table long before the count became negative.