Character mistake: In his argument with Leonard regarding Superman's ability to fly, Sheldon states that in Superman 1, Lois Lane is two feet off the ground when Superman catches her as she falls from the helicopter. Not so. Superman catches Lois at least ten stories up. Additionally, both Penny and Sheldon say Superman "swoops down" to save Lois. Penny can be forgiven, but Sheldon would know better: Superman flies UP from ground level to catch Lois.
Character mistake: In his conversation with Missy Sheldon makes it clear that his superior intelligence is a result of a random, mutated gene. Since Missy isn't similarly intelligent she obviously isn't carrying this mutated gene (which would be a billion to one shot anyway) so her offspring wouldn't inherit it. Sheldon would know this - his offspring would carry the mutated gene for superior intelligence, Missy's would not. Anyone knowing enough about genetics to use the term 'randomly mutated gene' understands enough to know that the mutated gene would only be expressed in a direct line from the carrier - Sheldon. Also bear in mind he has a model of the DNA molecule in his living room - it is obviously an interest.
Character mistake: When Sheldon arrives at the apartment with his Nintendo 64 in the box, he tells Leonard that they can play Super Mario 64. He then proceeds to say, upon finding a memory card, that they can pick up where he left off. Super Mario 64 does not make use of the memory card for progression, all of that data was saved on the cartridge.
Character mistake: Being the first to play in the "time machine," Leonard says he will set the date for March 10, 1876. Howard says "Good choice. Alexander Graham Bell invents the telephone and calls out for Doctor Watson." Bell's assistant, Thomas A. Watson, was an experienced electrical designer and mechanic, but he did not hold an advanced degree. It is well documented that the first words transmitted over the telephone identified him as Mister Watson.
Character mistake: When Howard finds the stack of paychecks in Sheldon's drawer and wonders why Sheldon hasn't cashed them, Sheldon explains that he's saving them to purchase items that have yet to be invented and he doesn't trust banks. But just keeping them in his drawer doesn't do anything, because checks are voided after a set period of time and they would eventually become worthless.
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Character mistake: After Sheldon has told Barry about the rules of Rock/Paper/Scissors/Lizard/Spock for the first time, Barry then replies "I'm sorry, can you repeat that." When he says "repeat", he correctly pronounces the "r." However, it has already been established throughout the series that Barry pronounces all his Rs like Ws due to his speech impediment.
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Character mistake: When Penny says she smells Chinese food and Sheldon corrects her by telling her she's slipping because it's Thai, Sheldon is eating the food with chopsticks, even though in a previous episode he uses a fork and spoon because that's how Thai people eat their food.
Character mistake: When Howard says they should put the time machine into the elevator in 1-14 "The Nerdvana Annihilation", Leonard states the elevator had been broken for two years, but in this episode, we found out the elevator broke seven years earlier.
Character mistake: Sheldon refers to using the spray bottle as negative reinforcement when it is in fact positive punishment. Reinforcement involves encouraging a behaviour by either adding or removing a stimulus whereas punishment involves discouraging a behaviour by adding or removing a stimulus. Sheldon would know the difference.
Character mistake: When Sheldon is placing his order near the beginning of the episode, he states "In the mid-18th century, King Rama IV of Siam..." King Rama IV was born 18th October 1804 and died 1st October 1868. His reign was during the mid-19th century (1851-1868).
Character mistake: In Amy's lab, when she is complaining to Sheldon how she gave him the simplest tasks, and that they're still not done, Sheldon replies with how Einstein failed math because he wasn't challenged, and that it is the same case with him. Einstein never failed math. Sheldon of all characters should know that.
Character mistake: When Howard is asking Sheldon engineering questions, the first question is "How do you quantify the strength of materials?" Sheldon answers "Young's Modulus," and Howard admits that is correct. But it's not correct - Young's Modulus quantifies a material's rigidity, not strength.