Character mistake: When the private investigator shows Castle and Beckett the VIP Liaisons website, "Liaison" is misspelled "Liason." (00:17:40)
Character mistake: In the scene where Castle and Kate are being told that the victim's SIM card is broken and the last two digits of the phone number are missing, Castle says that they have 99 possibilities, which mathematically is wrong. Obviously the writer didn't count the combination 00. The correct number of possibilities is 100. (00:08:40)
Character mistake: Castle claims that they are technically standing on Russian soil inside the consulate. Contrary to popular belief foreign diplomatic missions, while being afforded special privileges (ie. Diplomatic immunity), are not considered sovereign territory of the represented state. Because of that Castle and Beckett are in fact on U.S. soil.
Character mistake: When Beckett is bemoaning losing her job with the government, she says she hasn't been out of work since she was fifteen. But in "Always" at the end of season four, she is suspended by Captain Gates and chooses instead to resign from the force.
Character mistake: The killer's motive for the murder is getting kicked out of the band akin to Beatles drummer Pete Best, and Castle whispers to the camera that bass player Stu Sutcliffe would have been a more appropriate comparison. Sutcliffe quietly quit the Beatles voluntarily, while Best was fired, with his dismissal becoming widely-known public knowledge, so the comparison to Best actually is more applicable.
Character mistake: They find the DeLorean and the licence plate is shown: LFM-26W1. Esposito calls in and says the car needs towing: "'81-'82 DeLorean, New York plates, JLD..." Nothing like the actual licence plate.
Character mistake: Beckett identifies the area given for the Skype call as the whole city of Paris. It's hard to believe that Beckett knows the area of a foreign city by heart, the problem is also that she is wrong. Paris actually has an area of about 105 km² (41 sq mi).
Flowers for Your Grave - S1-E1
Character mistake: At the end of the episode, Castle finds his inspiration again and there's a word processor page fading in/out with text being written. Funnily enough, whoever typed in overlooked something, since there's a sentence that says "Begfore she felt safe." (00:41:30)
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.