Adjournment - S1-E6
Character mistake: When Beth is asked to solve a chess puzzle, she answers, "King to Queen 7." The white King is nowhere near the 7th rank. (00:10:38)
Suggested correction: You're looking at the board the wrong way. When he sets it up, he's using his side as the white side so Beth is sitting on the black side. So the white king is on Q6. Since the question is about white winning, she's playing as white and using white's notation, just from the opposite seating position.
Adjournment - S1-E6
Factual error: The song in the car when they were heading to New York was released in 1979, but this scene is set in 1967. (00:01:04)
Suggested correction: The song playing is "Stop Your Sobbing" by The Kinks, which was released in 1964. What you're talking about is The Pretender's cover version from 1979, but that's not the version heard.
Middle Game - S1-E4
Other mistake: Beth walks into the pool and the letters are mirrored. (00:19:53)
Suggested correction: If you're talking about the "Aztec Palace" sign, it's not a mistake. The sign is on top of the building in bright lights so that if you're facing the front of the building, the sign is the correct way. But since there's nothing on the back of the sign, if you look at it from behind (such as in this shot), the letters would be backwards.
Continuity mistake: The first time Beth plays Beltik she starts the game as black but finishes the game as white.
Suggested correction: Maybe you're thinking of a different scene. When she's waiting for Beltik, she's playing white. You even see she's moved her pawn and it's white. When Beltik shows, he moves his black pawn.