Continuity mistake: Beltik rolls up his right-hand sleeve. Says, "Hell no" to the offer of a draw. Other side tips King. When they shake hands, Beltik's sleeve is back down, then back up.
Character mistake: When Beth and her mother discuss the agent's fee of 15%, Beth says it is $49.54. However the prize money was $500, the expenses were $172.30, which leaves them $327.70. 15% of $327.70 equals $49.16. (00:10:00)
Factual error: The car accident scene takes place in Kentucky but the state troopers are wearing the uniforms of Pennsylvania state troopers evident by the distinctive keystone arm badge.
Character mistake: The American flag in the gymnasium of Beth's first competition is hung incorrectly. The correct way to hang the flag, if hanging it vertically, is to display the stars in field of blue in upper left corner, not as displayed in the film. (00:31:00)
Continuity mistake: The lady asks her for three packs of Chesterfields, at the store, the shopkeeper gives her just one pack, and she can be seen holding and walking with just one pack. When she returns to the house, she puts three packs of Chesterfields on the kitchen counter. (00:24:37 - 00:26:11)
Factual error: Just before Annette Packer and Beth begin, the host pumps some water out of a cooler. I don't recall seeing that kind of dispenser - usually for coffee or hot water - until many years later.
Factual error: The year is 1967 but the song Classical Gas wasn't out until 1968. Same is true for the song Venus which wasn't released until 1969.
Factual error: At the 1966 Chess Championship held in Las Vegas they show a brown 1967 Mustang.
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.
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.