When Beth is shaking hands with the man from the chess club, the camera changes angles and their hands are now both at their sides.
Mr. Ganz says there are a dozen members in the high school chess club and proposes that Beth play a simultaneous match against all 12. When Mrs. Deardorff insists on having a female chaperone for the trip, Ganz suggests Shirley Munson, the club treasurer and one of his top students. But when Beth plays at the high school, she goes against 12 male competitors and doesn't play Munson.
Beth visits the custodian's workshop and takes a large screwdriver. When she visits the dispensary, the screwdriver is smaller.
When Mr. Shaibel first introduces Beth to a Queen Pawn Game Opening (d4 in algebraic notation), he says the opening is the "Queen's Gambit". In fact, the opening is not considered the Queen's gambit unless it continues 1. d4 d5 2. c4 e6 (or c6). Until then, it could be any number of less popular continuations of the Queens Pawn Game opening, such as London System, Tompowsky Attack, Stonewall Attack, Colle System, etc.
In the opening scene, there is a flash back to a car accident on a bridge. There is a truck on top of a hood of a sedan. This type of accident would require the heavy truck to jump the hood of the car on a single lane bridge. Given the style of cars during this period and the weight they would not be able to ride up on the tire make this scene impossible.
At 32:00 the teacher is writing multiplication facts on the chalkboard and wrote 5x3=18.
Beth was invited to play against twelve. Eleven presented themselves into the room.
Before the first game with the high school coach, she chooses the black piece hidden in his hand. She is subsequently seen playing white. However, she actually chooses the white piece, which looks black because we mostly just see the felt on its bottom.
Orphanages in Kentucky weren't desegregated until the 1960s, so in 1957 there wouldn't have been black and white orphans together in the same one.
The cars at the opening scene are Kentucky State Police, but the uniforms are Pennsylvania patches. The states are not close enough together to share jurisdiction.
The phone on Ms. Dearborn's desk is ivory/beige. Phones in 1958 were almost always black, especially in an institution of modest means such as an orphanage.
At the beginning of the show, Mrs. Deardorf states that Beth is 9 years old. The admissions document is immediately shown which states Beth was born Nov 2, 1948 and the admissions document was crated on 7/23/57. This would have made Beth 8 years old.