When the Osterlitz Endgame is being explained to Nick, the full view of the board shows a pawn on the c5 square behind the tipped pawn on the c6 square. In the next shot, which is a closeup of that area of the board, the c5 pawn has vanished.
When Greg is explaining the Osterlitz Endgame to Nick, he points out a supposed two-move checkmate starting by sacrificing the pawn on the c6 square to the king on d7. The checkmate can actually be accomplished in one move by promoting the pawn on e7 to a queen.
When Greg explains the queen-side castling to Sara he says (as with the animation): "Now there is a chess move known as castling, in which the player moves two pieces at the same time: the rook from its original position to D1 and the King from E1 to C1." However the official chess rule states that one should first move the King 2 places then the rook, this to avoid confusion with a normal move of the rook.
When D.B. Executes a queen-side castle playing chess, he moves his king only one space. In a castle, the king always moves two spaces.
Rooks don't have white skin round their beaks, they have white beaks (only the adults) and short white feathers on their faces around the beak that's known as "bald' (like the eagles).
And one was in Nevada because it flew there, it's not that far from Eurasia to the Americas, and it's not unheard of to find rooks there. They're known as a "vagrant" species for this reason.
When DB and Sara are discussing castling they say (and the animation shows) that two pawns have to move, but, as shown on the actual board, only one pawn has to move for the Queen and the Bishop to pass through the empty space.
When Nick puts the article about Troy Parker down on the table in the investigation room, a reading of the article reveals that Karl Schrute's last name is referenced twice, but the second time, it is incorrectly spelled "Schute".