Deliberate mistake: Towards the end of the film, Josh is playing a game in the park with Vinnie, and Vinnie asks "What's that?", to which Josh replies "Schliemann Attack." However, this dialogue occurs while the game is still in standard opening theory and before Josh has played the move that introduces the Schliemann Attack (pawn to f5).
Deliberate mistake: When Josh is playing against Jonathan Poe for the national championship, the players should be notating their moves, but they aren't.
Continuity mistake: When Josh is playing against his father for the first time, he is able to shout moves to him from another room (i.e. from the tub in the bathroom) to checkmate him. Later when he is learning from his teacher Bruce, Josh claims he can't solve a puzzle without moving the pieces physically.
Continuity mistake: When Josh and his father play a game of chess Josh's father picks up a piece from the centre of the board and moves it diagonally towards one of the flanks. In the next shot it is visible that the piece is a white rook, which can't move that way. Plus the previous shot shows both white rooks still standing in their original positions on the board. (00:15:55)
Answer: Vasek Simek.