Deliberate "mistake": When Josh is playing against Jonathan Poe for the national championship, the players should be notating their moves, but they aren't.

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).