Sure, but the LLM is free to construct a representation of the chess board and update it as it goes along. It is not in any way banned from using a virtual boar

Sure, but the LLM is free to construct a representation of the chess board and update it as it goes along. It is not in any way banned from using a virtual board, or whatever representation of game state it pleases. AFAIK, current models will still sometimes make illegal moves even if given the entire game state (e.g. in FEN notation), so it is not purely an issue with the models’ ability to keep track of sequences of moves.