Theory 5: GPT-3.5-instruct plays chess by calling a traditional chess engine.
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Also, it makes a lot of sense if you expect people to play chess against the LLM, especially if you are later training future models on the chats.
You say it makes sense but how does it make sense for OpenAI to add overhead to all of its API calls for the super niche case of people playing 1800 ELO chess/chat bots? (that often play illegal moves, you can go try it yourself)