He doesn’t have to know how to do that. Any information he has regarding positions he’d never get in can be wrong and he’d still be unbeatable. This includes knowing whether or not it’s a position he can get in. The only way to reach a contradiction is if you show that he can lose from a given position, and that he can get there from a starting position.
You could try working backward by checking every position that might lead to this one and see if he moves so it does, but there might be no way to get to it, or multiple. You’d have to follow the entire tree back to prove that it doesn’t connect to the beginning. Worse, he isn’t even guaranteed to play deterministically, and just because he didn’t move to that position this time doesn’t mean he can’t.
He doesn’t have to know how to do that. Any information he has regarding positions he’d never get in can be wrong and he’d still be unbeatable. This includes knowing whether or not it’s a position he can get in. The only way to reach a contradiction is if you show that he can lose from a given position, and that he can get there from a starting position.
You could try working backward by checking every position that might lead to this one and see if he moves so it does, but there might be no way to get to it, or multiple. You’d have to follow the entire tree back to prove that it doesn’t connect to the beginning. Worse, he isn’t even guaranteed to play deterministically, and just because he didn’t move to that position this time doesn’t mean he can’t.