"Checking every time we return something might be a good idea or it might be a waste of time," Sora said thoughtfully, "If it is working like a video game, then it's possible that even if there is some progress there might not be anything useful yet. So... as far as I'm concerned, we can do it either way." Indeed, some games would let solving different parts of a puzzle show the opening of various gates or barriers without actually giving any access to anything. That part in particular reminded him of those emblem pieces in Hollow Bastion, making for a lock that the Keyblade actually can't open. Perhaps this was similar. The memories he had from Roxas pointed to a puzzle needing to be solved to allow access to the computer room. This was no different.

As to York's question, it was more or less what he's been thinking. "Most likely. Assuming the rooms hidden under the library aren't also... part of the puzzle."