im not too positive about this answer but im pretty sure you cant get to blackthorn city through the dark cave
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in the dark cave near blackthorn city
A man gives them to you in Dark Cave. He's in the northwest corner of the Blackthorn City entrance.
you don't go that way, instead you go though dark cave (you need rock smash) to reach blackthorn city
By beating the gym leader Clair in Blackthorn City.
You need to go to Blackthorn City, which is where the Blackthorn Gym resides and the final badge of the Johto region. Go back to Mahogany Town, past the place where the guy who was selling that RageCandyBar was, and go onto the path. Up ahead, there will be two routes, but both of them will take you to the Ice Cave. Navigate through the Ice Cave, and then you should be in Blackthorn City.