With these Rockets defeated, your next obstacle are the shifting walls. The easiest way past here is to activate the three switches in order from left to right (3-2-1). Activating them in a different order will let you reach the Full Heal to the right (1), and the Smoke Ball to the left (1-3-2-1-3). Go south, defeat the female Rocket, and walk through the doors into the next room.
Follow the hallway past the first Rocket, and pick up the Max Ether in the southeast corner. Fight the Rocket in the northeast, then take a detour upstairs to find an Amulet Coin in the Department Store's basement. Go west to battle one more Rocket, then grab the TM35 (Sleep Talk) and speak to the Director. He gives you the Card Key, which unlocks the shutter on the tower's third floor. Climb the northeast stairs and take the elevator up out of the basement.
The next destination is Blackthorn City, east of Mahogany Town. However, the feather that the Director found lets you track down the first of Johto's two legendary birds, so you may want to detour for it now.
Route 44 is a quiet mountain road that serves as the only path into Blackthorn City. Both the north and south paths lack any tall grass, but Pokémon like Tangela and Lickitung can be found in the grass between the two lakes. Be sure to trade phone numbers with Fisher Wilton for alerts about Remoraidswarms. Go east to reach the entrance to Ice Path.
Ignore the stairway for a moment and go east to find the next ice patch. This one is simpler, and there is an item, HM07 (Waterfall), lying on the far side of the ice.
Right, Up, Left, Down, Left, Up, Right
Climb the stairs and cross the rock platform to reach the ladder in the northeast corner.
There are four boulders scattered around this floor, and you need to drop each through a different hole to form a path on the floor below. Go north and loop around to find the first two boulders.
Right 1, Down 4, Left 2, Down 2, Left 1, Up 1, Left 1, Up 2
Left 1, Up 5, Left 2, Down 4
Left 1, Down 4, Right 1, Down 2, Left 5
Up 2, Left 1, Up 3, Right 1
Start off with the boulder farthest to the south. Use Strength to move it southward and drop it down the southwest hole. Go back and move the second boulder north, so it sits at the break in the wall. Head to the east side to reach the third boulder, and drop it through the southeast hole. Go north and drop the fourth boulder through the north hole. Finally, move the second boulder to the northwest hole. Climb down the ladder in the northeast.
The boulders from the floor above have formed a convenient spiral that leads to the center of the room. Check the four corners for items, including a Max Potion, a hidden Carbos, and a hidden Ice Heal. Follow the boulders to the room's center, pick up the Full Heal, and climb down the ladder.
Smash the rock, or loop around to the right, to reach the NeverMeltIce. Climb the east ladder.
Go east and slide across the smaller ice patch. Stand south of the stairway to the platform, and jump the ledge. Slide to the east to find TM44 (Rest). Slide west, then north, and return to the ledge. Walk up onto the platform, stand at the stairway to the left, and slide south to the ladder.
Head east to the stairway; from here, slide down, right, up, then left. There is an Iron across the ice to the southwest, while the next ladder is off to the northwest.
Go to the west wall and slide south to reach a PP Up. Return to the ladder and jump the ledge to slide to the exit. Welcome to Blackthorn City!