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Sahasrahla will speak to you telepathically across the void. He explains that you are now in what was once the Golden Land, which has been turned into the Dark World by evil forces. The seven wise men's seal is indeed broken, and a gate has been opened to the Dark World at Hyrule Castle. To save the Light World from destruction, you must win the Golden Power back from evil hands, and rescue the seven kidnapped maidens, who are being held deep within dark dungeons around the Dark World. The first such dungeon you must visit is called the Palace of Darkness. Sahasrahla (somehow) marks it on your map of the Dark World (which you somehow have..?).
Anyway, you may return to the Light World at any time by using the Magic Mirror. When you do, a warp point will be created in the Light World, and if you step into it, you return to the Dark World at that point. There are other ways to go between the worlds that will be revealed later on.
Before traveling to the Palace of Darkness, descend from the top of the pyramid and head down the eastern stairs. Go east from there and drop down the small ledge, and follow the path to the end to find a piece of heart.
Drop down to the base of the pyramid. You are faced with new enemies in this twisted land, including the bomb-throwing cyclopses, spear-throwing pigs, and man-eating plants you see here. To proceed, go the only way you can go: east.
Let's get you some items before you head to the Palace of Darkness, shall we?
Once east from the Pyramid of Power, head north, and continue north past the large boulder in your way (there is a broken bridge here). Head east, and follow the path to where Zora's Domain is in the Light World. You find the Lake of Ill Omen. There is a circle of stones here; throw something into it (a skull, a bush, anything). A large fish will come out and berate you for disturbing its nap, but will give you the Quake Medallion, which uses up a lot of magic to cause an earthquake to shake the ground, killing many enemies on the screen. You will need this item later.
Swim back towards the broken bridge and you find a regular whirlpool (the fireball-spewing monsters' whirlpools are a different shape). Swim into it to be warped elsewhere, then swim southwest to where Lake Hylia is in the Light World. There is an icy place you can't enter; ignore that. Swim to the northwest of the lake to find a shallow area surrounded by stones. Stand in the shallows and use your Magic Mirror to be transported back to the Light World. You end up on an island where a piece of heart sits! Return to the Dark World.
Back in the Dark World, swim northwest to a dock, and go back up to land. Head north.
Before going up to the Palace of Darkness, there is another piece of heart you can get, but it is in the Light World. Use your Magic Mirror to return to the Light World, then find the place where the lumberjacks were cutting the tree near the entrance to the Lost Woods. The tree they were cutting now has different colored leaves. Dash into the tree and the leaves will fall, revealing a hole in the stump. Fall into it. There are fairies up above. Go through the door to find a piece of heart on a ledge.
Return to the Dark World via the warp point you created when you used the Magic Mirror to get back to the Light World (or go into the front gate of Hyrule Castle to end up in front of the Pyramid of Power).
The Palace of Darkness is deep within a maze to the east of the Pyramid of Power, where the Eastern Palace was (the palace where you won the first pendant). Go there.
From the mountain pass that leads you to where the Eastern Palace was in the Light World, head east, past some horned statues, until you see some trees and an arrow pointing north. Follow the arrow into the hedge. You can see where you are through tiny holes in the top. Go through to the other side, around the back of a building (there's nothing interesting in there), and all the way west through a hedge. Go north, battling the green and orange monster as you go, then re-enter the hedge through the opening to your right. Enter the big hedge through the entrance below, then head south, then east to get out of the hedge.
Kiki the monkey will be tailing you when you emerge. It asks for 10 rupees, so be nice and give it the rupees. Kiki will follow you for a while. Continue east, through the hedge, then north, and finally east to the Palace of Darkness. Go up to the entrance, only to find that it's closed tight! Kiki will pipe up now and say that he will open the entrance for 100 rupees. It's a steep price, but it's the only way you can get into the Palace. Hey, things in the Dark World are pretty rough, okay?