First steps

The Maze is currently a dead end, but you're going to be able to change that in just a minute. Take the elevator up to balcony above you. There you can see The Maze — and its exits — in its entirety.

You have three goals in this maze, corresponding to the three exits. You need to exit to the right and straight across first, then to the left. To do this, you're going to do a lot of rotating, teleporting and backtracking.

Rotating

Take the elevator back down to ground level, then hit the down button again. Beneath each of the four, smaller circles in The Maze is a Swap Machine.

This Swap Machine takes you to Hunrath, where you'll find a pair of buttons to rotate The Maze section one quarter turn at a time. For sections you only have to rotate, you can spin the sphere just like you did before in the globe puzzle and The Gauntlet, then head back inside to take it back to Maray.

Swapping

If you look carefully at The Maze from above, you'll notice that you can't actually complete the paths you need to with the pieces arranged as they are. Wouldn't it be great if there were a secret fifth piece, say, hanging out in Hunrath that you could use too? Well good news!

At the top of the stairs, turn right. The path will bring you to a locked door that you can open from this side.

Opening this door will lead you back to the Swap Machines you used in The Gauntlet. You can use any of these to travel back to Maray and see the new section you swapped into The Maze. Doing this will exchange two sections of The Maze.

Unlock a shortcut to Hunrath

On one of your trips to Hunrath, go past the rotate button. There's a tunnel that leads to a locked door. Open it to find a shortcut to the main area of Hunrath. You're not done with The Maze yet, though, so this shortcut will have to wait for another time. Backtrack and finish whatever maze manipulations you were working on.

Solving The Maze

First, you have to solve The Maze for the left and right exits. Use the button on the balcony to rotate the entire Maze once (90 degrees) counterclockwise (or anticlockwise if you're Doctor Who) and swap the bent Maze section for the straight one.

In Maray, go back up to the balcony to make sure it's pointed toward the center of The Maze (so you can get into it). If it's oriented left-right, rotate it.

Using the button on the balcony, rotate The Maze twice (180 degrees) then swap the bent section for it's mirror which is still sitting in Hunrath.

Rotate this new section so that it lines up with the rest of The Maze. This may take a couple tries.

Rotate the entire Maze once (90 degrees) clockwise and rotate this section to open up the path through The Maze. The path to the exit on the right is now open.

Follow the path you've created through The Maze.

You'll pass through the dome wall and find yourself on a long path. Follow it to the end, where you'll find a Villein lock.

Set the lock to zero like you did before — either by connecting the last dot or by removing all the connections — to open a door above you.

Backtrack all the way along the path and through The Maze. Take the elevator up to the balcony. Use the button to rotate The Maze twice (180 degrees).

Take the elevator down twice and rotate the section by 180 degrees as well. This will let you walk straight across The Maze.

Use the Villein device you find there to open another door on the rails above you.

Go back across The Maze and undo the rotation you did on the section by rotating the section 180 degrees again. Then rotate The Maze counterclockwise (anticlockwise if you're Harry Potter) to point the path to the exit on the left.

Head through The Maze one last time and take the path up until you see a building.

Take the path to the right and follow it around. Eventually, it will lead to the beginning of The Gauntlet and let you build a bridge.

Backtrack to the building, and go up the ramp. At the first landing (before you get to the top level of the structure) there's a path around to the side. Take it.

Head down to receive a message from Mayor Faux-sef. Press the button either when he's done or when you've heard enough.

This will call a pod to the top level. Climb inside, and you'll ride the rails across Maray — passing through both of the doors you opened — to the next location.