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Lonely Heights is the gold puzzle in the Wooded Plateau area of The Talos Principle 2.
Overview
When the user enters, they find a driller freely available. There is a detachable fan and a pressure plate that activates it. The fan is pointed vertically at a copper ceiling. On the left, there is a blocked off area with a pressure plate and a hexahedron inside it, and further inside is another room blocked off with a copper wall with a connector in it. there are three red receptacles throughout the puzzle and one red energy source. The progress wheel is blocked off by a plasma gate that is turned off by one of those red receptacles.
Solution
A basic hint:
[ hint ]
Since the name of the puzzle is Lonely Heights, understand that at some point, you'd be putting something at a significant height. The word lonely is just for effect, it is not a hint.
If you're having trouble getting started:
[ hint ]
The only equipment you have is the driller. Remember, if you aim it properly, you can drop it onto the pressure plate while it is active.
If you made a hole but then don't know what to do next:
[ hint ]
The first objective is to get hold of the connector. How will you do it? Elevate yourself first.
If you got the connector, but don't understand what to do next:
[ hint ]
The final objective is to the use the connector to activate the receptacle close to the progress wheel. You will need significant elevation to achieve this.
If you're having trouble keeping the gate open
[ hint ]
Remember three things: 1. The fan is detachable. 2. There is nothing stopping you from making multiple trips to the rooftop. 3. If you open the gate once, it is possible for you to configure it to self-sustain.
Story
On solving the puzzle, we are rewarded with a line of spoken dialog from Cornelius (3):
We were happy here, in the beginning. It was a relief to be away from the politics of New Jerusalem. To be free to study and to create, to imagine a better future without all the exhausting nonsense of the city. Even re-using the technology from New Alexandria felt good. Like we were picking up a thread that should never have been abandoned. For the first time, we could be who we wanted to be. The best versions of ourselves.
—Cornelius (3)