Opening the West Gate in Blue Prince is one of the first permanent unlocks worth chasing. To do it, you need to draft the Utility Closet and the Garage in the same day, switch the Garage power on at the breaker box, place the Garage on the far-left side of the house, then head outside through it and unlatch the gate from behind.
Once you do that, the West Path stays unlocked for future runs.
This only works if the Utility Closet and the Garage show up in the same run.
The other thing to watch is Garage placement. The Garage has to be drafted on the far-left edge of the mansion layout. If it is not, this route will not work.
Start by getting the Utility Closet into your run. This room contains the breaker box you need to power the Garage.
Next, draft the Garage on the west-most column of the floor plan. This part matters. If the Garage is built anywhere else, you will not get the outside access needed for the gate.
Head back to the Utility Closet and use the breaker box to switch the Garage power on.
Return to the Garage and use the opener by the entrance. If the light is green, the Garage has power and the door should open.
It is worth checking the wall while you are there, since there are keys hanging beneath the opener.
Once the Garage door is open, head outside and keep following the path south. After you cross the river, take the left turn at the fork.
That will take you around to the back side of the West Gate.
Interact with the gate from the outside to open it.
This permanently unlocks the West Path, so you do not need to repeat the whole process on later days.
The main reward here is the West Path. That gives you access to the area used for Outer Rooms, which makes this one of the best early permanent unlocks in the game.
Some permanent unlocks are nice to have. This one keeps paying off.
Once the West Path is open, you have access to an area of the estate that leads into several useful room options and later progression steps. That makes it a strong early target instead of something to leave until much later.