Find the princess (Solution)

For the castle with 4 doors:

Label the doors 1-4 from left to right. I will pick open the doors in the following order: 2, 3, 3, 2 - and I guarantee you I will find the princess! So it will take me 4 nights.

For $n$ doors:

It will take me $2 (n-2)$ nights.

I will start at door 2 and then progress, in order, to $n-1$, and then I will open the exact same doors, but in reverse order (start at $n-1$ and work my way back down to 2).