7. Finding the Cable

Streight across our square piece of land (of side length $1$) leads a power line–a buried cable. We would like to find the cable and secretly connect to it. Unfortunately, we do not know precisely where it is. We need to find a way to dig on our land piece to surely find the cable, no matter the actual configuration.

We have presented on the lecture, that digging all sides of the square leads to a solution of length $4$. We further noticed that three sides suffice, hence obtaining a solution of length $3$. Another variant is to dig both diagonals of the land piece with length $2\sqrt{2}$. Your task is to find a solution of length $\leq \frac{1+2\sqrt{5}}{2}$. If your solution is significantly shorter than that, you might get to the hall of fame below. Remember to explain the computation of the length of your solution.