DK-GEORGMOHR 2009 Problem 5

Imagine a square scheme consisting of $n\times n$ fields with edge length $1$ , where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route).