Gii and His Club

In some faraway land, there is a man named Gii, who is often considered the coolest person that has ever lived. One Saturday evening, he decided to go to a club on the other side of his town. The town is shaped like a grid with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$), where each cell represents a neighbourhood in the city. Let's denote the cell in row $r$ an