Hamiltonian Paths in Projective Checkerboards

Place a checker in some square of an \(n \times n\) checkerboard. The checker is allowed to step either to the east or to the north, and is allowed to step off the edge of the board in a manner suggested by the usual identification of the edges of the square to form a projective plane. We give an explicit description of all the routes that can be taken by the checker to visit each square exactly once.