\(A\) and \(B\) are two opposite corners of an \(n \times n\) board \((n \geq 1)\) divided into \(n^2\) unit squares. Each square is divided into two triangles by a diagonal parallel to \(A B\), giving \(2 n^2\) triangles in total. A piece moves from \(A\) to \(B\) going along the sides of the triangles and, whenever it moves along a segment, it places a seed in each of the triangles having that segment as a side. The piece never moves along the same segment twice. It turns out that after the trip every triangle contains exactly two seeds. For which values of \(n\) is this possible?