Lattice n-gons with ordered side lengths 1,2,3,…,n @BernardoRecamánSantos: It seems likely that your conjecture is true (e.g. further search found $225$ such polygons for $n = 19$). But Gerhard Paseman is right that $n$ must be congruent to $0$ or $3$ modulo $4$. -- Consider a checkerboard coloring of the integer points in the Cartesian plane. Then the ends of a side have the same color if its length is even, and they have different color if its length is odd. This holds also for diagonal sides, as you can check. So a closed path must contain an even number of sides of odd length.