15 Find all positive integers $n$, such that it is possible to write +1 or -1 in each cell of an $n \times n$ grid, satisfying: each cell labeled +1 has exactly one adjacent cell labeled -1, and each cell labeled -1 has exactly one adjacent cell labeled +1.
15. Let $a_{ij}$ denote the number filled in the square at the $i$-th row and $j$-th column. If there exists a valid filling method, we can assume $a_{11}=1$ (otherwise, change the signs of all numbers in the table and then discuss), at this point, one of $a_{21}$ and $a_{12}$ must be -1, let's assume $a_{21}=-1$ (otherwise, interchange the 2nd row and the 2nd column and then discuss), then $a_{12}=1$, further discussion shows that $a_{22}=-1, a_{13}=1, \cdots$, it follows that all numbers in the 1st row are 1, all numbers in the 2nd row are -1, and so on, all numbers in the 3rd row are -1, all numbers in the 4th row are 1, and so forth, it is known that when and only when $i \equiv 1(\bmod 3)$, all numbers in the $i$-th row are 1, and all numbers in the other rows are -1. If $n \equiv 0(\bmod 3)$, then the numbers in the $n$-th row are -1, and each square in this row has adjacent squares with numbers -1, which does not meet the requirements. Direct verification shows that the other cases all meet the requirements.

Therefore, a valid filling method exists if and only if $3 \nmid n, n>1$.