Question 34
Find all positive integers $n$ such that we can place in each of the $n^2$ cells of an $n \times n$ board one of the four arrows $\uparrow, \downarrow, \leftarrow, \rightarrow$ so that the following three conditions are all satisfied:
(1) Starting from any cell and moving in the directions of the arrows, we always return to the starting cell, and we never leave the board on the way.
(2) In every row except the top and bottom rows, the numbers of $\uparrow$ and $\downarrow$ arrows are equal (but the numbers may differ from row to row).
(3) In every column except the leftmost and rightmost columns, the numbers of $\leftarrow$ and $\rightarrow$ arrows are equal (but the numbers may differ from column to column).
(1) Starting from any cell and moving in the directions of the arrows, we always return to the starting cell, and we never leave the board on the way.
(2) In every row except the top and bottom rows, the numbers of $\uparrow$ and $\downarrow$ arrows are equal (but the numbers may differ from row to row).
(3) In every column except the leftmost and rightmost columns, the numbers of $\leftarrow$ and $\rightarrow$ arrows are equal (but the numbers may differ from column to column).