A $2 n \times 2 n$ chessboard has been covered with $2 n^{2}$ dominoes of size $2 \times 1$. According to Exercise 2831, when $n=3$, there exists a horizontal or vertical line along which the board can be cut into two without cutting any domino. For which other values of $n$ is this statement true?
For $n=1$, the statement is obviously true. The case for $n=2$ can be handled in exactly the same way as the solution to Problem 2831, so we will not repeat it here.

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However, for the case $n \geq 4$, we provide a general tiling method that guarantees the non-existence of the desired line. This is visible in the diagram; the dotted areas should be continued in the same way as we started in the upper left and lower left corners of the board, while the hatched $(2 n-4) \times (2 n-5)$ and $(2 n-8) \times 3$ sub-boards can have the dominoes placed arbitrarily.

Among the vertical and horizontal lines, only those marked with arrows need to be checked (i.e., those that are "one full side of a domino" away from the edges of the board), since the other lines definitely intersect a domino. Let's first examine the vertical lines. The 1st, 2nd, ..., $(2 n-3)$-rd lines intersect one domino each from the top and bottom two rows, while the $(2 n-2)$-nd and $(2 n-1)$-st lines intersect the dominoes at the right edge of the 4th and 3rd rows, respectively.

Among the horizontal lines, the 2nd, 3rd, ..., $(2 n-2)$-nd lines intersect the dominoes in the left strip, while the 1st and $(2 n-1)$-st lines intersect the dominoes in the upper right and lower right corners of the board. This way, we have checked all lines, and the answer to the problem is $n=1$ and $n=2$.

Based on the work of Benjámin Lakatos (Esztergom, P. Temesvári Ferences Gymnasium II. grade)