A line in a square grid divides the grid into two parts. Because the grid is symmetrical, the line must be either horizontal or vertical and must pass through the center of the grid. Since each domino covers exactly two squares in the grid, it must intersect any line that passes through the center of the grid. Therefore, no line can intersect no dominoes. However, the line that passes through the center of the grid is not the only line that does not intersect any dominoes. There are many other lines that do not intersect any dominoes, such as a line that is parallel to one of the sides of the grid and is offset from the center by half a grid square. Therefore, the line that intersects no dominoes is not unique.

Why does it follow that the line must pass through the center of the grid? Is this a WLOG assumption?