2021 USAMO Problems/Problem 3

Let be an integer. An board is initially empty. Each minute, you may perform one of three moves: [list] [*] If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. [*] If all cells in a column have a stone, you may remove all stones from that column. [*] If all cells in a row have a stone, you may remove all stones from that row.

\[\begin{asy}
unitsize(20);
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0));
fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey);
draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2));
draw((0,2)--(4,2));
draw((2,4)--(2,0));
\end{asy}\] (Error compiling LaTeX. Unknown error_msg)

For which is it possible that, after some non-zero number of moves, the board has no stones?