An $n\times n$ board is initially empty. We are allowed to perform the following operations:

1. Place Stones: If there is an L-shaped tromino region of three empty cells (in any orientation), we can place a stone in each of these three cells.

2. Clear Column: If a column has all of its $n$ cells filled with a stone, we can remove all stones from that column.

3. Clear Row: If a row has all of its $n$ cells filled with a stone, we can remove all stones from that row.

Determine all integers $n\ge 2$ for which it is possible to reach a state with no stones on the board after a non-zero number of moves.

Step 1 (Necessity - Invariant): Let $\omega =e^{2\pi i/3}$ be a primitive cube root of unity. We assign a weight to each cell $(x,y)$ of the $n\times n$ board, defined as $W(x,y)={\omega }^{x+y}$ for $1\le x,y\le n$. The total weight of stones on the board is $S={\sum }_{(x,y)\in \text{Stones}}{\omega }^{x+y}$.

When an L-tromino is placed, the three cells covered have coordinates of the form $\{(x,y),(x+1,y),(x,y+1)\}$ (or other orientations). The change in the total weight $S$ modulo $3$ (or using the relation ${\omega }^2+\omega +1=0$) is always $0$, because ${\omega }^x{\omega }^y(1+\omega +{\omega }^2)=0$ or a similar combination of consecutive powers.

When a row or column is cleared, we remove $n$ stones. If $3\nmid n$, the sum of weights of a full row or column ${\sum }_{i=1}^n{\omega }^{i+k}={\omega }^k{\sum }_{i=1}^n{\omega }^i$ is non-zero (since $1+\omega +\cdots +{\omega }^{n-1}\ne 0$). This alters the invariant, making it impossible to return to $0$ if we started from an empty board. Thus, a non-zero number of clears and placements can only return to an empty board if $3\mid n$.

Step 2 (Sufficiency - Construction): For $n$ being a multiple of 3, we can tile the board with $3\times 3$ blocks. Within a $3\times 3$ block, we can place L-trominoes systematically to fill rows/columns completely, clear them, and eventually return to an empty state. By induction and coordinate-wise tiling, we construct a valid sequence of moves for any multiple of 3. $■$