Coin flipping problem on an $n \times m$ grid
This is just linear algebra. Let $x$ be the vector over $\mathbb{Z}_2$ representing the initial state of the board, and let $y$ be the goal state of the board. Let $v_i$ be the vectors corresponding to the allowed flips (flipping corresponds to adding some $v_i$). Then we want to check whether $x+y$ is in the span of the $v_i$.