Roll the Bar

You are given a $1 \times 1 \times 2$ bar (a cuboid) and a grid $A$ with $N$ rows (numbered $1$ through $N$) and $M$ columns (numbered $1$ through $M$). Let's denote the cell in row $r$ and column $c$ by $(r, c)$. Some cells of the grid are blocked, the remaining cells are free. Each cell has dimensions $1 \times 1$, the same as two opposite faces of the cuboid. When the bar is placed on the grid