A bishop on a grid

Question

Suppose that we have an $n\times m$ chessboard and bishop on the square $(1,1)$. It starts to move diagonally with the following rules:

1. If bishop is in any corner square except $(1,1)$, it stops moving.

2. If bishop meets a boundary square which is not a corner, it changes the direction of motion by continuing in the only direction that is not in the opposite direction the path just traveled (i.e. will travel along positions $(3,1) \to (2,2) \to (1,3) \to (2,4) \to \cdots $)

Is it possible to determine where the bishop will stop and its path? I found the case $\min\{n,m\}\not\mid\max \{n,m\}$ hard.

Answer 1

A hint:

Denote the centers of the squares by $(i,j)$ with $i$, $j$ integers $\geq0$. The size of the board is then $n'\times m'$ with $n'=n-1$, $\>m'=m-1$, and the starting point is $(0,0)$.

Replace the board with its universal cover, i.e. the integer lattice ${\mathbb Z}^2$, and mark all points $(k n', l m')$, $\ k,l\in{\mathbb Z}$ with at least one of $k$, $l$ odd, with a black star.

The bishp's complicated travel can now be "lifted" to ${\mathbb Z}^2$ and appears as a straight line. In this way it becomes easier to analyze under which circumstances this journey will hit a "forbidden corner" or will become periodic.

Answer 2

Hint: As is standard in this kind of problem, think of the bishop as moving along the line $y=x$, just that we have many chessboards (possibly reflected at the boundary).