  We consider Chess played on an $m \times n$ board (with $m$ and $n$ arbitrary
positive integers), with only the two kings and the white rook remaining, but
placed at arbitrary positions. Using the symbolic finite state method,
developed by Thanatipanonda and Zeilberger, we prove that on a $3 \times n$
board, for almost all initial positions, White can checkmate Black in $\leq
n+2$ moves, and that this upper bound is sharp. We also conjecture that for an
arbitrary $m \times n$ board, with $m,n \geq 4$ (except for $(m,n)=(4,4)$ when
it equals $7$), the number of needed moves is $\leq m+n$, and that this bound
is also sharp.
