Problem $1\mathrm{.}2.$ Suppose that $m$ and $n$ are positive odd integers.

$($a$)$ Does there exist a perfect cover of the $mn$ chessboard?

$($b$)$ If we remove $1$ square from the $mn$ chessboard, will it have a perfect cover? To help answer this

general question, try this out in the case $m=7$ and $n=9.$