  Let $S$ be a connected graph which contains an induced path of $n-1$
vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A
configuration of the puzzle is simply an $n$-dimensional column vector over
$\{0, 1\}$ with coordinates of the vector indexed by the vertex set $S$. For
each configuration $u$ with a coordinate $u_s=1$, there exists a move that
sends $u$ to the new configuration which flips the entries of the coordinates
adjacent to $s$ in $u.$ We completely determine if one configuration can move
to another in a sequence of finite steps.
