The flipping puzzle on a graph
  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. 
Hau-wen Huang and Chih-wen Weng