# A Ploughing Puzzle I recently came across the following puzzle on social media. > Is it possible to plough a plot of land $$ \mathbf{L} = \begin{bmatrix} \square & \square & \square & S \\ \square & \square & \square & \square \\ \square & \square & \square & E \end{bmatrix} $$ > from $S$ to $E$ using only up, down, left, and right moves without > backtracking? This puzzle can be solved by a simple parity argument. Suppose you equip a device $D$ that, at any position $l_{i,j}$, says if $i+j$ is odd or even. Clearly, any move from any position will flip what $D$ says. We know $D$ says "odd" at $S=l_{1,4}$, because $1+4=7$ is odd. To complete the plough, we must make $11=3(4)-1$ moves, and thus, flips. So if an ending position $E'$ exists, $D$ must say "even" at $E'$. But we know $D$ says "odd" at $E=l_{3,4}$, because $3+4=7$ is odd. Thus, the task is impossible.