# 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 \(1+4=5\) 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.