Jumping on the Coordinate lattice grid

Question

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show that if he starts at $(0, 1)$ he can never get to $(0, 0)$.

Answer 1

For a move from $(x,y)$ to one of the $(x', y')$, one has either $x'+y' \equiv x+y \pmod{5}$ or $x'+y' \equiv 3(x+y) \pmod{5}$. Initially it starts at $(0,1)$ with $0+1 \equiv 1 \pmod{5}$, hence it can't reach $(0,0)$ with $0+0 \equiv 0 \pmod{5}$.