You're surrounded by infinitely many zombies. You're at the origin, and zombies occupy the points $(100i,100j)$ for all integer $i, j$ except the origin, as shown below:
You and the zombies move alternatively. You all move in steps. Each step is of distance 1, and can only be taken in one of the four cardinal directions. In your move, you can move at most 100 steps. In a zombies move, the simple-minded zombies move according to the following simple rules:
- if $0\lt \vert Z_x - Y_x\vert \leq \vert Z_y- Y_y \vert$, the zombie moves 1 step horizontally towards you;
- if $0\lt \vert Z_y - Y_y\vert \lt \vert Z_x- Y_x \vert$, they move 1 step vertically towards you;
- if $Z_x=Y_x$ or $Z_y=Y_y$, they simply move 1 step closer to you.
where $Z_x$ and $Z_y$ are the zombie's x and y coordinates, and $Y_x$ and $Y_y$ your coordinates.
Question: Can you survive indefinitely? If so, how? If not, how many turns can you survive and how far (in steps) can you move away from the origin before being caught?
Edit: fixed the bug mentioned by @iBug.