Checkpoints

Chef has $N$ points (numbered $1$ through $N$) in a 2D Cartesian coordinate system. For each valid $i$, the $i$-th point is $(x_i, y_i)$. He also has a fixed integer $c$ and he may perform operations of the following type: choose a point $(x_i, y_i)$ and move it to $(x_i + c, y_i + c)$ or $(x_i - c, y_i - c)$. Now, Chef wants to set up one or more *checkpoints* (points in the same coordinate sy