Question 70
Let us call a point in the $xy$-plane a good point if each of its coordinates is an integer from $1$ to $2000$. Let us also call polyline $ABCD$ a Z-shaped polyline if four points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, $D(x_4, y_4)$ satisfy all the following conditions:
* $A$, $B$, $C$, $D$ are good points.
* $x_1 \lt x_2$, $y_1 = y_2$.
* $x_2 \gt x_3$, $y_2 - x_2 = y_3 - x_3$.
* $x_3 \lt x_4$, $y_3 = y_4$.
Determine the smallest possible positive integer $n$ such that, there exist Z-shaped polylines $Z_1, Z_2, \dots, Z_n$ which satisfy the following condition:
Any good point $P$ lies on $Z_i$ for some $1 \le i \le n$.
Note that polyline $ABCD$ is the union sets of line segments (including both endpoints) $AB$, $BC$ and $CD$.
* $A$, $B$, $C$, $D$ are good points.
* $x_1 \lt x_2$, $y_1 = y_2$.
* $x_2 \gt x_3$, $y_2 - x_2 = y_3 - x_3$.
* $x_3 \lt x_4$, $y_3 = y_4$.
Determine the smallest possible positive integer $n$ such that, there exist Z-shaped polylines $Z_1, Z_2, \dots, Z_n$ which satisfy the following condition:
Any good point $P$ lies on $Z_i$ for some $1 \le i \le n$.
Note that polyline $ABCD$ is the union sets of line segments (including both endpoints) $AB$, $BC$ and $CD$.