9.5. On an infinitely large chessboard, the distance between any two squares is defined as the minimum number of steps a King needs to move from one square to another. Given three squares that are pairwise 100 steps apart, find the number of squares that are 50 steps away from each of these three squares.
9.5. Let the side length of a small square be 1.

Represent a small square by the coordinates of its center. Let the absolute difference of the x-coordinates of two small squares be $x$, and the absolute difference of the y-coordinates be $y$.

It is not hard to see that a king can reach from one square to another in $\max \{x, y\}$ steps, but not in fewer steps. Therefore, the distance between these two squares is $\max \{x, y\}$.

Let $d(A, B)$ denote the distance between squares $A$ and $B$.

Suppose $A$, $B$, and $C$ are three small squares, each pair of which is 100 units apart. Then, each pair of these squares will have either x-coordinates or y-coordinates that differ by 100. Thus, among the three pairs of squares $(A, B)$, $(B, C)$, and $(A, C)$, there must be two pairs such that the two squares in each pair have x-coordinates or y-coordinates that differ by 100. Without loss of generality, assume that $A$ and $B$, as well as $A$ and $C$, have x-coordinates that differ by 100, then the x-coordinates of $B$ and $C$ differ by 0 or 200. Since $d(B, C) = 100$, $B$ and $C$ must have the same x-coordinate. Therefore, their y-coordinates differ by 100. Without loss of generality, let the coordinates of $A$, $B$, and $C$ be $B(0,0)$, $C(0,100)$, and $A(100, x) (0 \leqslant x \leqslant 100)$.

Considering the small square $X$ that is 50 units away from $A$, $B$, and $C$, its x-coordinate must be 50 (otherwise, $d(X, A) > 50$ or $d(X, B) > 50$).

Similarly, its y-coordinate must also be 50 (otherwise, $d(X, C) > 50$ or $d(X, B) > 50$).

Therefore, $X(50,50)$.
Hence, there is a unique small square that is 50 units away from $A$, $B$, and $C$.