The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells?
1. **Define the problem and set up coordinates:**
   We are given three cells on an infinite chessboard with pairwise distances of 100. We need to find the number of cells that are at a distance of 50 from each of these three cells. We start by placing one of the cells at the origin, \((0,0)\).

2. **Characterize the set of cells at a given distance:**
   For a cell \((x_0, y_0)\), the set \(S\) of cells at a distance \(d\) from it is given by:
   \[
   S = \{(x_0 + d, y_0 + k) : k \in \{-d, \ldots, d\}\} \cup \{(x_0 - d, y_0 + k) : k \in \{-d, \ldots, d\}\} \cup \{(x_0 + k, y_0 + d) : k \in \{-d, \ldots, d\}\} \cup \{(x_0 + k, y_0 - d) : k \in \{-d, \ldots, d\}\}
   \]
   This set forms a square-like figure centered at \((x_0, y_0)\) with side length \(2d\).

3. **Assume coordinates for the second cell:**
   Without loss of generality (WLOG), assume the second cell is at \((100, k)\) for some integer \(0 \leq k \leq 100\).

4. **Prove the lemma:**
   **Lemma:** Two of the three cells lie either on the same horizontal or vertical line.
   - If \(k = 0\), the second cell is \((100, 0)\), and the lemma is trivially true.
   - If \(k \neq 0\), the third cell must be at one of the intersections of the distance-100 squares centered at \((0,0)\) and \((100,k)\). These intersections are \((0, 100)\) and \((100, k - 100)\). In both cases, two cells lie on the same vertical line.

5. **Reposition the cells:**
   Using the lemma, we can reposition the cells so that two of them are at \((0,0)\) and \((0,100)\). Assume the third cell is at \((d, k)\) for some \(0 \leq k \leq 100\).

6. **Determine the set of points at distance 50:**
   - The set of points at distance 50 from \((0,0)\) forms a vertical bar: \((50, n)\) for \(-50 \leq n \leq 50\).
   - The set of points at distance 50 from \((0,100)\) forms another vertical bar: \((50, n)\) for \(50 \leq n \leq 150\).
   - The set of points at distance 50 from \((d, k)\) forms a square-like figure centered at \((d, k)\).

7. **Find the intersection:**
   - If \(k \neq 0\) and \(k \neq 100\), the intersection of the vertical bar \((50, n)\) with the square-like figure centered at \((d, k)\) will be a single point.
   - If \(k = 0\) or \(k = 100\), the intersection is still a single point due to the symmetry of the problem.

Thus, there is exactly one cell that is at a distance of 50 from each of the three cells.

The final answer is \(\boxed{1}\)