Eight tiles are located on an $8\times 8$ board in such a way that no pair of them
are in the same row or in the same column. Prove that, among the distances between each pair of tiles, we can find two of them that are equal (the distance between two tiles is the distance between the centers of the squares in which they are located).
1. **Define the Distance Function:**
   Let the distance between two adjacent center squares be 1. Define the distance function \( f(x, y) \) as:
   \[
   f(x, y) = \sqrt{x^2 + y^2}
   \]
   where \( x \) and \( y \) are the horizontal and vertical distances between two tiles, respectively.

2. **Constraints on Tile Placement:**
   Since no two tiles are in the same row or column, the horizontal and vertical distances between any two tiles must be at least 1.

3. **Possible Distances:**
   The possible distances between any two tiles can be written as:
   \[
   f(x, y) = \sqrt{x^2 + y^2}
   \]
   where \( x \) and \( y \) are integers ranging from 1 to 7 (since the board is \( 8 \times 8 \)).

4. **Counting Distinct Distances:**
   The number of distinct pairs \((x, y)\) where \( 1 \leq x, y \leq 7 \) is:
   \[
   \{(1,1), (1,2), \ldots, (1,7), (2,1), (2,2), \ldots, (2,7), \ldots, (7,1), (7,2), \ldots, (7,7)\}
   \]
   This gives us a total of \( 7 \times 7 = 49 \) pairs. However, many of these pairs will result in the same distance due to symmetry (e.g., \( f(1,2) = f(2,1) \)).

5. **Counting Unique Distances:**
   To find the number of unique distances, we need to consider the distinct values of \( \sqrt{x^2 + y^2} \). The maximum value of \( x^2 + y^2 \) is \( 7^2 + 7^2 = 98 \). However, not all values from 1 to 98 are perfect squares, and many will repeat due to different pairs \((x, y)\) giving the same distance.

6. **Applying the Pigeonhole Principle:**
   We have 8 tiles, and thus \( \binom{8}{2} = 28 \) pairs of tiles. Since there are at most 21 distinct distances (as calculated from the possible pairs), by the pigeonhole principle, at least two of these 28 pairs must have the same distance.

\[
\boxed{\text{True}}
\]