Consider a chessboard of size 8 units `xx8` units (i.e., each small square on the board has a side length of 1 unit). Let S be the set of all the 81 vertices of all the squares on the board. What is number of line segments whose vertices are in `S_(1)` and whose length is a positive integer? (The segments need not be parallel to the sides of the board).

Answer

To find the number of line segments whose vertices are in the set \( S \) (the set of all 81 vertices of the squares on an 8x8 chessboard) and whose length is a positive integer, we can break the problem down into several steps.

### Step-by-Step Solution:

1. **Understanding the Chessboard:**
The chessboard has dimensions of 8 units by 8 units, which means it has 9 vertices along each side (0 to 8). Thus, the total number of vertices \( |S| \) is \( 9 \times 9 = 81 \).

2. **Identifying Line Segments:**
We need to count the line segments formed by pairs of vertices in \( S \) that have integer lengths. The length of a line segment between two vertices \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For \( d \) to be a positive integer, \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \) must be a perfect square.

3. **Calculating Possible Lengths:**
The maximum distance between any two vertices on the chessboard occurs between the corners, which is:
\[ d_{\text{max}} = \sqrt{(8-0)^2 + (8-0)^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \]
However, we are only interested in integer lengths. The possible integer lengths can be from 1 to 8 (the maximum horizontal or vertical distance).

4. **Counting Segments for Each Integer Length:**
We will count the number of line segments for each integer length \( n \) from 1 to 8.

- **Length 1:**
- Segments parallel to axes: \( 8 \times 9 \times 2 = 144 \) (8 horizontal and 8 vertical segments).
- Diagonal segments: \( 8 \times 8 = 64 \) (each square contributes 1 diagonal).
- Total for length 1: \( 144 + 64 = 208 \).

- **Length 2:**
- Segments parallel to axes: \( 7 \times 9 \times 2 = 126 \).
- Diagonal segments: \( 7 \times 7 = 49 \).
- Total for length 2: \( 126 + 49 = 175 \).

- **Length 3:**
- Segments parallel to axes: \( 6 \times 9 \times 2 = 108 \).
- Diagonal segments: \( 6 \times 6 = 36 \).
- Total for length 3: \( 108 + 36 = 144 \).

- **Length 4:**
- Segments parallel to axes: \( 5 \times 9 \times 2 = 90 \).
- Diagonal segments: \( 5 \times 5 = 25 \).
- Total for length 4: \( 90 + 25 = 115 \).

- **Length 5:**
- Segments parallel to axes: \( 4 \times 9 \times 2 = 72 \).
- Diagonal segments: \( 4 \times 4 = 16 \).
- Total for length 5: \( 72 + 16 = 88 \).

- **Length 6:**
- Segments parallel to axes: \( 3 \times 9 \times 2 = 54 \).
- Diagonal segments: \( 3 \times 3 = 9 \).
- Total for length 6: \( 54 + 9 = 63 \).

- **Length 7:**
- Segments parallel to axes: \( 2 \times 9 \times 2 = 36 \).
- Diagonal segments: \( 2 \times 2 = 4 \).
- Total for length 7: \( 36 + 4 = 40 \).

- **Length 8:**
- Segments parallel to axes: \( 1 \times 9 \times 2 = 18 \).
- Diagonal segments: \( 1 \times 1 = 1 \).
- Total for length 8: \( 18 + 1 = 19 \).

5. **Summing All Lengths:**
Now, we sum the total segments for each length:
\[ 208 + 175 + 144 + 115 + 88 + 63 + 40 + 19 = 1080 \]

### Final Answer:
The total number of line segments whose vertices are in \( S \) and whose length is a positive integer is **1080**.