Find the number of ways of selecting pair of black squares in chessboard such that they have exactly one common corner

Answer

To find the number of ways to select a pair of black squares on a chessboard such that they have exactly one common corner, we can break down the problem into three cases based on the position of the black squares.

### Step-by-Step Solution:

1. **Identify the Structure of the Chessboard:**
A standard chessboard has 8 rows and 8 columns, alternating colors. The black squares are located at positions (i, j) where (i + j) is odd.

2. **Case 1: Selecting Corner Black Squares:**
- There are 2 corner black squares on the chessboard: (1, 2) and (2, 1).
- For each corner black square, there is exactly 1 adjacent black square that shares one corner.
- Therefore, the total for this case is:
\[ 2 \text{ (corner squares)} \times 1 \text{ (adjacent square)} = 2. \]

3. **Case 2: Selecting Side Black Squares:**
- Each side of the chessboard has 3 black squares (the squares in the middle of each side).
- There are 4 sides on the chessboard.
- For each of these side black squares, there are 2 adjacent black squares that share one corner.
- Therefore, the total for this case is:
\[ 4 \text{ (sides)} \times 3 \text{ (black squares per side)} \times 2 \text{ (adjacent squares)} = 24. \]

4. **Case 3: Selecting Middle Black Squares:**
- In the center of the chessboard, there are 6 rows with 3 black squares each.
- For each of these middle black squares, there are 4 adjacent black squares that share one corner.
- Therefore, the total for this case is:
\[ 6 \text{ (rows)} \times 3 \text{ (black squares per row)} \times 4 \text{ (adjacent squares)} = 72. \]

5. **Total Count of Pairs:**
- Adding all the cases together gives:
\[ 2 + 24 + 72 = 98. \]

6. **Adjust for Order of Selection:**
- Since the order of selection does not matter (i.e., selecting square A and then square B is the same as selecting square B and then square A), we divide the total by 2:
\[ \frac{98}{2} = 49. \]

### Final Answer:
The total number of ways to select a pair of black squares that share exactly one corner is **49**.