In how many ways can two squares be chosen on a `8 xx 8` chessboard such that they have only one corner in common?

Options

A. 98
B. 94
C. 108
D. None of these

Answer

To solve the problem of how many ways two squares can be chosen on an 8x8 chessboard such that they have only one corner in common, we can follow these steps:

### Step 1: Understanding the Chessboard Layout
An 8x8 chessboard consists of 64 squares arranged in 8 rows and 8 columns. Each square can be identified by its row and column coordinates.

**Hint**: Visualize the chessboard and label the rows and columns to understand the arrangement of squares.

### Step 2: Identifying Common Corners
When two squares share exactly one corner, they must be positioned such that they are adjacent diagonally. For example, if we consider a square at position (i, j), the squares that share one corner with it can be located at (i+1, j) and (i, j+1).

**Hint**: Think about the positions of squares that can share a corner based on their coordinates.

### Step 3: Counting Possible Pairs
For each square on the chessboard, we can identify pairs of squares that share a corner.

- For a square not on the edge of the board, there are typically two squares that can be chosen that share a corner.
- For squares on the edges or corners of the board, the number of pairs will be fewer.

### Step 4: Analyzing Rows and Columns
To systematically count the pairs:
1. **For each square in the first two rows**:
- Each square can form pairs with squares in the next row and the next column.
- The number of pairs can be calculated based on the position of the square.

2. **Calculate for all rows**:
- Since the chessboard is symmetric, the calculations for the first two rows can be multiplied by the number of rows.

### Step 5: Total Calculation
- For the first two rows, we find that there are 14 ways to select pairs of squares sharing a corner.
- Since there are 7 such pairs in each row (as they can be moved down to the next row), we multiply the number of ways by the number of rows.

Total ways = 14 (ways for two rows) * 7 (rows) = 98.

### Final Answer
Thus, the total number of ways to choose two squares on an 8x8 chessboard such that they have only one corner in common is **98**.

---