Two small squares on a chess board are choosen at random. Find the probability that they have a common side:

Options

A. `1/12`
B. `1/18`
C. `2/15`
D. `3/14`

Answer

To find the probability that two randomly chosen squares on a chessboard share a common side, we can follow these steps:

### Step 1: Calculate the Total Number of Ways to Choose 2 Squares
A standard chessboard has 64 squares. The total number of ways to choose 2 squares from 64 is given by the combination formula:

\[ \text{Total ways} = \binom{64}{2} = \frac{64 \times 63}{2} = 2016 \]

### Step 2: Identify the Squares that Share a Common Side
Next, we need to determine how many pairs of squares share a common side. We can categorize the squares based on their positions on the chessboard:

1. **Interior Squares**: These squares are surrounded by 4 other squares. There are \(6 \times 6 = 36\) interior squares (since they are located in a 6x6 grid within the 8x8 chessboard). Each of these squares can form 4 adjacent pairs (up, down, left, right).

\[ \text{Pairs from interior squares} = 36 \times 4 = 144 \]

2. **Edge Squares**: These squares are on the edge but not in the corners. There are \(4\) edges with \(6\) squares each (since corners are excluded), giving us \(4 \times 6 = 24\) edge squares. Each edge square can form 2 adjacent pairs (one towards the inside of the board and one towards the outside).

\[ \text{Pairs from edge squares} = 24 \times 2 = 48 \]

3. **Corner Squares**: There are 4 corner squares, and each corner square can form 1 adjacent pair (only towards the inside of the board).

\[ \text{Pairs from corner squares} = 4 \times 1 = 4 \]

### Step 3: Calculate the Total Number of Pairs that Share a Common Side
Now, we can sum the pairs from all categories:

\[ \text{Total pairs sharing a common side} = 144 + 48 + 4 = 196 \]

### Step 4: Calculate the Probability
Finally, the probability that two randomly chosen squares share a common side is the ratio of the number of favorable outcomes (pairs that share a side) to the total outcomes (all pairs of squares):

\[ \text{Probability} = \frac{\text{Number of pairs sharing a common side}}{\text{Total ways to choose 2 squares}} = \frac{196}{2016} \]

### Step 5: Simplify the Probability
To simplify \(\frac{196}{2016}\):

\[ \frac{196 \div 28}{2016 \div 28} = \frac{7}{72} \]

Thus, the final answer is:

\[ \text{Probability} = \frac{7}{72} \]