If the probability that two queens, placed at random on a chess-board, do not take on each other, is `K/207`, then K equals______

Answer

To solve the problem, we need to find the value of \( K \) such that the probability that two queens placed at random on a chessboard do not attack each other is \( \frac{K}{207} \).

### Step-by-Step Solution:

1. **Total Number of Ways to Place Two Queens**:
- A chessboard has 64 squares (8 rows and 8 columns).
- The total number of ways to place 2 queens on the chessboard is given by choosing 2 squares out of 64:
\[ \text{Total ways} = \binom{64}{2} = \frac{64 \times 63}{2} = 2016 \]

2. **Number of Ways Queens Can Attack Each Other**:
- Queens can attack each other if they are in the same row, column, or diagonal.
- **Same Row**: There are 8 rows, and in each row, we can choose 2 squares:
\[ \text{Ways in same row} = 8 \times \binom{8}{2} = 8 \times 28 = 224 \]
- **Same Column**: Similarly, there are 8 columns:
\[ \text{Ways in same column} = 8 \times \binom{8}{2} = 224 \]
- **Diagonals**: We need to consider both types of diagonals (positive and negative slope).
- For diagonals of length 2 to 8, we calculate:
- Length 2: 2 ways (2 diagonals)
- Length 3: 3 ways (2 diagonals)
- Length 4: 6 ways (2 diagonals)
- Length 5: 10 ways (2 diagonals)
- Length 6: 15 ways (2 diagonals)
- Length 7: 21 ways (2 diagonals)
- Length 8: 28 ways (1 diagonal)
- Total for both diagonals:
\[ \text{Total diagonal ways} = 2 + 2 + 6 + 10 + 15 + 21 + 28 = 84 \]
- Therefore, total ways queens can attack each other:
\[ \text{Total attacking ways} = 224 + 224 + 84 = 532 \]

3. **Calculating the Probability**:
- The probability that two queens attack each other is:
\[ P(\text{attack}) = \frac{\text{Ways to attack}}{\text{Total ways}} = \frac{532}{2016} \]
- The probability that they do not attack each other is:
\[ P(\text{not attack}) = 1 - P(\text{attack}) = 1 - \frac{532}{2016} = \frac{2016 - 532}{2016} = \frac{1484}{2016} \]

4. **Simplifying the Probability**:
- We simplify \( \frac{1484}{2016} \):
- The GCD of 1484 and 2016 is 148. Thus:
\[ \frac{1484 \div 148}{2016 \div 148} = \frac{10}{13.5} = \frac{74}{108} \]
- This simplifies to:
\[ \frac{37}{54} \]

5. **Setting Up the Equation**:
- According to the problem, we have:
\[ \frac{K}{207} = \frac{37}{54} \]
- Cross-multiplying gives:
\[ K \cdot 54 = 37 \cdot 207 \]
- Calculating \( 37 \cdot 207 \):
\[ 37 \cdot 207 = 7659 \]
- Thus:
\[ K = \frac{7659}{54} = 141 \]

### Final Answer:
The value of \( K \) is \( \boxed{141} \).