A queen on an English chessboard is able to attack in the same row, column and diagonal. The probability that 2 randomly placed queens on an 8 by 8 chessboard will be able to attack each other can be expressed as ab, where a and b are positive, coprime integers. What is the value of a+b?
To find the probability that two randomly placed queens on an 8 by 8 chessboard will be able to attack each other, we need to determine the total number of possible queen placements and the total number of queen placements where they can attack each other.
Total possible queen placements:
The first queen can be placed anywhere on the chessboard, so it has 8 options for the row and 8 options for the column. Therefore, there are a total of 8 x 8 = 64 possible placements for the first queen.
The second queen can also be placed anywhere on the chessboard, but we need to make sure it is not in the same row, column, or diagonal as the first queen.
Total number of queen placements where they can attack each other:
If the second queen is placed in the same row as the first queen, there is only 1 possible position, as the second queen cannot be in the same column.
If the second queen is placed in the same column as the first queen, there is only 1 possible position, as the second queen cannot be in the same row.
If the second queen is placed in a diagonal position relative to the first queen, there are a total of 14 possible positions (7 on each diagonal). This is because for each diagonal, we need to consider both the forward and backward directions.
Therefore, the total number of queen placements where they can attack each other is 1 (same row) + 1 (same column) + 14 (diagonal) = 16.
Probability calculation:
The probability is calculated by dividing the total number of queen placements where they can attack each other by the total number of possible queen placements.
Probability = (number of queen placements where they can attack each other) / (total number of possible queen placements)
= 16 / 64
= 1 / 4
Therefore, the probability that two randomly placed queens on an 8 by 8 chessboard will be able to attack each other is 1/4.
The value of a+b is 1 + 4 = 5.