The squares of a 2×60 chessboard are coloured black and white in the standard alternating pattern. At random, exactly half of the black squares are removed from the board. The expected number of white squares that have no neighbours after this process can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?
Details and assumptions:
>Note: The black squares are not removed with probability 12. Rather, it is given that exactly 30 black squares are removed.
>A square is a neighbour if it is located directly to the left, right, top or bottom of the initial square. Squares that are connected by exactly 1 vertex are not neighbors.
To solve this problem, let's break it down into smaller steps:
Step 1: Count the total number of white squares on the chessboard.
In a standard alternating pattern on a 2×60 chessboard, half of the squares will be white. So, we have (2 × 60) / 2 = 60 white squares.
Step 2: Count the number of white squares that have neighbors.
To determine the number of white squares with neighbors, we need to consider the possible configurations of white squares that can have neighbors.
i) The first row and the last row of the chessboard:
In these rows, each white square will have 3 neighbors: one on the left, one on the right, and one below (except for the corners, which will have only 2 neighbors). There are 60 white squares in each of these two rows, so the total number of white squares with neighbors in these rows is 60 × 3 = 180.
ii) The remaining rows:
In these rows, each white square will have 4 neighbors: two on the left and right, and one each above and below (except for the corners, which will have 3 neighbors). There are 2 × 60 - 4 = 116 white squares in each of these rows (subtracting 4 because the corners will have only 3 neighbors). So, the total number of white squares with neighbors in these rows is 116 × 4 = 464.
Therefore, the total number of white squares with neighbors is 180 + 464 = 644.
Step 3: Calculate the expected number of white squares that have no neighbors.
The expected value can be calculated by dividing the number of white squares with no neighbors by the total number of white squares.
Since exactly half of the black squares (30 in this case) are removed, we know that exactly half of the white squares will be removed as well. So, the remaining number of white squares is 60 - 30 = 30.
The number of white squares without neighbors is the difference between the total number of white squares and the number of white squares with neighbors: 30 - 644 = -614.
Therefore, the expected number of white squares that have no neighbors is -614 divided by the total number of white squares, which is 30.
Putting it all together, the expected value is -614/30 = -307/15.
Finally, the value of a + b is |-307| + 15 = 307 + 15 = 322.