No black neighbors

Discrete Mathematics Level 5

The squares of a \(2 \times 60\) 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 \(\frac{a}{b}\) 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 \( \frac{1}{2} \). 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.


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