# Removing squares from a chessboard

**Discrete Mathematics**Level 5

The squares of a \(2 \times 500\) chessboard are coloured black and white in the standard alternating pattern. \(k\) of the black squares are removed from the board at random. What is the minimum value of \(k\) such that the expected number of pieces the chessboard is divided into by this process is at least \(20\)?

**Details and assumptions**

The squares removed from the chessboard are not counted as pieces.

A piece of the chessboard is a set of squares joined together along edges. Being connected at corners of squares is not sufficient for two squares to be in the same piece.

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