Imagine a chessboard, consisting of \(64\) squares arranged in an \(8\times8\) grid. If you had \(32\) dominoes, each the size of two squares \( (2 \times 1 ) \), you could perfectly cover the entire board. Now, think about removing two corners from the chessboard that are diagonally opposite to each other. Can you still cover this modified board with \(31\) dominoes?