Imagine you have a standard 8×8 chessboard and 31 dominoes. Each domino can cover exactly two adjacent squares on the board. The challenge is to cover the entire board using the 31 dominoes.
Now, let’s add a twist.
Two corners of the chessboard (diagonally opposite corners) are cut out, leaving 62 squares.
Can you still use the 31 dominoes to cover every single square of this modified board without any overlap or overhang?
No, you can’t cover the modified board with 31 dominoes.
Explanation:
Each domino you place on the board will always cover one black square and one white square. Therefore, 31 dominoes will cover 31 black squares and 31 white squares exactly.
However, when you cut out two diagonally opposite corners of the chessboard, you are removing two same-colored squares (for instance, two black squares). This means the modified board has 32 squares of one colour (white, in this case) and only 30 squares of the other colour (black).
Since you can’t place a domino on the board such that it covers two squares of the same colour, you can’t cover the modified board with 31 dominoes.
