Consider a chessboard, from which we cut out the top-left square and the bottom-right square. Can the remaining 62 squares be paved with dominoes?
Problem 946
Official solution
A very classic problem: each domino covers one white square and one black square, so no matter how the dominos are arranged, they will cover as many white squares as black squares. However, if the two squares at the top left and bottom right of the chessboard are cut out, they will be of the same color, both black or both white. Therefore, there will be either 30 black squares and 32 white squares or 32 black squares and 30 white squares left. If it is possible to place 30 dominos (which remains to be proven), the last two squares will necessarily be of the same color, and it will not be possible to place another domino: it is therefore not possible to tile the entire chessboard in this way.