Solution to: Placing Bricks
Imagine the board to be colored like a chessboard. Each brick will cover a white and a black square of the board, so the number of bricks equals the number of white squares equals the number of black squares. However, by removing the two opposite corners of the chessboard (with an even numbers of rows and columns), two squares of the same color are removed, so there is an imbalance of white and black squares.
Conclusion: No, you cannot fill the board.