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Prove that an 8 x 8 board cannot be covered by 15 L-tetrominos and one square tetromino (an L-tetromino is a plane figure shown below, constructed from four unit squares arranged in the form of L; a square tetromino is a plane figure shown below, constructed from four unit squares arranged in the form of a square)
Do the following: color every cell on an odd row in white and color every cell on an even row in black. Suppose that the board can be covered with $15$ $L$-shaped tetrominos and one square-shaped tetromino. Notice now that since your $L$-shaped tetrominos will always cover an odd number of white cells and your square-shaped tetromino will alway cover an even number of white cells the total number of covered white cells will be always be odd. But this is not possible since the total number of white cells is $32$ which is an even number. Contradiction.
