From: meepches at 2018-01-11 04:52:58
you can’t solve any tetris puzzle where you put any pieces into a rectangle with an odd number of T’s
it’s a variation of the mutilated chessboard problem. Think about tiling a chessboard by putting dominoes on it, such that each domino covers two spaces. Because of the layout of a chessboard, a domino must cover one white square and one black square. Therefore, if you cut off two diagonal corners so that there are 62 squares, you might think that you can tile it with dominoes because that’s an even number of squares, and dominoes cover 2 squares each. But that’s not the case. In the chessboard with the removed diagonals there are 32 Black squares and 30 White squares. Because a domino placed on a chessboard must cover one black and one white square, you will always end up with at least two black squares showing once you’ve placed so many dominoes that you can no longer place another domino.
how this applies to the tetris problems is that you need to think of the rectangle as being tiled like a chessboard, and tetris pieces being tiled as well. Because a tetris piece can be placed anywhere on the board, it doesn’t have specific White or Black squares on it, but that doesn’t really matter. Every tetris piece, except for the T, will cover an equal number of black and white squares when placed on a chessboard. The T, however, when placed on a chessboard, will cover three of one color and one of another. Therefore, any tetris puzzle where you need to cover an equal number of “Black” and “White” squares can’t be solved if it has an odd number of T pieces, because the T will cause a disparity in numbers of each color covered.