In Tetris, players try to efficiently pack a rectangular space with pieces that can have seven different shapes. As it happens, if you have exactly one of each of those seven possible pieces, then it is impossible to arrange them into a rectangle. But there is one piece you can discard that will allow you to make such an arrangement. Which piece must be discarded? Find any rectangular arrangement of the remaining six pieces.
Just as in Tetris, you may rotate pieces but not reflect them. In other words, if the pieces were cutouts on a table, then you could slide them and spin them but never lift them off the table. Tip: you can determine the problematic piece in a principled way; it does not require trial and error.
The T-shaped piece must be discarded to form a rectangle with the remaining pieces. To see why, note that each Tetris piece comprises four squares. Imagine the rectangle that you’re trying to form has a black-and-white checkerboard pattern, where each black or white square is the same size as the squares in the Tetris pieces. Each piece, no matter how it is placed, will cover exactly two white squares and two black squares of the rectangle’s checkerboard except for the T-shaped piece, which will cover three of one shade and one of the other. Any rectangle comprised of Tetris pieces must contain an even number of squares total (four squares per piece). This means the rectangle will have an equal number of black squares and white squares. The T-shaped piece will create an imbalance that none of the other pieces can correct.
Discarding the T-shaped piece, we can create a 3 × 8 or a 4 × 6 rectangle from the other pieces like so:
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