![]() The tetracubes can be packed into two-layer 3D boxes in several different ways, based on the dimensions of the box and criteria for inclusion. This was discovered by exhausting all possibilities in a computer search. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. By extension, any odd number of sets for either type cannot fit in a rectangle. Similarly, a 7×4 rectangle has 28 squares, containing 14 squares of each shade, but the set of one-sided tetrominoes has either 15 dark squares and 13 light squares, or 15 light squares and 13 dark squares. This is due to the T tetromino having either 3 dark squares and one light square, or 3 light squares and one dark square, while all other tetrominoes each have 2 dark squares and 2 light squares. A 5×4 rectangle with a checkerboard pattern has 20 squares, containing 10 light squares and 10 dark squares, but a complete set of free tetrominoes has either 11 dark squares and 9 light squares, or 11 light squares and 9 dark squares. This can be shown with a proof similar to the mutilated chessboard argument. Tiling a rectangle Filling a rectangle with one set of tetrominoes Ī single set of free tetrominoes or one-sided tetrominoes cannot fit in a rectangle.
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