to the mutilated chessboard argument. 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 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. 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. By extension, any odd number of sets for either type cannot fit in a rectangle. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. This was discovered by exhausting all possibilities in a computer search. Filling a modified rectangle with one set of tetrominoes However, all three sets of tetrominoes can fit rectangles with holes: All 5 free tetrominoes fit a 7×3 rectangle with a hole. All 7 one-sided tetrominoes fit a 6×5 rectangle with two holes of the same "checkerboard color". All 19 fixed tetrominoes fit a 11×7 rectangle with a hole. Filling a rectangle with two sets of tetrominoes Two sets of free or one-sided tetrominoes can fit into a rectangle in different ways, as shown below: Etymology The name "tetromino" is a combination of the prefix tetra- 'four' (from Ancient Greek ), and "domino". The name was introduced by Solomon W. Golomb in 1953 along with other nomenclature related to polyominos. Filling a box with tetracubes Each of the five free tetrominoes has a corresponding tetracube, which is the tetromino extruded by one unit. J and L are the same tetracube, as are S and Z, because one may be rotated around an axis parallel to the tetromino's plane to form the other. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube: 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. They are shown in both a pictorial diagram and a text diagram. For boxes using two sets of the same pieces, the pictorial diagram depicts each set as a lighter or darker