4. Two dominoes are glued along their longer sides to form a "tetromino," such that the midpoint of one domino's longer side is a vertex of the other domino. This results in two types of tetrominoes based on their orientation, referred to as S-tetromino and Z-tetromino (as shown in Figure 1).
If a lattice polygon can be covered by S-tetrominoes (without overlap), prove: regardless of how is covered by S-tetrominoes and Z-tetrominoes (without overlap), the number of Z-tetrominoes is always even.
4. Suppose the polygon is the union of some squares on an infinite chessboard, and each square of the chessboard is colored black or white as shown in Figure 2.
Regardless of how the S-tetrominoes and Z-tetrominoes cover the polygon , each S-tetromino covers an even number of black squares, while each Z-tetromino covers an odd number of black squares.
Since the polygon can be covered by S-tetrominoes, the polygon contains an even number of black squares.
If S-tetrominoes and Z-tetrominoes cover an even number of black squares, then the number of Z-tetrominoes must be even.