ISL 2014 C4

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them Sand Z-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with S-tetrominoes. Prove than no matter how we tile $P$ using only S - and Z -tetrominoes, we always use an even number of Z -tetrominoes. (Hungary)