2014 IMO Shortlist 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. S-tetrominoes Z-tetrominoes 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)

通过沿其较长边连接两个 $2 \times 1$ 多米诺骨牌来构建四块骨牌，使得一个多米诺骨牌较长边的中点是另一个多米诺骨牌的角。这种结构产生了两种方向相反的四格骨牌。让我们分别称它们为 Sand Z-tetrominoes。 S-四格骨牌 Z-四格骨牌 假设格子多边形 $P$ 可以用 S-四格骨牌平铺。证明无论我们如何仅使用 S- 和 Z-四联骨牌来平铺 $P$，我们总是使用偶数个 Z-四联骨牌。 (匈牙利)