Question 87
A tetromino can be made from two $2\times1$ dominoes by placing them together along their longer sides so that the midpoint of the longer side of one is a vertex of the other. This produces two differently oriented types, called $S$- and $Z$-tetrominoes.
$S$-tetrominoes:
$Z$-tetrominoes:
A lattice polygon $P$ is a simply connected region whose boundary lies entirely along lines of the plane integer-coordinate grid. A tiling of $P$ is a complete, nonoverlapping covering by pieces no part of which lies outside $P$.
Suppose $P$ can be tiled using only $S$-tetrominoes. Prove that if $P$ can also be tiled with $S$- and $Z$-tetrominoes, the number of $Z$-tetrominoes used is always even.
$S$-tetrominoes:
$Z$-tetrominoes:
A lattice polygon $P$ is a simply connected region whose boundary lies entirely along lines of the plane integer-coordinate grid. A tiling of $P$ is a complete, nonoverlapping covering by pieces no part of which lies outside $P$.
Suppose $P$ can be tiled using only $S$-tetrominoes. Prove that if $P$ can also be tiled with $S$- and $Z$-tetrominoes, the number of $Z$-tetrominoes used is always even.