IMO 2014 Shortlist C4
Construct a tetromino by attaching two 2 ˆ 1 dominoes along their longer sides such that the midpoint of the longer side...
Category: Combinatorics
Problem
Construct a tetromino by attaching two 2 ˆ 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 S- and 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)