2021 ICO Advanced P5
Source:
August 9, 2021
combinatoricsTilingpolyomino
Problem Statement
By a \emph{tile} we mean a polyomino (i.e. a finite edge-connected set of cells in the infinite grid). There are many ways to place a tile in the infinite table (rotation is allowed but we cannot flip the tile). We call a tile special if we can place a permutation of the positive integers on all cells of the infinite table in such a way that each number would be maximum between all the numbers that tile covers in at most one placement of the tile.1. Prove that each square is a special tile.
2. Prove that each non-square rectangle is not a special tile.
3. Prove that tile is special if and only if it looks the same after rotation.