Every Tiling of the First Quadrant by Ribbon <i>L n</i>-Ominoes Follows the Rectangular Pattern

Abstract

Let and let be the set of four ribbon L -shaped n -ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L -shaped n -ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n . Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n /2 vertical rectangles, with a tiling by n /2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L -shaped n -ominoes. We show that none of these results are valid for any odd n . The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L -shaped n -omino, n even, is not a replicating tile of order k 2 for any odd k .