Every Tiling of the First Quadrant by Ribbon L n-Ominoes Follows the Rectangular Pattern

Let n 4 ≥ and let n  be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by n  . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by n  , n even, reduces to a tiling by n 2× and n 2 × 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  , n even: a rectangle can be tiled by n  , 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  , 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 n n × 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 nominoes. 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 2 2 × tile to n  , 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 k2 for any odd k.