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 k2 for any odd k.
Highlights
We study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino
The L-shaped polyomino we study is placed in a square lattice and is made out of n, n ≥ 3, unit squares, or cells (see Figure 1(a))
A ribbon polyomino [3] is a connected polyomino without two unit squares lying along a line parallel to the first bisector y = x
Summary
We study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino. Not much is known about tiling integer sides rectangles with an odd side if we allow in the set of tiles all 8 orientations of an L-shaped n-omino, n even. It is known, and can be proved via a coloring argument, that if n ≡ 0 mod 4, the area of a rectangle that can be tiled is a multiple of 2n. No definitive results are known about tiling odd integer sides rectangles if the set of tiles consists of all 8 orientations of an L-shaped n-omino, n odd, despite serious computational effort invested in this question by various authors. When the height of the dissected rectangle is odd, the problem is completely solved by showing that always there exist tilings by the tile set that do not follow the rectangular pattern. More evidence supporting the conjecture is shown in the recent paper [14], were certain dissections of rectangles of base equal to 3 and higher are considered, as well as tilings of all four quadrants
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