The Tilings of Deficient Squares by Ribbon <i>L</i>-Tetrominoes Are Diagonally Cracked

Abstract

We consider tilings of deficient rectangles by the set T 4 of ribbon L -tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3) . The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T 4 is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T 4 is twice the number of tilings by dominoes of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T 4 , we call the new tile set T⁺<sub style="margin-left:-6px;">4</sub> . A tiling of a deficient rectangle by T⁺<sub style="margin-left:-6px;">4</sub> exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3) . The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T⁺<sub style="margin-left:-6px;">4</sub> is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T⁺<sub style="margin-left:-6px;">4</sub> is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T 4 and T⁺<sub style="margin-left:-6px;">4</sub> of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T⁺<sub style="margin-left:-6px;">4</sub> . Therefore T 4 and T⁺<sub style="margin-left:-6px;">4</sub> give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.