%0 Journal Article %T The Tilings of Deficient Squares by Ribbon L-Tetrominoes Are Diagonally Cracked %A Viorel Nitica %J Open Journal of Discrete Mathematics %P 165-176 %@ 2161-7643 %D 2017 %I Scientific Research Publishing %R 10.4236/ojdm.2017.73015 %X We consider tilings of deficient rectangles by the set T₄ 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)¡Á(4m+1) and in an even position if the square is (4m+3)¡Á(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2¡Á4 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¡Á3 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)¡Á(4m+1) deficient square by T₄ is equal to the number of tilings by dominoes of a 2m¡Á2m square. The number of tilings of a (4m+3)¡Á(4m+3) deficient square by T₄ is twice the number of tilings by dominoes of a (2m+1)¡Á(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¡Á2 tile is added to T₄ , we call the new tile set T+4. A tiling of a deficient rectangle by T+4 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)¡Á(4m+1) and in an even position if the square is