Tiling Deficient Rectangles with Trominoes

Tromino tilings of domino-deficient rectangles

“…, (4, 7)} make the cornermost squares (1, 1), (1,8), (4, 1), (4,8) (1,6), (4,3), and (4, 6) make the squares (1,8), (4,1) and (4,8) inaccessible. So, we conclude that these pairs are also bad (refer Figure 5 (1, 2), then the square (1, 1) becomes inaccessible.…”

“…Theorem 2 (Deficient Rectangle Theorem [6]) An m × n deficient rectangle, 2 ≤ m ≤ n, 3|(mn − 1), has a tiling, regardless of the position of the missing square, if and only if (a) neither side has length 2 unless both of them do, and (b) m = 5.…”

Tiling Rectangles with Gaps by Ribbon Right Trominoes

“…3 and are placed on NW-SE diagonals. The missing cell has to be the third or fifth cell on a diagonal of length 7, or be on a diagonal of length greater than 7 and not among the first two cells or the last two cells on that diagonal.…”

“…If a deficient rectangle ( ) ( ) 6 1 6 4 , , 1 p q p q + × + ≥ can be tiled by ∑ , then the missing cell has the sum of the coordinates congruent to 0 modulo 3 and are placed on NW-SE diagonals. The missing cell has to be the third or fifth cell on a diagonal of length 7, or be on a diagonal of length greater than 7 and not among the first two cells or the last two cells on that diagonal.…”

“…Chu-Johnsonbaugh [5] [6] studied the general case of boards and deficient boards: any deficient n n × board is tilable if and only if n is not divisible by 3. Later Ash-Golomb [7] found that an m n × deficient rectangle 2 ;3 | 1 m n mn ≤ ≤ − , has a tiling if and only if neither side has length 2, unless both of them do; or 5 m ≠ . Furthermore, in all the exceptional cases, they found all missing cells for which tiling is possible.…”