## 2 Polyomino Tiling Theory
: _A deficient Aztec Diamond, AZ(k) can be tiled using 'T' set of tromino tiling pieces if, and only if \(k\) = (\(3n-2\)), where \(n\) can be a positive integer._  
An Aztec diamond of order n is the region obtained from staircase shapes of height n by gluing them together along the straight edges. According to Theorem 4, a non-deficient Aztec diamond can be tiled using the 'T' set of tromino pieces if and only if n (\(n+1\)) \(\equiv\) 0 mod 3. Hence, it is clear that the tabbed \(n\)th order must be divisible by 3 after applying it to \(n\) (\(n+1\)), were n can be any integer. Similarly, with Theorem 5, it is clear that, we can tile a defected Aztec diamond (with one square removed) using 'T' tromino set only if the \(n\)th order is equivalent to \(3n-2\), where n can be any integer. Below mentioned Lemma 5 and 6 supports the conditions of Theorems 4 and 5. **Lemma 5**. : _Note that the only values for which \(n\) (\(n+1\)) \(\equiv\) 0 (mod 3) holds are \(n=3\) k or \(n=3k-1\) for some unique positive integer k. Thus, the statement is equivalent to say that for all positive integers k there is a tiling for AZ (\(3k\)) and AZ (\(3k-1\)) as shown in Figure 7 but there is no tiling for AZ (\(3k-2\)). The tiling of Aztech diamond can be achieved by tiling the edges of considered space. We used stairs concept in order to tile the edges of the Aztec diamond. A stair is a polyomino made-up of tromino pieces wherein their \(180^{\circ}\) rotations are connected as steps shown in Figure 8. The height of the stair was computed under the formulation of \(3k+2\), for any positive integer k. The height of the stair is equal to the order of Aztec diamond. If the order of Aztec diamond is \(n\)\(\leq\) 4, then AZ (\(n\)) can be tiled using 'T' trominoes through tiling partial or half stairs as shown in Figure 7 (Top Right & Left). If the order \(n\)\(\geq\) 5, then the AZ (\(n\)) can be tiled using K-stair of 'T' tromino pieces. The image argument for order \(n\) = 5 is shown in Figure 7 (bottom), green colored tromino pieces tiled the edges using 1-stair._  
Figure 6: Image Argument for Lemma 4. (**a**) Decomposing a Modified rectangle with a 4 \(\times\) 4 tiled modified rectangle; (**b**) A 10 \(\times\) 10 modified decomposed rectangle. **Lemma 6**. : _To tile AZ (3k \(-\) 2) with one defect a fringe appearing has been used as shown in Figure 9 left. It is easy to check that if a fringe has exactly one defect, then it can be covered with T\({}^{\prime}\) frominoes. In particular, a possible rectangle with a fringe that can be tiled by T\({}^{\prime}\) fromino is 2 \(\times\) 2 square. Similarly, as lemma 5, the stair patterns were used to tile the edges of the concern defected Aztec diamond. The space with fringe placed is considered as a 2 \(\times\) 2 square plot and can be tiled using T\({}^{\prime}\) frominoes Figure 9 left. The other space of defected Aztec diamond was tiled similarly to Lemma 5. In another approach, tiling pattern of Figure 8 was used wherein the k-stairs laid above and below the fringe in order to tile the Aztec diamond with T\({}^{\prime}\) frominoes, as shown in Figure 9 Right._  
Figure 8: Stair Edge Tiling. Figure 7: Image Argument for Lemma 5.