case 4: the group is collectively y -monotone 

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... treat the U as a 2 3 rectangle, which is y -monotone. In the second case, there are two polyominoes in the group, and it is orthogonally convex. In this case, the group is by definition y -monotone. In the third case, there are two polyominoes in the group, which is not orthogonally convex. It can be shown by a case analysis (see Figure 6) that all of these groups are y monotone, although they are not x -monotone. In the last case, there are two U ’s grouped with the same third polyomino. In order to touch a U ’s pocket, a polyomino must have a square in which three edges are exposed. Since any pentomino can have at most two of these partially exposed squares oriented in the y -direction, the largest number of U ’s that be attached to any pentomino is two. After these three pentominoes are grouped together, no more U ’s can attach, because the group no longer has any vertically oriented partially exposed squares. We further claim that any group formed by two U ’s and a third pentomino must be y -monotone. The accuracy of these claims is verified by the case analysis shown in Figure 7. Thus, we have proved the claim that pentominoes cannot interlock. Next, we find a lower bound for polyomino interlockedness. We show that hexominoes can interlock. We also investigate the computational complexity of deciding interlockedness of systems containing ...