Rigid Tilings of Quadrants by L-Shaped n-ominoes and Notched Rectangles

In this paper, we examine rigid tilings of the four quadrants in a Cartesian coordinate system by tiling sets consisting of L-shaped polyominoes and notched rectangles. The first tiling sets we consider consist of an L-shaped polyomino and a notched rectangle, appearing from the dissection of an n×n square, and of their symmetries about the first diagonal. In this case, a tiling of a quadrant is called rigid if it reduces to a tiling by n× n squares, each of the squares in turn tiled by an L-shaped polyomino and a notched rectangle. We further determine the rigidity of tilings of the quadrants with tiling sets appearing from similar dissections of mn × n rectangles. Our technique of proof is to use induction along a staircase line built out of n × n squares and to show that the existence of a tile in irregular position propagates further towards the edges of the quadrant eventually leading to a contradiction. Further generalizing, we examine sets of tiles appearing from dissections of rectangles of co-prime dimension into an L-shaped polyomino and a notched rectangle. These tilings are never rigid. We give descriptions of nonrigid tilings for each quadrant and for each tiling set of this type. Finally we record some general conjectures about problems of this type.