Many L-Shaped Polyominoes Have Odd Rectangular Packings

Abstract

A polyomino is called odd if it can tile a rectangle using an odd number of copies. We give a very general family of odd polyominoes. Specifically, consider an L-shaped polyomino, i.e., a rectangle that has a rectangular piece removed from one corner. For some of these polyominoes, two copies tile a rectangle, called a basic rectangle. We prove that such a polyomino is odd if its basic rectangle has relatively prime side lengths. This general family encompasses several previously known families of odd polyominoes, as well as many individual examples. We prove a stronger result for a narrower family of polyominoes. Let L _n denote the polyomino formed by removing a 1 ×  (n−2) corner from a 2 ×  (n−1) rectangle. We show that when n is odd, L _n tiles all rectangles both of whose sides are at least 8n ³, and whose area is a multiple of n. If we only allow L _n to be rotated, but not reflected, then the same is true, provided that both sides of the rectangle are at least 16n ⁴. We also give several isolated examples of odd polyominoes.