Tiling the unit square with squares and rectangles

Abstract

Call a rectangle small if it will fit inside the unit square; call a rectangle binary if its dimensions are powers of 2. • Theorem 1. If S is a set of small binary squares having total area greater than 1, some finite subset of S will tile the unit square . • Theorem 2. If R is a set of small binary rectangles having total area greater than 5 2 , some finite subset of R will tile the unit square . The above bounds are sharp. Various corollaries are drawn. In particular we show that if R is a collection of rectangles, then R can cover the plane iff Σ{min( a, 1) · min( b, 1) | R ϵ, R }; = ∞, where for a given rectangle, a is its width and b is its length.