Tiling a unit square with 8 squares

Abstract

Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote the maximum values of the sum of the edge lengths of the n small squares, where in the case of f(n) the maximum is taken over all arbitrary packings of the unit square, and in the case of g(n) it is taken over all tilings of the unit square (i.e., the total area of the n small squares is 1). Benton and Tyler asked for which values of n we have f(n)=g(n). We show that f(8)>g(8). More precisely, we show that g(8)=13/5; it is known that f(8) is at least 8/3.

8 pages; revised (shorter) proof