Tiling a unit square with 8 squares

TL;DR

This paper investigates the maximum total edge length of 8 nonoverlapping squares within a unit square, demonstrating that the maximum sum over all packings exceeds that over all tilings, with specific bounds established.

Contribution

It establishes that for n=8, the maximum total edge length in packings exceeds that in tilings, providing exact and lower bound values.

Findings

g(8)=13/5

f(8)≥8/3

f(8)>g(8)

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.