Tiling a unit square with 8 squares

摘要

Place n non-overlapping small squares of side lengths s_1, s_2,...,s_n inside a unit square; such a configuration is called a packing of the unit square. We define ψ_1(n) = max ∑_i~n=1 s_i, where the maximum is taken over all possible packings of the unit square. Not a lot is known about the function ψ_1. Erdos [1] asked whether ψ_1(k~2 + 1) = k. More generally, Erdos and Soifer [2] presented explicit packings that provided lower bounds of ψ_1(n) for all n; they mentioned that these lower bounds appear to be good. Thus we have tentative values for ψ_1(n).