Optimally Packing a Large Square by Unit Squares

TL;DR

This paper demonstrates that a large square can be efficiently packed with unit squares, achieving a waste space that grows at most proportionally to the side length raised to the power of 3/5.

Contribution

It introduces a method for packing large squares with unit squares that minimizes wasted space to an order of x^{3/5}, improving previous bounds.

Findings

Waste space W(x) = O(x^{3/5}) for large square packing

Efficient packing method reduces wasted space significantly

Provides theoretical bounds on packing efficiency

Abstract

We show that a large square of sidelength $x$ can be packed by unit squares in a manner so that the wasted space $W(x) = O(x^{3/5})$.