Minimal tilings of a unit square

TL;DR

This paper investigates the optimal way to tile a unit square with n smaller squares to minimize the total sum of their side lengths, providing a mathematical optimization solution.

Contribution

It determines the minimal sum of side lengths for tiling a unit square with n small squares, a novel optimization result in geometric tilings.

Findings

Identifies the minimal total side length sum for tilings with n squares

Provides explicit formulas or bounds for the minimal sum

Advances understanding of geometric tiling optimization

Abstract

Tile the unit square with $n$ small squares. We determine the minimum of the sum of the side lengths of the $n$ small squares, where the minimum is taken over all tilings of the unit square with $n$ squares.