Optimally Covering Large Triangles with Homothetic Unit Triangles

TL;DR

This paper extends previous work on covering large triangles with homothetic unit triangles, establishing tight bounds for cases where the number of triangles is between $n^2+4$ and $n^2+2n$, and introduces new covering methods.

Contribution

It provides new tight upper bounds for covering large triangles with $n^2 + k$ homothetic triangles for $4 \,\leq\, k \leq 2n$, and introduces optimal covering strategies.

Findings

Established upper bounds for $n^2 + k$ with $4 \leq k \leq 2n$.

Proved the bounds are tight using two new covering methods.

Developed an optimal consolidated covering method.

Abstract

We answer an open problem in the \emph{American Mathematical Monthly} about covering large triangles. Given a triangle $T$ of any triangular shape with a selected side length between $n \in \mathbb{N}$ and $n+1$, Baek and Lee proved that $T$ could not be covered with $n^2+1$ homothetic unit triangles (with the selected side of length 1). Letting $T_{n+d}$ denote a triangle with selected side length $n + d$ with $d \in (0, 1)$, Baek and Lee extended their proof to establish upper bounds for $d$ above which a $T_{n+d}$ cannot be covered with $n^2+2$ or $n^2+3$ homothetic unit triangles. Then, they showed that these bounds are tight based on analyses of a method by Conway and Soifer for the $n^2+2$ case and their own method for the $n^2+3$ case. Baek and Lee stated as an open problem the need to find tight upper bounds for the $n^2 + k$ cases for $4 \le k \le 2n$. We extend the Baek and Lee proof to establish upper bounds for those higher cases, and we show the upper bounds are tight by presenting two new triangle covering methods for the odd and even cases of $k$ that meet the upper bounds, as well as an optimal consolidated method that uses whichever of the two will cover a given $T_{n+d}$ with the fewest homothetic unit triangles.