Optimally Covering Large Triangles with Homothetic Unit Triangles
Solves an open problem in combinatorial geometry about covering triangles with smaller homothetic copies, but the result is incremental as it extends known techniques to higher cases.
Baek and Lee proved that a triangle with side length between n and n+1 cannot be covered with n^2+1 homothetic unit triangles, and gave tight bounds for n^2+2 and n^2+3. This paper extends their results to all n^2+k for 4 ≤ k ≤ 2n, providing tight upper bounds and two new covering methods that meet these bounds, plus an optimal consolidated method.
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.