On a Covering Problem for Equilateral Triangles

Adrian Dumitrescu, Minghui Jiang

Abstract


Let $T$ be a unit equilateral triangle, and $T_1,\dots,T_n$ be $n$ equilateral triangles that cover $T$ and satisfy the following two conditions: (i) $T_i$ has side length $t_i$ ($0 < t_i < 1$); (ii) $T_i$ is placed with each side parallel to a side of $T$. We prove a conjecture of Zhang and Fan asserting that any covering that meets the above two conditions (i) and (ii) satisfies $\sum_{i=1}^n t_i \geq 2$. We also show that this bound cannot be improved.


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