An Equilateral Triangle of Side > n Cannot be Covered by n 2 + 1 Unit Equilateral Triangles Homothetic to it
Jineon Baek, Seewoo Lee · American Mathematical Monthly · 2024
John Conway and Alexander Soifer showed that an equilateral triangle T of side slightly longer than n can be covered by n2+2 unit equilateral triangles. They also conjectured that it is impossible to cover T with n2+1 unit equilateral triangles, no matter how close the side of T is to n.While the Conway–Soifer conjecture remains open, we prove an important case where the sides of the triangles used for covering are parallel to the sides of T (e.g., △ and ▽). That is, we show that if all unit equilateral triangles are required to be homothetic to T, then the minimum number of unit equilateral triangles that can cover T of side slightly longer than n is exactly n2+2.Our proof generalizes to covering T by (not necessarily equilateral) triangles of base one parallel to the x-axis and height equal to that of a unit equilateral triangle. Using our method, we also determine the largest side length n+1/(n+1) (resp. n+1/n) of T such that the equilateral triangle T can be covered by n2+2 (respectively n2+3) unit equilateral triangles homothetic to T.