An Equilateral Triangle of Side > n Cannot be Covered by n 2 + 1 Unit Equilateral Triangles Homothetic to it

Abstract

John Conway and Alexander Soifer showed that an equilateral triangle T of side slightly longer than n can be covered by n 2 + 2 unit equilateral triangles. They also conjectured that it is impossible to cover T with n 2 + 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 n 2 + 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 n 2 + 2 (respectively n 2 + 3 ) unit equilateral triangles homothetic to T.