Tilings of convex polygons by equilateral triangles of many different sizes

Abstract

An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles [Tutte 1948]. Therefore Tuza [Tuza 1991] asked for the largest number $s=s(n)$ such that there is a tiling of an equilateral triangle by $n$ equilateral triangles of $s(n)$ different sizes. We solve that problem completely and consider the analogous questions for dissections of convex $k$-gons into equilateral triangles, $k=4,5,6$. Moreover, we discuss all these questions for the subclass of tilings such that no two tiles are translates of each other.