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.
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