  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.
