Abstract
We show that every tiling of a convex set in the Euclidean plane $$\mathbb {R}^2$$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane $$\mathbb {R}^2$$, which is based on a surprising connection to a random walk on a directed graph.
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