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

Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles