Classification of possible tile counts in the 2π/3-angle cases

Classify all tile counts N arising when a triangle T is tiled by congruent copies of a triangle R having an angle equal to 2π/3, particularly for the six sporadic non-commensurable-angle configurations, and prove that the construction families described in the paper are the only possible values for sufficiently large N.

Background

The paper studies six sporadic non-commensurable-angle configurations in which the tile has an angle of 2π/3. It constructs infinite families of tilings for these cases, but explicitly conjectures that no other sufficiently large tile counts occur.

References

For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles.

— Tiling Triangles with $2π/3$ Angles  (2512.22696 - Zhang, 27 Dec 2025) in Abstract and Section 1, Introduction