Nonexistence of the 105- and 120-tile cases for (3,5,7)

Determine whether equilateral triangles can be tiled by 105 or 120 congruent copies of the triangle with side lengths (3,5,7), and prove that neither tile count is possible if the conjecture is true.

Background

For the tile (3,5,7), the paper’s construction and prior exclusions leave only the cases with 105 and 120 tiles below the established infinite family. The authors explicitly conjecture that these two remaining cases do not exist.

References

Separate work by Beeson shows that $X < 105$ is known to not exist , so this reduces understanding all equi-constructible $X$ for $(3,5,7)$ to only the cases $105$ and $120$, which we conjecture to not exist.

— Tiling Triangles with $2π/3$ Angles  (2512.22696 - Zhang, 27 Dec 2025) in Section 3.1, Equilateral Triangles