Problem 634

Erdős Problem #634: declared status 'open'. Formalized: no. Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles. Current best: Zhang [Zh25], among other results, has proved that for any integers $a \geq b$, if$$n\geq 3\left\lceil \frac{a^2+b^2+ab-a-b}{ab}\right\rceil$$then $n^2ab$ has this property (indeed, they explicitly show that an equilateral triangle can be tiled with $n^2ab$ many triangles of side lengths $a,b,\sqrt{a^2+b^2+2+ab}$). Prize: $25. Tags: geometry.