OPEN - $25

Classify those triangles which can only be cut into a square number of congruent triangles.

Erdős' question was reported by Soifer [So09c]. It is easy to see (see for example [So09]) that any triangle can be cut into $n^2$ congruent triangles (for any $n\geq 1$). Soifer [So09b] proved that there exists at least one triangle (e.g. one with sides $\sqrt{2},\sqrt{3},\sqrt{4}$) which can only be cut into a square number of congruent triangles. (In fact Soifer proves that any triangle for which the angles and sides are both integrally independent has this property.)

Soifer proved [So09] that if we relax congruence to similarity then every triangle can be cut into $n$ similar triangles when $n\neq 2,3,5$ and there exists a triangle that cannot be cut into $2$, $3$, or $5$ similar triangles.

See also [634].