Erd\H{o}s' question was reported by Soifer \cite{So09c}. It is easy to see (see for example \cite{So09}) that any triangle can be cut into $n^2$ congruent triangles (for any $n\geq 1$). Soifer \cite{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 \cite{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].


References


[So09] Soifer, Alexander, How Does One Cut a Triangle? I. (2009), 15-23.

[So09b] Soifer, Alexander, How Does One Cut a Triangle? II. (2009), 37-39.

[So09c] Soifer, Alexander, Is there anything beyond the solution?. (2009), 47-50.