OPEN - $25 Find all$N$such that there is at least one triangle which can be cut into$n$congruent triangles. Erdős' question was reported by Soifer [So09c]. It is easy to see that all square numbers have this property (in fact for square numbers any triangle will do). Soifer [So09c] has shown that numbers of the form$2n^2,3n^2,6n^2,n^2+m^2$also have this property. Beeson has shown (see the slides below) that$7$and$11$do not have this property. It is possible than any prime of the form$4n+3$does not have this property. In particular, it is not known if$19$has this property (i.e. are there$19$congruent triangles which can be assembled into a triangle?). For more on this problem see these slides from a talk by Michael Beeson. As a demonstration of this problem we include a picture of a cutting of an equilateral triangle into$27$congruent triangles from these slides. Soifer proved [So09] that if we relax congruence to similarity then every triangle can be cut into$N$similar triangles when$N\neq 2,3,5$. If one requires the smaller triangles to be similar to the larger triangle then the only possible values of$N$are$n^2,n^2+m^2,3n^2\$, proved by Snover, Waiveris, and Williams [SWW91].

See also [633].

Additional thanks to: Boris Alexeev, Yan Zhang