Constructing a Platonic solid

There are five Platonic solids: Octahedron, Cube, Tetrahedron, Icosahedron, and Dodecahedron.

The only two Platonic solids that can be formed from different Platonic solids are the tetrahedron and the octahedron. (The cube is out, since 364 isn't a cubic number)

The formulae for constructing these solids (thanks @Michael Mendrin for providing these!) are as follows:

To make tetrahedra of linear size $n$ , you need

• $\frac{1}{6}n(n^2 - 1)$ octahedra
• $\frac{1}{3}n(n^2 + 2)$ tetrahedra

For example for $n=2$ we have this construction:

To make octahedra of linear size $n$ , you need

• $\frac{1}{3}n(2n^2 + 1)$ octahedra
• $\frac{4}{3}n(n^2 -1)$ tetrahedra

For example, for $n=2$ , we have this construction:

For integer $n$ , the only equation which gives $364$ of any of these is the first one.

This is for $n = 13$ , which implies you have $364$ ocathedra and $741$ tetrahedra and you are constructing a large tetrahedron.

So the total number of Platonic solids you begin with is $364+741=\boxed{1105}$