Truncated Tetrahedron, Octahedron and Cube Designs

We prove that the complete graph \( K_v \) can be decomposed into truncated tetrahedra if and only if \( v \equiv 1 \text{ or } 28 \pmod{36} \), into truncated octahedra if and only if \( v \equiv 1 \text{ or } 64 \pmod{72} \), and into truncated cubes if and only if \( v \equiv 1 \text{ or } 64 \pmod{72} \).