Question 12
Given a triangle, its *midpoint triangle* is obtained by joining the midpoints of its sides. A sequence of polyhedra $\mathcal{P}_i$ is defined recursively as follows: $\mathcal{P}_0$ is a regular tetrahedron whose volume is $1.$ To obtain $\mathcal{P}_{i+1},$ replace the midpoint triangle of every face of $\mathcal{P}_i$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $\mathcal{P}_3$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$