HMMT-Nov 2009 gen, Problem 5

A polyhedron has faces that are all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?

Solution: Let $s$ be the number of square faces and $t$ be the number of triangular faces. Every edge is adjacent to exactly one square face and one triangular face. Therefore, the number of edges is equal to $4 s$, and it is also equal to $3 t$. Thus $4 s=3 t$ and $\frac{t}{s}=\frac{4}{3}$ $\fbox{\frac{4}{3}}$.

Answer: \frac{4}{3}