Question

Euler’s formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2.$ A particular convex polyhedron has $32$ faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P+10T+V?$

Answer

Answer: 250

Concepts:double counting Euler’s Polyhedron Formula polyhedron

Difficulty rating: 2500

Small Hint:

Let $x$ be the number of triangular faces and count face-edge and face-vertex incidences

Big Hint:

Use Euler’s formula to express $x$ in terms of $V$ , then obtain two divisibility conditions on $V$

Solution:

Let $x$ be the number of triangular faces, so there are $32-x$ pentagons and $E=\frac{3x+5(32-x)}2=80-x.$ Euler’s formula gives $V+x=50.$ Counting face-vertex incidences yields $\begin{aligned}TV&=3x=150-3V,\\PV&=5(32-x)=5V-90.\end{aligned}$ Hence $V(T+3)=150$ and $V(5-P)=90.$ Also $18\leq V\leq50,$ so the only common divisor of $150$ and $90$ in that range is $V=30.$ Then $T=2$ and $P=2,$ giving $100P+10T+V=250.$

1993 AIME Problem 10

Answer: 250

Difficulty rating: 2500

Small Hint:

Big Hint:

Solution:

Problem 10 in Other Years