Math Gold Medalist

Lor

2023 AMC 10A

Problem 18

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?

$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Double Counting

Graph

Divisibility

Euler’s Formula

t=number of vertices at which exactly 3 edges meet

f=number of vertices at which exactly 3 edges meet

e=number of edges

3t+4f=2e

e=24

3|f

4|t

.

   Solution