AMC 12 2009b Problem 17

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?

\textbf{(A)}\ \frac {1}{8} \textbf{(B)}\ \frac {3}{16} \textbf{(C)}\ \frac {1}{4} \textbf{(D)}\ \frac {3}{8} \textbf{(E)}\ \frac {1}{2}