2009
Problem - 1584
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the 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?
There are two possible orientation of the strip on each surface. Therefore the total number of cases is $2^6=64$.
There are $3$ pairs of parallel surfaces, therefore there are $3$ possible encircling strips. Meanwhile, the orientation of the strips on those two surfaces which are not part of the encircling strip decides the encircling strip. Each of the two faces has two possibilities, hence the total is $2\times 2=4$ which means there are $3\times 4=12$ qualified cases.
It follows that the answer is $$\frac{12}{64}=\boxed{\frac{3}{16}}$$