2004 AMC 12B Problem 20

There are $2^6 = 64$ colorings. A suitable orientation exists when all six faces are one color ($2$ ways), exactly five faces are one color ($2 \cdot 6 = 12$ ways), or four faces are one color with the other color on a pair of opposite faces ($2 \cdot 3 = 6$ ways). That is $2 + 12 + 6 = 20$ favorable colorings, so the probability is $\dfrac{20}{64} = \dfrac{5}{16}.$ Thus, the correct answer is B.