2004 AMC 10B Problem 23

Fixing the orientation, there are $2^6 = 64$ colorings. A coloring works if all six faces match ($2$ ways), exactly five match ($\binom{6}{5} \cdot 2 = 12$ ways), or four faces share a color with the remaining pair being opposite faces of the other color ($3$ opposite pairs, $2$ colors, giving $6$ ways). The total is $2 + 12 + 6 = 20,$ so the probability is $\dfrac{20}{64} = \dfrac{5}{16}.$ Thus, the correct answer is B.