Question

Each face of a cube is painted either red or blue, each with probability $\tfrac12.$ The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

$\dfrac{1}{4}$

$\dfrac{5}{16}$

$\dfrac{3}{8}$

$\dfrac{7}{16}$

$\dfrac{1}{2}$

Solution:

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.

2004 AMC 12B Problem 20

Solution:

Problem 20 in Other Years