2009 AMC 12B Problem 17

Question

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?

\dfrac{1}{8}

\dfrac{3}{16}

\dfrac{1}{4}

\dfrac{3}{8}

\dfrac{1}{2}

Answer: B

Concepts:basic probability cube geometry casework

Difficulty rating: 1890

Small hint:

Each face’s stripe has

2

orientations, giving

2^6 = 64

equally likely configurations

Big hint:

An encircling stripe runs around one of

3

pairs of opposite faces; count the configurations forming a loop

Solution:

Each of the

6

faces has

2

equally likely stripe orientations, for

2^6 = 64

configurations.

An encircling stripe runs around one of the

3

pairs of opposite faces. Fixing such a band, the four faces it passes through must be aligned, with probability

\left(\dfrac{1}{2}\right)^4 = \dfrac{1}{16},

while the two remaining faces are free. The

3

possible bands are disjoint events, so the probability is

3 \cdot \dfrac{1}{16} = \dfrac{3}{16}.

Thus, the correct answer is B.