Problem:
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?
Answer Choices:
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Solution:
The stripe on each face of the cube will be oriented in one of two possible directions, so there are possible stripe combinations on the cube. There are pairs of parallel faces so, if there is an encircling stripe, then the pair of faces that do not contribute uniquely determine the stripe orientation for the remaining faces. In addition, the stripe on each face that does not contribute may be oriented in different ways. Thus a total of stripe combinations on the cube result in a continuous stripe around the cube, and the requested probability is .
Without loss of generality, orient the cube so that the stripe on the top face goes from front to back. There are two mutually exclusive ways for there to be an encircling stripe: either the front, bottom, and back faces are painted to complete an encircling stripe with the top face's stripe, or the front, right, back, and left faces are painted to form an encircling stripe. The probability of the first cases is , and the probability of the second case is , so the answer is .
There are three possible orientations of an encircling stripe. For any one of these to appear, the four faces through which the stripe is to pass must be properly aligned. The probability of one such stripe alignment is . Because there are such possibilities, and these events are disjoint, the total probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions