Problem:
Twenty-seven unit cubes are each painted orange on a set of four faces so that the two unpainted faces share an edge. The cubes are then randomly arranged to form a cube. Given that the probability that the entire surface of the larger cube is orange is , where , and are distinct primes and , and are positive integers, find .
Solution:
A cube can be oriented in ways because each of the six faces can be on top and each of the top face's four edges can be at the front. There are eight corner cubes in the large cube. For the corner cubes, six orientations will expose three orange faces. This is because there are two sets of three orange faces that can be exposed. For each such set, each of the three orange faces can appear in a given position, and the positions of the other two are then determined. Thus the probability that all corner cubes expose three orange faces is . For cubes at the center of an edge, there are orientations that expose two orange faces. This is because there are five sets of two orange faces that share an edge, and each such set can appear in two orientations. The probability that all of these edge cubes expose two orange faces is . A cube that is in the center of a face can have any of the four orange faces outward in four orientations, and thus there is a probability of that each center cube exposes an orange face. Thus the probability that the entire surface of the larger cube is orange is
and .
The large cube contains eight corner unit cubes, twelve unit cubes at the center of an edge, and six unit cubes at the center of a face. All visible faces of a unit cube are orange if and only if the shared edge of its two unpainted faces, except perhaps for an endpoint, is in the interior of the large cube. The number of edges interior to the large cube is three for a corner cube, five for a cube at the center of an edge, and eight for a cube at the center of a face. Thus the probability that the entire surface of the large cube is orange is
and, as above, .
The problems on this page are the property of the MAA's American Mathematics Competitions