Question number seven. The outside of the box is painted with gray and white squares in such a way that it appears as if the box is made using smaller gray and white cubes, as shown over here. Which of the following diagrams shows the surface of the box? So the first difficulty is to understand that we are not sure which face on the box corresponds with which face in the diagram. We don't know what the orientation is. So let's study the faces of the box carefully and decide which choices are not possible. We notice that on each of the visible faces, two shaded squares share a corner, and so we must be able to find that arrangement in each of the diagrams. And in A, we find the three that we need, so A is fine in this respect. In B, we don't have any, because although it appears there is one over here, we have maybe a second one there. There is no third one, and these do not belong to a face. They're actually on the edges. Now in C, we have plenty such things, so C is okay. In D, again, it appears we have some, but none of them belong to a face, so we discount D. And then in E, we find at least three that we need. Now the second thing we notice is that the three shaded squares here that share two edges and appear as a cube in the corner must look like this arrangement in the unfolded diagram, and we must have two of them, because after a rotation, the opposite view here that we are not shown would look exactly the same. And so we find such an arrangement in E, and we find such a thing in A. And in C, unfortunately, we don't have such a thing so we must discount C. And finally, the last thing to observe is that taking opposite faces one at a time, we see that here there is a white cube, but the opposite face would have a gray cube, a shaded square. And so likewise over here, the colors are opposite. And if we look from this point of view, we have the same situation. A white face corresponds to a gray face, and finally, likewise over here. So what we notice is that opposite faces have opposite coloration, and none of them have gray squares that share an edge, which is precisely what happens in A over here. This face would be opposite that face in the cube, and we cannot have this edge being shared by two gray cubes. So we must discount A as well, and the only remaining possibility is answer E.

box diagram

painted surface

shaded squares

cube arrangement

diagram selection