Problem number 15, John had a large number of identical cubes. He made the structure on the right by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way, so that each face of his original structure will have a cube stuck to it. How many extra cubes will he need to complete his extended structure? So let's make a larger diagram, and notice that on this diagram, we need to stick a cube so that all of the exterior gray faces are covered. First, notice that we have a lot of these angles, which I've highlighted in red here, and these angles only need one cube to cover both of those faces. How many angles are there? Well, notice that each angle has a unique edge that it corresponds to of the center dark gray cube. So that means that the number of angles is equal to the number of edges of a cube, and we know that a cube has 12 edges. So that means that there is one cube that covers each of the angles, and we know that since there are 12 edges in blue, that that means that there will be 12 angles total. So that means we need 12 cubes to cover all of the angles of the shape. However, there are slightly more faces than the angles, because we also need to cover these protruding faces, essentially. And we notice that there is one of these protruding faces for each of the six faces of the original cube. So that means that there must be six faces, and that means there are six cubes that need to be stacked on the six outside protruding faces that are colored in teal here. And so that means the total number of cubes we need is 6 plus 12 is 18, and so the answer is A, 18.

cube structure

extra cubes

exterior faces

shared edges

protruding faces