John makes a sequence of structures on a table, beginning with one cube. He makes the next structure by adding five cubes which hide the visible faces of the initial cube, as shown. What is the smallest number of cubes he needs to add to the second structure so that all the visible faces of the second structure are hidden? For this question, let's go layer by layer. First, let's consider the top of the topmost cube. It takes one cube to cover that face. Then, let's consider the side faces of this same topmost cube. We need four cubes to cover all four of those faces. Now, we only need to cover the sides of the bottom four cubes because all of these top faces are already covered by the blocks we just added. Then we have to put a cube here, here, in the back here, and here for these two faces. And then a cube here, here, on this side face, and on this back face to cover each of these outwards facing faces. Therefore, in total we need eight cubes on the bottom level, and it took us 13 cubes in total.

cubes

structure

visible faces

covering

sides