Question number 23. The picture shows the same cube from two different views. It is made up of 27 smaller cubes, some of which are gray and some of which are white. What is the largest number of gray cubes there could be? First, let's count the number of gray cubes that we can see. And so we have this corner cube here. Now, I will label that one as the first one. Then opposite of it, we have cubes two and three. And the center of this face here is another cube. That's cube four. And we observe that if we do the following, and that is, imagine that there is an axis here passing through the center of the cube and then out the bottom like that. And we are rotating this cube like so. This face containing cubes one, two, and three is rotated and becomes the face in the picture on the right. So the cube that is labeled one rotates over here, cube three is here, and this is cube two. And we have not seen the last cube here in the right corner, and that is the fifth one. So there are already five gray cubes here. And there can be several more. So, for example, which face is hidden from view in both diagrams? We have the back face here. Let me draw that in and how it moves about. So this back face in blue here becomes this face. And that one we cannot see in both pictures. So in that face, there could be hidden two cubes. So on the blue face, it can contain two gray cubes. And those would be the cubes that are here opposite of this one that I will mark in blue. So this would be number six and also the cube next to it. So that would be, for example, cube seven, but on the opposite of that, on the blue face. Okay, six would be actually directly opposite across and seven opposite through the middle. So the only other face we have not considered is the bottom face. And on the bottom, it can also contain two gray cubes. And let's draw those in, but I have no choice but to draw them on the top face. So cube here eight would be the center one. We cannot see that through rotation. And cube number nine would be the corner one. So that one over there in the corner. And that corner we cannot see through rotation. So a gray cube could be hiding there. And those are the only possibilities. So together we have the five that we have. And then there is space for four more that will remain hidden from view no matter how the cube is rotated here. Okay, so the answer is then five plus two plus two. So nine cubes is the maximum.

gray cubes

larger cube

maximum number

hidden positions

visible areas