Problem 5.8. There are 7 completely identical cubes, on which 3 points are marked on one face, 2 points on two faces, and 1 point on the rest. These cubes were glued together to form a figure in the shape of the letter "P", as shown in the image, with the number of points on any two touching faces being the same.

![](https://cdn.mathpix.com/cropped/2024_05_06_e7f9c87b6a37ffba3564g-13.jpg?height=400&width=519&top_left_y=448&top_left_x=467)

What is on the three left faces $A, B$, and $C$?
Answer: $A-2, B-2, C-3$.

Solution. For a complete solution to this problem, it is necessary not only to understand how many points are located on the faces of the cube and which of them are adjacent. It is also important to figure out along which exact diagonals the points on the faces of the cube are directed.

Consider the middle right cube (above and below which there are other cubes). On its top face, as well as on its bottom face, there cannot be 3 points (otherwise, the top or bottom cube would have at least two faces with 3 points). Therefore, the face with 3 points is one of the two remaining faces (which are not visible and do not touch anyone). Then, in our cube, opposite the face with 3 points is a face with 2 points, one of the four remaining faces has 2 points, and the other three have 1 point each.

Consider the middle top cube. On its top face, there are 3 points, and on the front face, there are 2 points. It is clear that then on the bottom face of this cube there are 2 points, and on the left face, there is 1 point (Fig. 2).

Consider the middle top cube and the left middle cube. They are absolutely identical, so they can be superimposed in space (overlaid on each other). In this case, the faces with three points must coincide; considering the directions of these points, it is clear that this can be done in only two ways; one of these ways is impossible, as otherwise the visible face with 2 points would coincide with the visible face with 1 point. Therefore, the position of the left middle cube is uniquely determined, and on its left face $B$, as well as on the back face, there are 2 points. Then on its top and bottom faces, there is 1 point each.

For the top cube, the top, front, and bottom faces are determined. The right face is also determined - it has 1 point, because the left face of the middle top cube has 1 point. Then on face $A$ there must be 2 points, and on the back face, there are 3 points.

For the bottom cube, there is 1 point on the front and top faces, and 2 points on the right face. Since opposite the face with 3 points cannot be a face with 1 point, we get that on face $C$ there are 3 points.

Thus, on face $A$ there are 2 points, on face $B-2$ points, and on face $C-3$ points. It is not difficult to verify that all 7 cubes can be arranged as shown in the figure so that all the conditions of the problem are satisfied.

## 6th Grade

![](https://cdn.mathpix.com/cropped/2024_05_06_e7f9c87b6a37ffba3564g-14.jpg?height=397&width=422&top_left_y=84&top_left_x=516)

Fig. 2: Solution to problem 5.8; black represents the three "real" faces of the cube; gray represents the projections of the other three faces.