Problem 5.6. There are 4 absolutely identical cubes, each of which has 6 dots on one face, 5 on another, ..., and 1 on the remaining face. It is known that on any two opposite faces of the cube, the total number of dots is 7.

These 4 cubes were glued together to form the figure shown in the image. It is known that on each pair of glued faces, the same number of dots is marked. How many dots are on the faces $A, B, C$?

![](https://cdn.mathpix.com/cropped/2024_05_06_32e5791ee8081231aabeg-5.jpg?height=428&width=654&top_left_y=644&top_left_x=396)
Answer: $A=2, B=2, C=6$.

Solution. First, we use the condition that the sum of the number of dots on a pair of opposite faces of the die is 7, to understand how the die, three faces of which we know, looks. Its hidden faces are shown in Fig. 2 (up to the arrangement of dots on the face with 2 dots, which cannot be uniquely determined from the condition but does not play a role in solving this problem).

We know that the other dice are exactly the same. For each die with faces $A$ and $C$, we see one face, and we know another face since it is glued to the known face of the first die. By the position of two adjacent faces, we can uniquely determine the orientation of the die in space. After this, the two faces of die $B$, with which it is glued to dice $A$ and $C$, will become known, and we can similarly determine its orientation. The dice in the unfolded form are shown in Fig. 3 (again, up to the arrangement of dots on the faces with two dots).