Problem 4. Misha built a $3 \times 3 \times 3$ cube from smaller cubes. Then he glued some adjacent cubes together. The resulting solid structure consisted of 16 cubes, and Misha removed the rest. After dipping the structure in ink, he pressed it against a piece of paper with three different faces. The result was the word КОТ (see the image). What will be the result if the face opposite the letter "O" is printed?
![](https://cdn.mathpix.com/cropped/2024_05_06_a681d444f3973745158cg-5.jpg?height=150&width=494&top_left_y=968&top_left_x=480)

[8 points]
(M.A. Evdokimov, O.A. Zaslavsky, A.V. Shapovalov) Answer. See the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_a681d444f3973745158cg-5.jpg?height=147&width=151&top_left_y=1255&top_left_x=655)

Solution. Since the letter T can be printed, two corner cubes are removed. The remaining six corner cubes must remain, otherwise it would be impossible to print K and O. From this, we conclude that the letters K and O are located on adjacent faces, and all three cubes connecting these faces are present. Now the letter T can only be placed on the face opposite the letter K. Thus, the positions of 13 out of 16 cubes are determined (see Fig. 1). The remaining three cubes must stabilize the structure. The cubes labeled 1 and 2 are currently "hanging in the air" - they are not glued to any other cubes by any of their faces. Moreover, if they were to be glued by any face, it would only be the faces on which we wrote the numbers 1 and 2. Therefore, Misha inevitably glued a cube to each of these faces. However, these cubes still do not make the structure rigid: the pairs of newly glued cubes continue to "hang in the air." Misha managed to stabilize everything with exactly one additional cube - meaning he glued it to both pairs. Without spoiling the letter K, this can be done in only one way (see Fig. 2).

Now we can look at the face opposite the face "O" and draw the

![](https://cdn.mathpix.com/cropped/2024_05_06_a681d444f3973745158cg-6.jpg?height=204&width=214&top_left_y=261&top_left_x=1062)

Fig. 1

![](https://cdn.mathpix.com/cropped/2024_05_06_a681d444f3973745158cg-6.jpg?height=214&width=216&top_left_y=547&top_left_x=1063)

Fig. 2 answer (up to a rotation of the face).