The cube was composed of 64 smaller cubes with an edge of $2 \mathrm{~cm}$. Then, several smaller cubes were removed from the visible side, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_04_17_e23bba49e38f63edc998g-3.jpg?height=337&width=351&top_left_y=1548&top_left_x=884)

1. What is the volume and surface area of the resulting solid?
2. The solid was painted red on its entire surface, then disassembled into the original smaller cubes. How many of them had 6, 5, 4, 3, 2, 1, or no red faces?

(M. Volfová)
1. The surface area of the body is the same as the surface area of the original cube, i.e.,

$$
6 \cdot 8 \cdot 8 = 384 \left(\mathrm{~cm}^{2}\right) \text {. }
$$

From the original cube, $3+5+9=17$ small cubes (counted by layers from the bottom) were removed, and the volume of the original cube was $8 \cdot 8 \cdot 8 = 512 \left(\mathrm{~cm}^{3}\right)$. The volume of the resulting body is therefore

$$
512 - 17 \cdot (2 \cdot 2 \cdot 2) = 512 - 136 = 376 \left(\mathrm{~cm}^{3}\right)
$$

2. None of the small cubes have 5 or more faces painted red; other cases are discussed in the following table. In each layer (numbered from bottom to top), we count the small cubes that have $4, 3, 2, 1$, or no faces painted red. The answer is in the last row, and the last column is filled in for verification:

|  | 4 | 3 | 2 | 1 | 0 | total |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1st layer | 1 | 5 | 6 | 4 | 0 | 16 |
| 2nd layer | 0 | 2 | 3 | 6 | 2 | 13 |
| 3rd layer | 0 | 3 | 3 | 5 | 0 | 11 |
| 4th layer | 2 | 5 | 0 | 0 | 0 | 7 |
| total | **3** | **15** | **12** | **15** | **2** | 47 |