Problem 6. A wooden block was cut into eight smaller blocks with three cuts. On Fig. 5, the surface area of seven blocks is indicated. What is the surface area of the invisible block?

$[8$ points] (A. V. Shapovalov)
Answer: 22.

Solution: For each small block, the surface of the cuts constitutes half of its entire surface. We will only consider this. We will color the small blocks in black and white as shown in Fig. 6 (the invisible block is black). Then, every two identical rectangles touching on the cut are of different colors. Therefore, the sum of the areas of the black cuts is equal to the sum of the areas of the white ones.

![](https://cdn.mathpix.com/cropped/2024_05_06_ad0910d2ff8aaa6a488fg-4.jpg?height=476&width=503&top_left_y=83&top_left_x=257)

Fig. 5

![](https://cdn.mathpix.com/cropped/2024_05_06_ad0910d2ff8aaa6a488fg-4.jpg?height=394&width=388&top_left_y=158&top_left_x=849)

Fig. 6

Thus, the sum of the areas of the surfaces of the white blocks is equal to the sum of the areas of the surfaces of the black ones. Therefore, the area of the surface of the invisible black block is

$$
(148+46+72+28)-(88+126+58)=22
$$