In the picture, there is a structure made of 14 identical cubes. We want to paint the structure from all sides, including the bottom. What will be the paint consumption if $10 \mathrm{ml}$ of paint is enough to paint one entire cube?

(M. Mach)

![](https://cdn.mathpix.com/cropped/2024_04_17_24a3eca61d5f78ac2719g-1.jpg?height=483&width=422&top_left_y=1369&top_left_x=820)
There are six different directions to look at the construction, or rather at the individual cubes of the construction. For each direction, we calculate how many walls of the cubes will need to be painted:

- When we look at the picture from above, we see exactly 7 walls that will need to be painted. The same number is counted when looking from below.
- When looking from the front, we see a total of 8 walls. The same number is counted when looking from the back.
- When looking from the left, we see 7 walls, but we will also need to paint another 5 walls that are hidden - these are the cubes in the right pillar of the construction. In total, we counted 12 walls. The situation is the same when looking from the right.

In total, we need to paint $2 \cdot(7+8+12)=2 \cdot 27=54$ walls, which is the same as painting 9 whole cubes $(9 \cdot 6=54)$. To paint one whole cube, 10 ml of paint is needed, so in total we need $9 \cdot 10=90(\mathrm{ml})$ of paint.

Another solution. We can go through all the cubes in the construction one by one and determine how many of their walls we will be painting. We proceed layer by layer from top to bottom, in each layer row by row from front to back, in each row from left to right:

- In the 1st layer, there is only one cube, of which we will paint 5 walls.
- In the 2nd layer, there are three cubes — we count them sequentially $4+3+4=11$ walls.
- In the 3rd layer, there are four cubes — we count them sequentially $4+4+3+3=14$ walls.
- In the 4th layer, there are six cubes — we count them sequentially $5+5+3+3+4+4=24$ walls. In total, we need to paint $5+11+14+24=54$ walls. The subsequent procedure can be the same as in the previous solution.

Evaluation. 4 points for the number of walls that need to be painted; 2 points for determining the required amount of paint.