We received a cube with the length of its edge expressed in centimeters as a whole number. We painted all its sides red and then cut it into smaller cubes with an edge of $1 \mathrm{~cm}$.

- Lukáš claims that the number of smaller cubes with two painted sides is ten times greater than those with three painted sides.
- Martina says that the number of smaller cubes with two painted sides is fifteen times greater than those with three painted sides.

However, only one of them is right — who is it? And what was the length of the edge of the original cube?

(L. Simünek)
From the formulation of the task, it follows that the edge of the original cube measured at least $2 \mathrm{~cm}$.

A small cube has three colored faces if its vertex was originally a vertex of the large cube. Therefore, there are as many such small cubes as there are vertices of the cube, which is 8.

A small cube has exactly two colored faces if one of its edges originally formed an edge of the large cube and none of its vertices were originally vertices of the large cube. Since the large cube had 12 edges, the number of small cubes with exactly two colored faces is a multiple of twelve. According to Lukáš, there are $10 \cdot 8=80$ such small cubes, which is not possible because 80 is not a multiple of twelve. Martina is right, who claims that there are $15 \cdot 8=120$ such small cubes.

On each edge of the large cube, we obtained 120 : $12=10$ small cubes with two colored faces. However, the edge of the original cube also included two small cubes with three colored faces, so the length of the edge corresponded to twelve small cubes. The edge of the original cube measured $12 \mathrm{~cm}$.