We received a cube, the length of whose edge was expressed in centimeters as an integer greater than 2. We painted all its faces yellow and then cut it into smaller cubes, each with an edge length of $1 \mathrm{~cm}$. We sorted these small cubes into four piles. The first pile contained cubes with one yellow face, the second pile contained cubes with two yellow faces, and the third pile contained cubes with three yellow faces. The fourth pile contained cubes with no yellow faces. Determine the length of the edge of the original cube, given that at least one of the following statements is true:

- The numbers of cubes in the first and fourth piles were in the ratio $4: 9$.
- There were three times as many cubes in the first pile as in the second.
Let the length of the edge of the original cube in centimeters be denoted by $a+2$, where $a$ is a natural number. Each face of the original cube corresponds to $a^{2}$ small cubes with exactly one colored face, so there are a total of $6 a^{2}$ such small cubes. On each edge of the original cube, we have $a$ small cubes with exactly two colored faces. Since the original cube had 12 edges, there are a total of $12 a$ such small cubes. The number of small cubes that have no colored faces is $a^{3}$.

The first statement in the problem is expressed by this equation:

$$
\frac{6 a^{2}}{a^{3}}=\frac{4}{9}
$$

After simplifying the fraction by the non-zero expression $a^{2}$, we get

$$
\frac{6}{a}=\frac{4}{9},
$$

thus $a=13.5$. The problem assumes an integer length of the cube's edge, but here the length of the edge comes out to be $13.5+2=15.5(\mathrm{~cm})$. We see that the given ratio of the number of small cubes can never be obtained by cutting any cube.

The first statement in the problem is not true, so the second one must be true, which is expressed by the equation

$$
\frac{6 a^{2}}{12 a}=\frac{3}{1}
$$

After simplifying the fraction by the non-zero expression $6 a$, we get

$$
\frac{a}{2}=\frac{3}{1}
$$

thus $a=6$. The length of the edge of the original cube was $6+2=8(\mathrm{~cm})$.

Scoring. 3 points for expressing the number of small cubes in the first, second, and fourth piles; 2 points for the edge lengths according to the first and second statements; 1 point for the correct conclusion.