## Task A-2.4.

Five sides of a wooden cube are painted blue, while one side remains unpainted. The cube is then cut into smaller, congruent cubes, of which 649 have exactly one blue side. How many of the smaller cubes have exactly two blue sides?
## Solution.

Let the cube be divided into $n^{3}$ congruent smaller cubes.

Each of the 5 blue-painted faces of the cube corresponds to exactly $(n-2)^{2}$ smaller cubes with exactly one blue face. Additionally, on each of the 4 edges of the unpainted faces of the cube, there are another $n-2$ smaller cubes with exactly one blue face. Therefore, there are a total of $5(n-2)^{2}+4(n-2)$ smaller cubes with exactly one blue face.

The solutions to the quadratic equation $5(n-2)^{2}+4(n-2)=649$ are 13 and $-\frac{98}{10}$.

We are only interested in the first one, as it is a natural number. Thus, $n=13$.

Using a similar argument as above, with 8 edges of the cube that are common to two painted faces, there are a total of $8(n-2)$ smaller cubes with two blue faces. Furthermore, at the 4 vertices of the unpainted face, there are another 4 such smaller cubes. Therefore, the total number is

$$
8(n-2)+4=8 \cdot 11+4=92
$$

smaller cubes that have exactly two blue faces.

Note: The equation $5(n-2)^{2}+4(n-2)$ can be solved in multiple ways in natural numbers. For example, by noticing that the expression $5 m^{2}+4 m$ is increasing in $m:=n-2$, and $m=11$ is some solution, and thus the only one, or by using the factorization $649=11 \cdot 59$ and the equality $m(5 m+4)=649$. Of the 2 points from the above scoring scheme, finding the solution $n=13$ earns 1 point, and any justified method that it is the only solution earns 1 point.