Problem 5. (6 points) Peter has $n^{3}$ unit white cubes, from which he wants to assemble a cube $n \times n \times n$ with a completely white surface. How many of the faces of the unit cubes does Emil need to color to prevent Peter? Solve the problem for:
a) (2 points) $n=2$;
b) (4 points) $n=3$.
Solution. a) It is clearly not enough for Emil to color one wall. However, it is sufficient to color two opposite walls of one small cube: when assembling the $2 \times 2 \times 2$ cube, one of them will always be external.

b) It is sufficient for Emil to color the walls of two small cubes, as when assembling the $3 \times 3 \times 3$ cube, only one small cube can be completely hidden.

We will show that if no more than 11 walls are colored, Peter can assemble a $3 \times 3 \times 3$ cube with uncolored walls. Emil colors at least two walls of a small cube, because one colored wall can be hidden in any position of the small cube. Therefore, at most 5 small cubes have colored walls. Peter will place the small cube with the most colored walls in the center of the large cube, and the other small cubes that have colored walls in the centers of the walls of the large cube, with their external walls being uncolored. (Each of them has at least one uncolored wall, since only one small cube can be completely colored, and if there is such a cube, it is in the center of the cube.)

Therefore, Emil must color at least 12 walls.