## Task 4 - 080514

Five faces of a cube with an edge length of $3 \mathrm{~cm}$ are painted red, the sixth face remains unpainted. Afterwards, this cube is cut into exactly 27 smaller cubes, each with an edge length of $1 \mathrm{~cm}$. How many of these small cubes have $0 ; 1 ; 2 ; 3$ or 4 red painted faces?

Hint: You can also build a cube with an edge length of $3 \mathrm{~cm}$ for visualization. Divide each face into square centimeters!
Only the cubes that reach the colored outer sides of the large cube have a colored face, i.e., the small cube inside has no paint, nor does the middle cube of the sixth face.

One colored face have the middle cubes of the colored side faces and the 4 cubes of the sixth face that lie in the middle of the 4 edges. In total, 9 cubes.

Two colored faces have the middle edge cubes that are bounded by both colored side faces of the large cube, a total of 8. The 4 corner cubes of the sixth face are also colored on two sides. In total, there are 12 cubes with two colored faces.

Since there are no cubes with 4 colored sides, the remaining small cubes with 3 painted sides are $27-2$ - 9 - $12=4$.

Thus, there are small cubes with 2 times 0, 9 times 1, 12 times 2, 4 times 3, and 0 times 4 painted sides.