\section*{Problem 3 - 251033}

From 27 cubes with edge length \(a\), a cube with edge length \(3a\) is assembled. Each of the 27 small cubes is either completely white or completely black.

When assembling, a pattern should be created on each of the six square faces of the large cube, which in each row and each column contains exactly one black and exactly two white squares.

Determine

a) the smallest, \quad b) the largest number of black cubes, with which these requirements can be met!
}

For the patterns required on the faces of the large cube, there are exactly the two non-congruent possibilities as depicted in the following figure:
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1503.jpg?height=642&width=992&top_left_y=798&top_left_x=293)

Furthermore, the following holds:

(1) A composition of the required type is not possible with fewer than eight black cubes.

Proof: Suppose there were such a composition with seven or fewer black cubes. In the "front" (i.e., the square \(A B C\) filling) layer, there would then be exactly three black cubes, and similarly in the "back" (the square \(E F G H\) filling) layer.

For the "middle" (the square \(P Q R S\) filling) layer, at most one black cube would remain. If it were present, it would be one of the three black cubes in the row \(P Q\) as per the assumption, so the row \(R S\) would contain no black cube. Since this contradicts the conditions, statement (1) is proven.

(2) A composition of the required type is not possible with more than eleven black cubes.

Proof: Suppose there were such a composition with twelve or more black cubes. In the front and back layers, there would be exactly three black cubes each, and possibly the small cube containing the center of the large cube would be black.

Among the eight cubes that fill the boundary line of the square \(P Q R S\), there would therefore be at least five black cubes. Among them, there could only be four that do not contain any of the points \(P, Q, R, S\). Thus, at least one of these points, w.l.o.g. the point \(P\), would have to be contained in a black cube. There would still be at least four black cubes remaining; none of them could reach the line segment \(S P Q\) according to the conditions.

This leads to the contradiction that for these at least four black cubes, only the three positions at \(U, R\), and \(V\) would be free; thus, statement (2) is also proven.

(3) A composition of the required type is possible with eight cubes, as shown in the following figure.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1503.jpg?height=268&width=274&top_left_y=2259&top_left_x=571)

\(A B C D\)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1503.jpg?height=251&width=260&top_left_y=2259&top_left_x=904)

\(P Q R S\)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1503.jpg?height=254&width=260&top_left_y=2260&top_left_x=1229)

\(E F G H\)

(4) A composition of the required type is possible with eleven cubes, as shown in the following figure.
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1504.jpg?height=308&width=918&top_left_y=230&top_left_x=570)

From (1) and (3) it follows: The smallest number sought in a) is 8. From (2) and (4) it follows: The largest number sought in b) is 11.