## Task 2 - 230932

In the figure, a body $K$ is sketched. It consists of three cubes with an edge length of $1 \mathrm{~cm}$, which are firmly joined in the specified arrangement.

From a sufficient number of bodies of this shape $K$, a (completely filled) cube $W$ (edge length $n$ centimeters) is to be assembled.

Determine all natural numbers $n>0$ for which this is possible!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1052.jpg?height=368&width=374&top_left_y=1044&top_left_x=1326)
Since the volume of $K 3$ is given, it follows directly that only for $n=3 k$ can a cube of edge length $n$ be filled. A cube of edge length $3 k$ decomposes into $k^{3}$ cubes of edge length 3. Therefore, it suffices to provide a solution for $n=3$. One possibility is

| 1 | 1 | 3 |  | 5 | 6 | 3 |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | 2 | 3 |  | 5 | 7 |  |  |  |
| 6 | 6 | 7 |  | 8 | 9 | 7 |  |  |
| 2 | 2 | 4 |  |  |  |  |  |  |
|  |  | 8 | 4 | 4 |  |  |  |  |

where the three layers are indicated and a digit represents a $K$-block.