## Task A-4.7.

A figure placed on the surface of a cube $K_{n}$ of dimensions $n \times n \times n$ on a side attacks all the fields in the row and column in which it is located, similar to a chess rook, but also the fields on the other sides in the extensions of those rows/columns. (In the figure, the visible fields on the cube $K_{4}$ that the placed figure attacks are marked.)

How many figures can we place on the surface of the cube $K_{50}$ so that they do not attack each other?

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-28.jpg?height=323&width=303&top_left_y=2026&top_left_x=1385)
Solution.

The answer is 75.

1 point

We will call a ring a row or column along with its extensions.

Note that a figure attacks all the fields in the ring in which it is located, and that each figure is located in exactly two rings.

1 point

Since there are a total of 150 rings, and no two figures can be in the same ring, we conclude that on the surface of the cube $K_{50}$, we can place at most 75 figures.

It remains to find a configuration where we can place exactly 75 figures.

Choose three faces of the cube that share a common vertex and divide each into four squares of dimensions $25 \times 25$.

Now, on each of these three faces, select one square and place 25 figures on their diagonals, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-29.jpg?height=654&width=639&top_left_y=975&top_left_x=640)