4. An equilateral triangle $T$, with side 2022, is divided by lines parallel to its sides into equilateral triangles with side 1. The triangle is covered with the figures shown in the diagrams below, each composed of 4 equilateral triangles with side 1, and during the covering, they can be rotated by an angle of $k \cdot 60^{\circ}, k=1,2,3,4,5$.

![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=130&width=103&top_left_y=252&top_left_x=540)
1
![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=144&width=528&top_left_y=250&top_left_x=722)

The covering satisfies the following conditions:

- There can be no figures of a certain type and there can be more figures of another type.
- The triangles of the figures match the triangles into which triangle $T$ is divided.
- No two figures overlap and no figure extends outside triangle $T$.
- The entire triangle $T$ is covered.

What is the smallest possible number of figures of type 1 used in such a covering?
Solution. We color the triangles into which $T$ is divided alternately with black and white, with the triangles at the vertices of $T$ being colored black (see the diagram on the right for a triangle with side 10). In this case, figures of types 2, 3, and 4 will cover 2 white and 2 black fields, while the figure of type 1 covers 3 fields of one color and 1 field of the other color. The latter can be seen from the diagrams below, where the shaded fields are of one color, and the white fields are of the other color.

![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=452&width=514&top_left_y=937&top_left_x=952)
This can be seen from the diagrams below, where the shaded fields are of one color, and the white fields are of the other color.

![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=151&width=716&top_left_y=1501&top_left_x=534)

On the other hand, in each row of $T$, the number of black triangles is 1 more than the number of white triangles (each row starts and ends with a black triangle, and the triangles alternate in color). Therefore, in $T$, there are 2022 more black fields than white ones, so at least $\frac{2022}{2}=1011$ figures of type 1 must be used. In the following diagram, the coverage of two rows with one figure of type 1 and figures of type 2 is shown.
![](https://cdn.mathpix.com/cropped/2024_06_05_5a2d96b029bcdb4a14f1g-3.jpg?height=110&width=614&top_left_y=2043&top_left_x=576)

Thus, there exists a covering with exactly 1011 figures of type 1.