A Négyszögletű Kerek Erdő fái egy szabályos háromszögrács rácspontjain állnak. El lehet-e keríteni az erdőből egy téglalap alakú részt úgy, hogy a téglalap csúcsai rácspontok legyenek, és a téglalap határán ugyanannyi rácspont legyen, mint a belsejében?

The trees of the Square Round Forest stand on the points of a regular triangular grid. Can a rectangular part be cut out of the forest such that the vertices of the rectangle are grid points, and there are as many grid points on the boundary of the rectangle as there are inside it?
Solution. We will show that such a rectangle can be drawn.

Let the distance between the grid points be a unit distance. Then the distance between the points on the horizontal side of the rectangle is one unit, and on the vertical side, it is $\sqrt{3}$ units. Let the length of the horizontal side be $m$, and the length of the vertical side be $n \sqrt{3}$, where $n$ and $m$ are positive integers. The number of points on the perimeter of the rectangle is then $2(m+n)$.

In the interior of the rectangle, on those horizontal grid lines that do not intersect the vertical sides at grid points, there are $m$ points, and there are $n$ such lines. Thus, the number of these points is $m \cdot n$.

![](https://cdn.mathpix.com/cropped/2024_05_02_8d8deaf30e718d872653g-1.jpg?height=420&width=504&top_left_y=430&top_left_x=800)

On those horizontal grid lines that intersect the vertical sides at grid points, there are $m-1$ interior points, and there are $n-1$ such lines. Thus, the number of these points is $(m-1) \cdot(n-1)$.

Therefore, in the interior of the rectangle, there are $m \cdot n+(m-1) \cdot(n-1)=2 m n-m-n+1$ grid points.

Our condition is: $2(m+n)=2 m n-m-n+1$, or $2 m n-3 m-3 n+1=0$. Multiplying by 2 and factoring:

$$
\begin{aligned}
4 m n-6 m-6 n+2 & =0 \\
(2 m-3)(2 n-3)-9+2 & =0 \\
(2 m-3)(2 n-3) & =7
\end{aligned}
$$

Here, $2 m-3=1$ and $2 n-3=7$, or vice versa. Therefore, the sides of the rectangle that meet the condition are $m=2$ and $n=5$, or $m=5$ and $n=2$.

In both rectangles, there are 14 trees on the perimeter and in the interior.