The forty-first question: In a square with a side length of 4, if n points are placed arbitrarily, there will always be two points whose distance does not exceed $\sqrt{5}$. Find the minimum value of the integer n.

In a square with a side length of 4, if n points are placed arbitrarily, there will always be two points whose distance does not exceed $\sqrt{5}$. Find the minimum value of the integer n.
Question 41:
Solution: As shown in the figure, establish a rectangular coordinate system with $AB$ as the $x$-axis and $AD$ as the $y$-axis. Take six points $(2,0)$, $(2,2.75)$, $(0,4)$, $(4,4)$, $(0,1.25)$, and $(4,1.25)$. It can be verified that the distance between any two of these six points is greater than $\sqrt{5}$. Therefore, $n \geq 7$.

Next, we prove that when $n \geq 7$, there must be two points whose distance is no more than $\sqrt{5}$. It is easy to prove that, in the "2x2" shape and the "3-slope" shape shown in the figure below, if three points are placed, then there must be two points whose distance is no more than $\sqrt{5}$.
If the conclusion does not hold, i.e., there exists a way to place seven points such that the distance between any two points is greater than $\sqrt{5}$.
Divide the original square with side length 4 (i.e., the $4 \times 4$ grid) into four $2 \times 2$ shapes. According to the above conclusion, in the four $2 \times 2$ shapes, exactly three of them each have two points, and the other one has one point. Assume without loss of generality that the $2 \times 2$ shape in the lower right corner has one point, and the other three $2 \times 2$ shapes each have two points.
(1) If the points in the upper left $2 \times 2$ shape are placed as shown in the figure below (points placed in the shaded area), then the points in the upper right and lower left $2 \times 2$ shapes can only be placed as shown in the figure below, at which point a "3-slope" shape appears in the table, contradicting the above conclusion.
(2) If the points in the upper left $2 \times 2$ shape are placed as shown in the figure below (points placed in the shaded area), then the points in the upper right and lower left $2 \times 2$ shapes can only be placed as shown in the figure below, at which point a "3-slope" shape appears in the table, contradicting the above conclusion.
In summary, the minimum value of $n$ is 7.