Three, on each vertex of a regular $n$-sided polygon, there is a magpie. When startled, all the magpies fly away. After some time, they all return to these vertices, with one magpie on each vertex, but not necessarily to their original vertices. Find all positive integers $n$, such that there must exist 3 magpies, for which the triangles formed by their respective initial and final positions are either all acute triangles, all right triangles, or all obtuse triangles.
(Su Chun, Contributed)
Three, Solution 1: When $n=3,4$, the conclusion is obviously true.
When $n=5$, in a regular pentagon with 5 diagonals, there are exactly 5 obtuse triangles and 5 acute triangles. When 5 magpies $A$, $B$, $C$, $D$, and $E$ land in the states shown in Figure 1, acute triangles and obtuse triangles transform into each other, which does not meet the requirements of the problem. Therefore, $n=5$ does not work.

When $n=2m, m \geqslant 3, m \in \mathbf{N}$, take a pair of opposite vertices $A$ and $B$ of the regular $n$-sided polygon and examine the landing points of the two magpies (still called $A$ and $B$) that fly from these points. If the landing points of magpies $A$ and $B$ are still a pair of opposite vertices, then we can choose another magpie $C$, and thus, the vertices occupied by magpies $A$, $B$, and $C$ before and after flying form right triangles. If the landing points $A'$ and $B'$ of magpies $A$ and $B$ are not opposite vertices, then we can denote the opposite vertex of $A'$ as $C'$, and the magpie that lands on $C'$ after returning is from point $C$. Thus, the vertices occupied by magpies $A$, $B$, and $C$ before and after flying form right triangles. This proves that all even numbers $n$ not less than 4 meet the requirements.
Let $n=2m+1, m \geqslant 3, m \in \mathbf{N}$. In this case, there are no right triangles in Figure 1. Let us prove the following lemma.
Lemma: When $n=2m+1, m \geqslant 3$, in a regular $n$-sided polygon with all diagonals connected, the number of obtuse triangles is more than half of the total number of triangles.
Proof: Let the regular $n$-sided polygon be $A_{1} A_{2} \cdots A_{n}$. At this time, the number of obtuse triangles with diagonal $A_{1} A_{j+1}$ as the longest side is $j-1$. Therefore, the total number of obtuse triangles is
\[
\begin{aligned}
\mathrm{S} & =(2 m+1)(1+2+\cdots+(m-1)) \\
& =\frac{1}{2}(2 m+1) m(m-1) .
\end{aligned}
\]

The total number of triangles in the regular $n$-sided polygon is
\[
t=\mathrm{C}_{n}^{3}=\frac{1}{3}(2 m+1) m(2 m-1) \text{. }
\]

Since
\[
\begin{array}{l}
m-1>\frac{1}{3}(2 m-1) \\
\Leftrightarrow 3 m-3>2 m-1 \\
\Leftrightarrow m>3-1=2 .
\end{array}
\]

The latter is obviously true, hence the lemma is proved.
By the lemma, when $n=2m+1, m \geqslant 3$, there always exist 3 magpies such that the 3 vertices they occupy before and after flying form obtuse triangles. This indicates that all odd numbers not less than 7 meet the requirements of the problem.
In summary, all natural numbers that meet the requirements of the problem are $n \geqslant 3, n \neq 5$.
Solution 2: The previous proofs are the same as Solution 1, except for the case of $n=2m+1, m \geqslant 3$ which is proved as follows.

Since the shape of the triangle remains unchanged when the regular $n$-sided polygon and the magpies on the vertices are rotated together, we can assume that a magpie $A$ returns to the same place after flying away. Draw a diameter $A A'$ of the circumcircle of the regular $n$-sided polygon through the vertex where $A$ is located. Thus, there are $m$ magpies on each side of this diameter. It is easy to see that any two magpies on the same side of the diameter, together with $A$, form an obtuse triangle. Since $m \geqslant 3$, among the $m$ magpies on the same side of the diameter, there are always two, $B$ and $C$, that remain on the same side after flying and returning. Thus, the three magpies $A$, $B$, and $C$ form an obtuse triangle before and after flying. This proves that all odd numbers not less than 7 meet the requirements of the problem.
Therefore, all natural numbers that meet the requirements of the problem are $n \geqslant 3, n \neq 5$.