Let's reflect a square in some order across each of its four sides. How many different transformations result from the sequence of four reflections?
I. solution. Let us fix a Cartesian coordinate system in the plane of the square such that the sides of the square are parallel to the axes. We will prove that after four reflections, the image of any point $P\left(x_{p} ; y_{p}\right)$ is a translation of the point $P$ by a certain (independent of the position of the point $P$) vector $\vec{v}$.

Let the equation of the side of the square parallel to the $x$-axis, which we reflect $P$ over first, be $y=y_{1}$, and the equation of the side we reflect over second be $y=y_{2}$. Similarly, let the equation of the side of the square parallel to the $y$-axis, which we reflect over first, be $x=x_{1}$, and the equation of the side we reflect over second be $x=x_{2}$. The $y$-coordinate of the point $P$ changes as follows:

$$
\begin{aligned}
y_{p} & \longrightarrow y_{1}+\left(y_{1}-y_{p}\right)=2 y_{1}-y_{p} \longrightarrow \\
& \longrightarrow y_{2}+\left(y_{2}-\left(2 y_{1}-y_{p}\right)\right)=2 y_{2}-2 y_{1}+y_{p}=y_{p}+2\left(y_{2}-y_{1}\right)
\end{aligned}
$$

Similarly, the $x$-coordinate of the point $P$ changes as follows:

$$
\begin{aligned}
x_{p} & \longrightarrow x_{1}+\left(x_{1}-x_{p}\right)=2 x_{1}-x_{p} \longrightarrow \\
& \longrightarrow x_{2}+\left(x_{2}-\left(2 x_{1}-x_{p}\right)\right)=2 x_{2}-2 x_{1}+x_{p}=x_{p}+2\left(x_{2}-x_{1}\right)
\end{aligned}
$$

Thus, regardless of where the point $P$ is, the four transformations are equivalent to a translation by the vector

$$
\vec{v}\left(2\left(x_{2}-x_{1}\right) ; 2\left(y_{2}-y_{1}\right)\right)
$$

This vector depends on the order (determined in the coordinates), so there are four possible vectors.

Remark. Since the reflection of the point $P$ over the sides of the square parallel to the $y$-axis (e.g., $a, c$) only changes the first coordinate, and the reflection over the sides parallel to the $x$-axis (e.g., $b, d$) only changes the second coordinate, the reflections in the two different "directions" are independent of each other. Only the order of reflections over $a$, $c$, and over $b$, $d$ matters.

II. solution. We can break the solution into two cases: 1. First, we reflect over two parallel lines. 2. First, we reflect over two perpendicular lines.

1. case. The composition of two reflections over parallel lines is equivalent to a translation by twice the distance between the two parallel lines, with the vector pointing from the first line to the second, which in our case is twice the directed side of the square. Therefore, the composition of the four reflections is equivalent to the composition of two translations, whose sum is twice the vector of one of the diagonals. There are four possibilities for this.
2. case. The composition of two reflections over perpendicular lines is equivalent to a central reflection, whose center is the intersection of the lines. If we reflect over two pairs of perpendicular lines, it corresponds to two central reflections. The centers can be chosen in two ways (as the endpoints of one of the diagonals), and the distance between them is the diagonal of the square. The composition of two reflections over points is equivalent to a translation by twice the distance between the points. Just like in the first case, this is a translation by twice the vector of one of the diagonals. This case thus gives the same results as the first.

In total, there are four different transformations.