On a 5x5 chessboard, all 36 corner points must be traversed along the edges of the squares, such that each corner point is only passed through once, and after the traversal, we return to the starting point. Provide all the paths that are symmetrical about the board's center lines.
We pre-empt the following statements, which arise from the requirements of the task. Due to symmetry, it is sufficient to examine the traversal of a quarter board. To traverse, we must enter and exit the quarter, otherwise, the path would be closed without having visited all corner points. Upon entry and exit, our path crosses the board's different midlines; it cannot cross the same one twice, as that would result in a closed path on one half of the board. Each midline can only be crossed once, as crossing it twice would split our path into two disconnected segments.

![](https://cdn.mathpix.com/cropped/2024_05_02_f70f5dde9d781b151376g-1.jpg?height=220&width=984&top_left_y=386&top_left_x=496)

$0 \quad 0$

Figure 1

Next, we select the upper left quarter of the board for further examination. The corner point in the upper left has two edges leading to it, one for entering and one for exiting. Therefore, these two edges will be part of every possible path. We examine how many ways we can reach these two edges. There are 3 possibilities (Figure 1). Let's call these $A$, $B$, and $C$ type paths. Every traversal must include one of these. In each case, another $A$, $B$, or $C$ type path can connect from the other direction. This results in 9 possibilities: $A A, A B, A C, B A, B B, B C, C A, C B, C C$. Due to the diagonal symmetry of the square, $A B$ and $B A$, $B C$ and $C B$, $A C$ and $C A$ are mirror images of each other and do not provide new solutions.

The remaining 6 cases are as shown in Figure 2.

![](https://cdn.mathpix.com/cropped/2024_05_02_f70f5dde9d781b151376g-1.jpg?height=238&width=1331&top_left_y=1041&top_left_x=386)

Among these, $A A, B B, C C$ clearly do not meet the conditions. In the $A A$ and $B B$ cases, we exit without having visited all points, and in the $C C$ case, there is a closed path. In the other 3 cases, there is a path that meets the conditions, and it is easily verifiable that there is only one such path.

This gives us all possible paths. Reflecting the obtained paths across the diagonal of the square yields 3 more solutions, but we do not consider these fundamentally different.
![](https://cdn.mathpix.com/cropped/2024_05_02_f70f5dde9d781b151376g-1.jpg?height=362&width=1264&top_left_y=1602&top_left_x=424)