An ant starts from a vertex of a cube, walking only along the edges, until it returns to the initial vertex, without passing through any vertex twice. What is the longest walk that this ant can make?
In the figure, we have a path consisting of eight edges that the ant can take starting from the vertex identified as 1.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-084.jpg?height=314&width=309&top_left_y=1970&top_left_x=905)

Would it be possible for her to make a path passing through nine edges? To make this path, she would have to pass through nine vertices, as the arrival vertex is the same as the starting vertex, since the little ant returns to the initial vertex.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-084.jpg?height=226&width=829&top_left_y=2491&top_left_x=702)

Since the cube only has eight vertices, this walk is not possible. Therefore, the longest walk covers eight edges.