## Task A-4.3.

On a cube with side length 1, a network is highlighted consisting of 14 points and 36 segments. The points are the vertices of the cube and the centers of its faces. The segments are all the edges of the cube and four additional segments on each face of the cube that connect the center of that face to its vertices.

What is the length of the shortest path along this network that passes through all 14 points?
## Solution.

The given network consists of 14 points, 12 segments of length 1, and 24 segments of length $\frac{\sqrt{2}}{2}$. $\quad 1$ point

Any path that passes through all 14 points must consist of at least 13 segments.

There are 6 vertices at the centers of the faces of the cube, and since each must be passed exactly once, we conclude that the path must contain exactly 12 segments of length $\frac{\sqrt{2}}{2}$.

Therefore, the length of the shortest path that passes through all 14 points must be at least $6 \sqrt{2}+1$.

It remains to show that such a path exists. Label the vertices of the cube as $A B C D E F G H$.

An example of a path of length $6 \sqrt{2}+1$ that passes through all the points of the network is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_14eaa2cf500c23a22c24g-19.jpg?height=400&width=806&top_left_y=865&top_left_x=545)