0.13. a) Vertices $M$ and $N$ of a cube with edge length are asymmetric relative to the center of the cube. Find the length of the shortest path from $M$ to $N$ along the surface of the cube.

b) A box has the shape of a rectangular parallelepiped with dimensions $30 \times 12 \times 12$. Point $A$ is on the $12 \times 12$ face, and it is 1 unit away from one side of this face and equidistant from the two other parallel sides. What is the length of the shortest path along the surface of the box from point $A$ to the point symmetric to it relative to the center of the parallelepiped?
0.13. a) Any path going along the surface of a cube can be unfolded onto a plane. In Fig. 13, a, six shortest paths from $M$ to $N$ are shown unfolded; the length of each is $\sqrt{5}$.
![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-017.jpg?height=552&width=968&top_left_y=299&top_left_x=146)

Fig. 13

b) As in problem a), it is sufficient to choose the smallest of the segments $A B, A C, A D$, and $A E$ (Fig. 13, b). Clearly, $A B^{2}=42^{2}=1764, \quad A C^{2}=37^{2}+17^{2}=1658, \quad A D^{2}=32^{2}+24^{2}=1600 \quad$ and $A B > A D$. The shortest path $A D$ has a length of $40$.

Draw this path on the parallelepiped yourself.

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-017.jpg?height=333&width=338&top_left_y=1240&top_left_x=211)

Fig. 14

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-017.jpg?height=409&width=420&top_left_y=1177&top_left_x=633)

Fig. 15