Anička set out from the hotel for a walk, walked $5 \mathrm{~km}$ north, then $2 \mathrm{~km}$ east, $3 \mathrm{~km}$ south, and finally $4 \mathrm{~km}$ west. Thus, she arrived at a pond where she took a bath. Vojta set out from the camp, walked $3 \mathrm{~km}$ south, $4 \mathrm{~km}$ west, and $1 \mathrm{~km}$ north. Thus, he arrived at a rock which was $5 \mathrm{~km}$ west of the pond where Anička had bathed.

Another day, Anička set out from her hotel and Vojta from his camp, both wanting to reach the pond where Anička had bathed earlier, and both proceeded only in the directions of the four cardinal points.

Determine how many kilometers Anička and Vojta had to walk at a minimum. (E. Semerádová)

![](https://cdn.mathpix.com/cropped/2024_04_17_a328ac2838c6e0b4d9c6g-3.jpg?height=342&width=345&top_left_y=680&top_left_x=861)
According to the task, we will sequentially illustrate Anička's route, Vojta's route, and their mutual relationship (each marked tile represents $1 \mathrm{~km}$):

![](https://cdn.mathpix.com/cropped/2024_04_17_a328ac2838c6e0b4d9c6g-3.jpg?height=534&width=926&top_left_y=1252&top_left_x=565)

Now we see that Anička has at least $4 \mathrm{~km}$ from the hotel to the pond, for example, $2 \mathrm{~km}$ north and $2 \mathrm{~km}$ west. Vojta has at least $3 \mathrm{~km}$ from the camp to the pond, for example, $2 \mathrm{~km}$ south and $1 \mathrm{~km}$ east.

Evaluation. 1 point for illustrating Anička's and Vojta's routes; 2 points for their mutual relationship; 1 point for the shortest distances of Anička and Vojta to the pond.

Note on the second part. Routes of the same length between two places are not unique according to the given rules and can be devised in many ways (for example, Anička could have walked her $4 \mathrm{~km}$ from the hotel to the pond as follows: $1.5 \mathrm{~km}$ west, $2 \mathrm{~km}$ north, and $0.5 \mathrm{~km}$ west). We do not expect such an analysis from the solver, but a complete solution must include some specific realization.