S. Rukhin

A closed eight-segment broken line is drawn on the surface of a cube, with its vertices coinciding with the vertices of the cube.

What is the minimum number of segments of this broken line that can coincide with the edges of the cube?

#
The segments of the broken line are either edges of the cube or diagonals of its faces. To do this, let's color the vertices of the cube in two colors in a checkerboard pattern (see fig.). Note that an edge of the cube connects vertices of different colors, while a diagonal connects vertices of the same color. For the broken line to pass through all vertices, it must change color at some point, and to be closed, it must change color again. This means that the segments of the broken line must coincide with the edges of the cube at least twice.

This is sufficient, see the example in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_93426a107eaf3f76289cg-25.jpg?height=591&width=598&top_left_y=1048&top_left_x=730)

## Answer

Two segments.

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