On each face of a cube, we marked the points that divide the midlines in the ratio $1: 3$. How does the volume of the convex body determined by these points compare to the volume of the cube?
Planes passing through the center of a cube and parallel to its faces divide both the cube and the resulting convex body into eight congruent parts, because the individual parts are mirror images of each other with respect to these planes. Therefore, it is sufficient to examine one eighth of the original cube.

In one eighth of the cube, the points dividing the midlines of the cube's faces in the ratio $1:3$ exactly bisect the edges of the small cube (Figure 1), and these six midpoints form a regular hexagon (proof of this can be found, for example, in the 1848th problem of the first volume of the Collection of Geometric Problems). This regular hexagon divides the volume of the small cube in half, because the two parts of the cube defined by the plane of the hexagon are interchanged by reflection in the center of the cube (on Figure 1, vertices marked with the same letter are mirror images of each other with respect to the center of the cube).

Since in each eighth of the original cube exactly half of its volume belongs to the new body, the body defined by the points dividing the midlines in the ratio $1:3$ - which is bounded by 6 squares and 8 regular hexagons as shown in Figure 2 - has a volume that is half of the original cube.

Based on the work of Bence Seress (Budapest, Árpád Gymnasium, 8th grade)
![](https://cdn.mathpix.com/cropped/2024_05_02_cc8376967bf524a2b8b0g-1.jpg?height=756&width=412&top_left_y=652&top_left_x=867)