XXXIX OM - I - Problem 8

For a given cube with edge length 1, find the set of segments with a total length not exceeding $1 + 3\sqrt{3}$, having the property that any two vertices of the cube are the endpoints of some broken line composed of segments from this set.
We will provide an example of a set of segments with the required property. This example can be considered the spatial counterpart of the "optimal road network connecting the vertices of a rectangle," constructed in preparatory task $C$. (A hint for the construction is the observation that it is purposeful to search for such an arrangement of segments so that in each "network node" exactly three segments meet, lying in the same plane and forming angles of $120^\circ$, because the existence of a node of any other form allows for a local modification of the network in the vicinity of such a node, leading to a shortening of the network.)
Let $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ be two opposite faces of a cube, labeled so that segments $A_1A_2$, $B_1B_2$, $C_1C_2$, $D_1D_2$ are four parallel edges of the cube. Denote the midpoints of edges $A_1B_1$, $C_1D_1$, $A_2B_2$, $C_2D_2$ by $K_1$, $L_1$, $K_2$, $L_2$, respectively, and the midpoints of segments $K_1L_1$ and $K_2L_2$ (i.e., the midpoints of the two mentioned faces) by $O_1$ and $O_2$ (see Figure 4).
om39_1r_img_4.jpg
On segment $O_1O_2$, we find points $P_1$ and $P_2$ such that $|O_1P_1| = |O_2P_2| = \sqrt{3}/6$. Then $|P_1P_2| = 1 - \sqrt{3}/3$. (Segments $K_1P_1$, $L_1P_1$, $P_1P_2$, $P_2K_2$, $P_2L_2$ form a network in the square $K_1L_1L_2K_2$ as described in task $C$; the angles at points $P_1$ and $P_2$ are each $120^\circ$.)
The length of segment $K_1P_1$ is $\sqrt{3}/3$. Let $M_1$ be the midpoint of this segment; thus, $|K_1M_1| = |M_1P_1| = \sqrt{3}/6$. (It is not difficult to verify that now segments $M_1A_1$ and $M_1B_1$ form angles of $120^\circ$ with segment $M_1P_1$). Segments $A_1M_1$ and $B_1M_1$ have lengths equal to $\sqrt{3}/3$. Similarly, let $N_1$, $M_2$, $N_2$ be the midpoints of segments $L_1P_1$, $K_2P_2$, $L_2P_2$, respectively. We consider the set of the following thirteen segments:

Sum of their lengths is $1 + 3\sqrt{3}$. In this way, we have found a set of segments satisfying the given conditions.