Let Q be the cube with the set of vertices $$\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0, 1\}\}$$. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0, 0, 0) and (1, 1, 1) is in S. For lines $$\ell_1$$ and $$\ell_2$$, let $$d(\ell_1, \ell_2)$$ denote the shortest distance between them. Then the maximum value of $$d(\ell_1, \ell_2)$$, as $$\ell_1$$ varies over F and $$\ell_2$$ varies over S, is