Let $e$ be one of the edges of a cube. How many lines are there in space that intersect exactly those 12 edges of the cube which are skew to $e$?
Solution. Let the line $e$ be the line $AB$. The lines that are skew to this one are accordingly the (dashed lines in the drawing) $CG, DH, FG$, and $EH$.

![](https://cdn.mathpix.com/cropped/2024_05_02_cc319685fde4ac76bb0cg-1.jpg?height=553&width=588&top_left_y=304&top_left_x=758)

The lines passing through a point on each of the parallel lines $EH$ and $FG$ are certainly in the plane containing these lines, the plane $EFGH$. The lines passing through a point on each of the parallel lines $CG$ and $DH$ are in the plane $CGHD$. These two planes have only one common line: their intersection, which is the line $HG$. This line is parallel to the edge line $AB$.

Therefore, there is one line in space, the line $HG$, which intersects exactly those of the 12 edge lines of the cube that are skew to $e$.