How many PAIRS of skew lines can be found in a cube?

Question

I just took a test with this question on it and im wondering if i got it right........please answer if u can.........thanks!!!!!!

Answer 1

There are 24 pairs of skew lines in a cube.

A cube has 12 edges, which if they are extended form lines. Because of symmetry, for each line there are:

3 parallel lines
4 intersecting lines
4 skew lines

The trick here is to eliminate double counting of skew lines. When we do that we discover that there are 24 pairs of skew lines in a cube. The fomula is: 4*12/2 = 24.

Answer 2

A cube has 12 edges. Pick one of the cube's edges. Of the remaining eleven, 4 intersect with the edge, 3 are parallel, and 4 are skew. So each edge has 4 skew lines. Since there are 12 edges, this gives 12*4 = 36 pairs of skew lines. BUT by doing this we're counting each pair twice, so the correct number is 36/2 = 18.