Observation: The number N^2 can be constructed from N lines, as follows.

Draw N lines in the plane so that:

1) Every pair of lines intersects once.

and

2) Three lines never intersect at the same point.

Now, if you cut each line at any intersections it has with other lines, then the total number of line segments that results is N^2.

For instance, 2 lines intersecting at one point yields 4 line segments. 3 lines intersecting at three points yields 9 line segments. 4 lines intersecting at six points yields 16 line segments. And so on.

To prove this, it may help to bend the lines, so that they form an orderly pattern. This would not affect the theorem, and it illustrates the basic idea of Topology.

Try to generalize this theorem.