Highest number of intersection

Start with one line. There are zero intersections so far. The second line can intersect the first one once. The third one can intersect the first two once each. The fourth line can intersect the first three at most three times, and so on. So, $n$ lines can intersect at most $0 + 1 + 2 + \cdots + n -1 = \frac{(n-1)n}{2}$ times. For 12 lines, the maximum number of intersections is 66.

It's not hard to draw $n$ lines that intersect maximally: For $k = 1, 2, \dots, n$ draw the line connecting the points $(0, k)$ and $(n-k+1, 0)$ .