What is the greatest number of intersection points four coplanar lines can have?

The answer is six. Each line can intersect with each other line. If we call the lines A, B, C and D, imagine the intersections. Line A can intersect B, C and D - that's three. B can intersect A, C and D, but we have already counted where A and B intersect, so we just get another two. Then C can intersect A, B and D, but we have counted A-C and B-C already so we just get one more, where C and D intersect. So the answer is 3+2+1=6.