To maximize the number of pieces, you want the nth line to cross as many regions as possible. Another way of putting this is to cross as many previous lines as possible.
In general it will cross one more region than the number of previous lines it crosses. And every region it crosses (divides), adds an additional piece.
With zero lines, there is one piece.
The first line crosses zero other lines, and one region, so it creates 2 pieces.
The second line crosses one other line, and two regions, so it adds 2 for a total of 4 pieces.
The third line crosses two other lines, and three regions, so it adds 3 for a total.
In general, the nth line will add n pieces:
So:
<PRE>
lines pieces
0 1
1 2
2 4
3 7
4 11
5 16
6 22
7 29
<PRE>
etc...
perform the little trick of calculus of differences (or any little trick you know) to determine the equation that generates this sequence.
If you do so, you'll find out that the equation is:
x = (n² + n + 2) / 2
full solution
