In the book «Concrete Mathematics: A Foundation for Computer Science» by Donald Ervin Knuth, Oren Patashnik and Ronald Lewis Graham (Copyright ⓒ 1994, 1989 by Addison-Wesley Publishing Company, Inc.) there is a following problem (ch.1, No.13, page 19):
What's the maximum number of regions definable by n zig-zag lines, each of which consists of two parallel infinite half-lines joined by a straight segment?
The main idea of the solution is to consider very narrow zig-zags (almost straight lines) and then to use the result from the main text of the book, which gives the formula of regions definable by n straight lines.
My question is: does this «configuration» of zig-zags give us the real maximum of regions and why?
Original solution: given n straight lines that define $L_n$ regions, we can replace them by extremely narrow zig-zags with segments sufficiently long that there are nine intersections between each pair of zig-zags. This shows that $ZZ_n = ZZ_{n-1} + 9n - 8$.
In this example (given in the book) with 2 zig-zag lines we have $ZZ_{2}=12$.