I'm having trouble understanding how the book came up with the formula for calculating $Z\_n$, the maximum number of regions created using $n$ zig lines (a zig line is a line containing just one sharp bend). Earlier in the [book](https://notendur.hi.is/pgg/(ebook-pdf)%20-%20Mathematics%20-%20Concrete%20Mathematics.pdf) (Page 5), we derived $L\_n$, the maximum number of regions created using $n$ straight lines. What follows is the derivation of $Z\_n$:

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> [![](https://i.stack.imgur.com/1zFU0.png)](https://i.stack.imgur.com/1zFU0.png)
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> [![](https://i.stack.imgur.com/7E7LT.png)](https://i.stack.imgur.com/7E7LT.png)
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First, I'm not exactly sure what the underlined sentence means. From the picture of $Z\_2$, my only guess at its meaning is that none of the zigs can contain the zig point of another zig.

My other question is about the assumptions that the book seems to be making in its argument (written in color):

1) Imagine $n$ zigs, and assume they split the plane into $Z\_n$ regions. Extend all the lines as in the example. This creates a plane split by $2n$ lines $\color{red}{\text{into } L\_{2n} \text{regions}}.$

2) Now, erase the extensions. For each extension we erase, $\color{blue}{\text{we lose 2 regions}}.$

3) Therefore $Z\_n$ zigs split the plane into $L\_{2n}-2n$ regions.

The blue assumption surely has something to do with my underlined sentence, but its not obvious to me that in very complicated figure with many intersections, we lose exactly two regions per deletion of one extensions.

Could someone help make the justification for these assumptions more clear? 

Thanks.