Zig-Zag

And here is my purely intuitive (but correct) guess: In problems like this, the number of extra regions (or intersections, etc.) created in each step tends to be a linear function of the number of lines present.

We already have $Z_1 = 2$ and $Z_2 = 7$ , and a quick sketch showed me that $Z_3 = 16$ . Thus the increment is $\Delta Z_{1,2} = 5$ and $\Delta Z_{2,3} = 9$ . Thus I assumed that the increments form an arithmetic progression with step size 4: $\Delta Z_{1,2}, \Delta Z_{2,3}, \dots, \Delta Z_{99,100} = 5, 9, \dots, 397\ \ (= 5+98\cdot 4).$ (The progression is confirmed by the fact that $\Delta Z_0 = 1$ so that $\Delta Z_{0,1} = 1$ .)

To find the final number of regions, we must add all these increments to the initial value $Z_1 = 2$ . Note that the average value of the 99 increments is $(5 + 397)/2 = 201$ , so that the answer is $2 + 99\cdot 201 = \boxed{19901}.$ Note: since the increments $\Delta Z_{n,n+1}$ are a linear function of $n$ , the number of regions is a quadratic function of $n$ . It is not difficult to see that $Z_n = 2n^2 - n + 1.$

The general expression is nC2+1 where n is number of 'straight' lines $200C2+1=19901$

first lets look at $Z_{1}$ , 2 lines divide the space into 2 regions: inner and outer. (2C2+1=2) but this doesn't explain it all.

So lets look at $Z_{2}$ , now if we select any 2 lines from 4 [4C2] we will get a unique inner region corresponding to each. (regions 2,3,4,5,6,7) also one outer region will be mutual to all (region 1) So the solution becomes 4C2+1=7

for $Z_{100}$ , we will be selecting 2 lines from 200 lines for corresponding inner region (200C2) and one mutual outer region. 200C2+1=19901

NOTE: If there is a confusion that when we select the 2 lines (those equivalent to 2 lines in $Z_{1}$ ) 3 regions are formed (6,7,4) then consider it this way, those 2 lines form only region 6. region 4 will be formed when we select the "vertical right" line and "horizontal lower" line. Similarly, region 3 will be formed when we select the "vertical right" line and "horizontal upper" line, region 7 by "vertical left" line and "horizontal lower" line, region 2 by "vertical left" line and "horizontal upper" line, region 5 by 2 vertical lines.