Art of Problem Solving: Constructive Counting Part 2

Art of Problem Solving's Richard Rusczyk tackles some challenging problems with constructive counting.

[M_Chen]

5:25 LOL the moment he told us we wouldn't count it I counted it.

[experimentalgaming]

Notice the Fibonacci sequence at the beginning

[ansenliu]

for finding the number of sections using circles, there are a few factors to consider. how big is the circle? is the outer fence a square or rectangle?

[franklinlee]

Notice the pattern developing in on the fence map?

[tndkpgp]

Answer for lines is simply n(n+1)/2+1
Answer for circles is simply 2^n- tricked it's n(n+1)+2

[alexshein]

Would the formula for the numbers of sections in strait line be : (In LaTeX of course) ;

$\frac{(n)(n+1)}{2} + 1$, where $n$ denotes the number of straight lines?

Still working with circles.