374 Frequently Asked Questions In Quantitative Finance
The area increases by one third after the first iteration.
After the second iteration you add an area that is num-
ber of sides multiplied by area of a single small triangle
which is one ninth of the previously added triangle.
If we useAnto be the area afterniterations (when
multiplied by the area of initial triangle) then
An=An− 1 +
1
3
(
4
9
)n− 1
.
So
An= 1 +
1
3
∑∞
i= 0
(
4
9
)i
=
8
5
.
The final calculation exploits the binomial expansion.
This is the famous Koch snowflake, first described in
1904, and is an example of a fractal.
The doors
There are one hundred closed doors in a corridor. The
first person who walks along the corridor opens all
of the doors. The second person changes the current
state of every second door starting from the second
door by opening closed doors and closing open doors.
The third person who comes along changes the current
state of every third door starting from the third door.
This continues until the 100th person. At the end how
many doors are closed and how many open?
(Thanks to zilch.)
Solution
This is a question about how many divisors a number
has. For example the 15th door is divisible by one,
three, five and fifteen. So it will be opened, closed,
