Frequently Asked Questions In Quantitative Finance

Chapter 10: Brainteasers 373

What is the probability that a randomly selected point
is closer to the center than to the edge?

(Thanks to OMD.)

Solution
Many people will think that the required probability is
the same as the probability of landing in the circle with
diameter half the side of the square. But this is not the
case. The line separating closer to centre from closer to
edge is a parabola. The answer is
(
− 1 +

√

2

) 2

+

2

3

(

3 − 2

√

2

) 3 / 2

.

Snowflake

Start with an equilateral triangle. Now stick on to the
middle of each side equilateral triangles with side one
third of the side of the original triangle. This gives you
a Star of David, with six points. Now add on to the sides
of the six triangles yet smaller triangles, with side one
third of the ‘parent’ triangle and so onad infinitum.
What are the perimeter and area of the final snowflake?

(Thanks to Gerasimos.)

Solution
First count how many sides there are as a function of
number of iterations. Initially there are three sides, and
then 3×4. Every iteration one side turns into four. So
there will be 3 4nafterniterations. The length of each
side is one third what it was originally. Therefore after
niterations the perimeter will be
(
4
3

)n

multiplied by the original perimeter. It is unbounded as
ntends to infinity.