# Solution of;
# Project Euler Problem 564: Maximal polygons
# https://projecteuler.net/problem=564
# 
# A line segment of length $2n-3$ is randomly split into $n$ segments of 
# integer length ($n \ge 3$). In the sequence given by this split, the 
# segments are then used as consecutive sides of a convex $n$-polygon, formed 
# in such a way that its area is maximal. All of the $\binom{2n-4} {n-1}$ 
# possibilities for splitting up the initial line segment occur with the same 
# probability. Let $E(n)$ be the expected value of the area that is obtained 
# by this procedure. For example, for $n=3$ the only possible split of the 
# line segment of length $3$ results in three line segments with length $1$, 
# that form an equilateral triangle with an area of $\frac 1 4 \sqrt{3}$. 
# Therefore $E(3)=0. 433013$, rounded to $6$ decimal places. For $n=4$ you can 
# find $4$ different possible splits, each of which is composed of three line 
# segments with length $1$ and one line segment with length $2$. All of these 
# splits lead to the same maximal quadrilateral with an area of $\frac 3 4 
# \sqrt{3}$, thus $E(4)=1. 299038$, rounded to $6$ decimal places. Let 
# $S(k)=\displaystyle \sum_{n=3}^k E(n)$. For example, $S(3)=0. 433013$, 
# $S(4)=1. 732051$, $S(5)=4. 604767$ and $S(10)=66. 955511$, rounded to $6$ 
# decimal places each. Find $S(50)$, rounded to $6$ decimal places.
# 
# by lcsm29 http://github.com/lcsm29/project-euler
import timed


def dummy(n):
    pass


if __name__ == '__main__':
    n = 1000
    i = 10000
    prob_id = 564
    timed.caller(dummy, n, i, prob_id)
