[z_beeblebrox]

Originally Posted by Ether

What's the average length of all the line segments whose endpoints both lie on the unit square's perimeter?

Modified above Monte Carlo code to solve this:

import numpy as np

iterations = 100000

avg = 0

for i in range(1000):
    print(i)
    for j in range(iterations):
        perimeter = 4*np.random.rand(2)

        if perimeter[0] < 1:
            y1 = 0
            x1 = perimeter[0]
        elif perimeter [0] < 2:
            y1 = perimeter[0] - 1
            x1 = 1
        elif perimeter[0] < 3:
            y1 = 1
            x1 = 3 - perimeter[0]
        else:
            y1 = 4 - perimeter[0]
            x1 = 0

        if perimeter[1] < 1:
            y2 = 0
            x2 = perimeter[1]
        elif perimeter[1] < 2:
            y2 = perimeter[1] - 1
            x2 = 1
        elif perimeter[1] < 3:
            y2 = 1
            x2 = 3 - perimeter[1]
        else:
            y2 = 4 - perimeter[1]
            x2 = 0

        length = np.sqrt((x1-x2)**2+(y1-y2)**2)
        avg = (avg*j + length)/(j+1)
    with open("9.1.txt", "a") as f:
        f.write(str(avg)+'\n')

2.1e7 iterations yields 0.73511, standard deviation 0.00095 (between blocks averaging 1e6 iterations). More iterations will give a more accurate answer.