Average Distance

Here's my Python code. It runs 10 times to check for consistent results. Average distance is 0.521.

Since the problem is in computer science category, let me provide the "computer solution". Below is the simple Python snippet which gives the required value. It is helpful to run this several times just to make sure that the answer doesn't change up to the third decimal. I got $0.521$

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# import two modules that we need
import random
import math

n = 10000000 # This is the total number of data points. Take this as large as possible to get accurate enough answer
ave = 0 # This is the average distance

for i in range(n):
    # Generate two random points inside the square of side length 1
    p1 = (random.random(), random.random())
    p2 = (random.random(), random.random())

    ave += math.sqrt((p1[0]-p2[0])**2 + (p1[1]-p2[1])**2)


ave = ave/n

# Print the value of the average
print(ave)

We can place the square in the coordinate plane. From there we can use the distance formula and set the $x$ and $y$ values to random values. Then we set up a loop to compute the distance a certain number of times, and store those values to a list, in which we take the mean of. Here is the code:

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#import the random and math modules 
import random
import math

a = []

#set up loop for 10000000 trials
for i in range(0,10000000):

    #Distance formula
    x = math.sqrt(((random.random()-random.random())**2)+((random.random()-random.random())**2)) 

    #stores value to list
    a.append(x)

#prints average
print sum(a) / float(len(a)) 

This returns approximately $\boxed{0.521}$ . You can run the code here .