[1] What is the mean distance between two random points on the perimeter of a unit square?

[2] What is the mean distance between two random points on the interior of a unit square?

First way:  I began by finding the mean distance between two points on a unit segment by what is basically a double integral.  I do not doubt the result I got:  1/3.
Next I made an assumption that I don't know is valid:  I simply found the distance between two points whose x and y coordinated each differ by 1/3:  √(2)/3 ≈ .4714

This does not agree with Charlie's finding so I doubted it and decided to try a different method.

Second way:
I broke the unit square into n² congruent squares and considered the center point of each small square.  For n=2, 3 & 4 I calculated the distance between each of the (n²)² possible selections of two of these points.  The mean distances are as follows.
n=2
(8+4√2)/32 ≈ .4268
n=3
(48+24√2+16√5)/243 ≈ .4844
n=4
(136+80√2+48√5+24√10+16√13)/1024 ≈ .4785
This may be taken as evidence supporting my first solution  (as opposed to Charlie's.)  It may also be that both of my methods suffer a similar error.

Anyone care to write a program to increase n?  I can flesh out the details is needed.

Posted by Jer on 2013-06-02 01:10:39