Average distance between two randomly chosen points in unit square (without calculus) Imagine that you choose two random points within a 1 by 1 square.  What is the average distance between those two points?  Using a random number generator, I'm getting a value of ~0.521402... can anyone explain why I'm getting this value, or what this number means? 
More importantly, is there a way to solve this without using calculus and/or large random sampling?

When you say "without calculus", then presumably you are talking about just using something like basic algebra or Euclidean geometry. However, as you have seen from the up-voted posted answers and comments, the exact answer involves natural logarithm. It is basically impossible to define natural logarithm without some sort of calculus and/or real analysis. So it seems you need calculus (or its more formal version, real analysis) to come up with the formula for this answer.
