This is from Paul Zeitz, Art and Craft of Problem Solving. Problem 3.1.28, 3rd edition, pg 69.
I quote the problem verbatim: "Imagine dropping three pins at random on the unit interval [0,1]. They separate the interval into four pieces. What is the average length of the piece.
I am fully aware of two/three solution on Stack exchange that address the above problem. But I would like to understand the solution presented by the author of the book.
The solution goes as follows: "Imagine that we are not actually dropping three pins on a line segment, but instead dropping 4 pins of a circle with circumference 1 unit. Wherever the fourth point lands, cut the circle there and 'unwrap' it to form the unit interval."
I would like to understand where is symmetry coming into play in the above solution, and how does one arrive at the answer 1/4 by invoking symmetry.