Problem 547
Distance of random points within hollow square laminae
Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine the expected value of the distance between these two points.
For example, the expected distance between two random points in a unit square is about 0.521405, while the expected distance between two random points in a rectangle with side lengths 2 and 3 is about 1.317067.
Now we define a hollow square lamina of size n to be an integer sized square with side length n ≥ 3 consisting of n2 unit squares from which a rectangle consisting of x × y unit squares (1 ≤ x,y ≤ n - 2) within the original square has been removed.
For n = 3 there exits only one hollow square lamina:
For n = 4 you can find 9 distinct hollow square laminae, allowing shapes to reappear in rotated or mirrored form:
Let S(n) be the sum of the expected distance between two points chosen randomly within each of the possible hollow square laminae of size n. The two points have to lie within the area left after removing the inner rectangle, i.e. the gray-colored areas in the illustrations above.
For example, S(3) = 1.6514 and S(4) = 19.6564, rounded to four digits after the decimal point.
Find S(40) rounded to four digits after the decimal point.
带洞正方形内任意点间距离
在长方形中（均匀地）随机选择两个点，可以算出两点间距离的期望值。
例如，在单位正方形中任意两点间距离的期望约为0.521405，而在宽为2长为3的长方形中任意两点间距离的期望约为1.317067。
从边长为整数n ≥ 3、含有n2个单位正方形的大正方形中，去掉一个由x × y个单位正方形组成的长方形（1 ≤ x,y ≤ n - 2），我们称剩下的图形为边长为n的带洞正方形。
当n = 3时，只有一种带洞正方形：
当n = 4时，不考虑旋转和翻转，你能找出9种不同的带洞正方形：
记S(n)是所有可能的边长为n的带洞正方形中任意两点间距离的期望之和。这里的任意两点必须在大正方形去除内部长方形后剩下的区域，也即上图中的灰色区域。
例如，S(3) = 1.6514而S(4) = 19.6564，均保留小数点后4位小数。
求S(40)并保留小数点后4位小数。