## Distance of random points within hollow square laminae

### Problem 547

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 `n`^{2} 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 exists 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.