Problem 547

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.




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:



例如,S(3) = 1.6514而S(4) = 19.6564,均保留小数点后4位小数。