#547 Distance of Random Points Within Hollow Square Laminae - Project Euler

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 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 w

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