Originally Posted by Garnabby View Post
Which sort of makes this one, and the last one, about averages, in terms of 1961, and 2061, with 7/2, then being about the midpoint of space-time, and, etc., as it's the midpoint of non-leap years (but with the hour of 9 am factored in to become the thus midpoint overall). Ha.

0.777665653586267115337934094617819509962882724417 13305802344596486505735315926540114615165689316818 846... is about (1602/2061 + 0.00037307716705303745) ---> 1602/2061, and 1691/1961 from 37 = [(1 + 6) + √9 * 10], and, 307 = [100 * √9 + (6 + 1)], with 7167 ---> 7 * 2^4 * 7 ---> 7/2/7. I mean, 37307 sort of does go its way because it is about (69 * 540.6811594202899), which is about [69 * (90 * 6 + 1)] = [(10*6 + 9) * (10*9 * 6 + 1)] ---> 1691/1961.

Oh, and to get the two thus 7/2's, I guess that the number of my posts has to be again factored in, at 1129 = (27^2 + 20^2) ---> 2/7/2. Something like this.
What's interesting is that the answer in the 4-dimensional case now involves even more irrational numbers such as arctans of certain numbers, Clausen functions evaluated at certain numbers, and Catalan's constant, which nobody has proved is irrational, but is most likely irrational.

Now go back to the 2-dimensional unit square case, and instead of finding the average length of a line segment between two random points inside it, find the average area of a triangle formed by three random points inside it. Miraculously the answer is exactly 11/144.