It came to mind (not sure why), that 81 and 18 add up to 99. This was kind of intriguing since the numbers are similar.
But then I realized there was more to it than that. Both those numbers were not just multiples of 9, but were in fact 9^2 and 9*2 adding up to give 99 (two nines). Chatting with a friend, we wondered: was this a coincidence, or did it hold for higher numbers?

Let’s see:
99^2 + 99*2 = 9801 + 198 = 9999
999^2 + 999*2 = 998001 + 1998 = 999999

Basically, if x is a series of nines, then x^2 + 2x gives a series of nines twice as long.

So I think to myself “This can’t just be a coincidence. What is going on here?”

I came across the answer through the way I had been squaring the numbers. To figure out 999^2, for example, all I did was figure out 1000^2 (1000*1000 = 1000000), find 999*1000 by subtracting 1000, then find 999*999 by subtracting 999. This holds whenever you want the square of an integer given the square of the integer plus 1, as shown by:

(x+1)^2 - x^2 = x^2 - x^2 + x + (x+1) = 2x + 1

But that can simply be rearranged as:
(x+1)^2 - 1 = x^2 + 2x

So, if x is a series of nines, then x^2 + 2x is simply equal to a power of ten, squared, minus one.
Recognizing that the powers of ten always square to a one followed by a series of zeroes, it is clear that subtracting one will give us a series of nines.

And so the trend does hold, for all numbers made up entirely of nines.

Loving the chronicles by the way, and the multiplication method in the previous post. It works quite well for me.