[quote=@Alex]
So, this thread made me think of the interesting problem of exactly how many possible usernames there are!

I've come up with 342,708,664,283,810,176,729,802 - that's 342 [b]sextillion[/b], 3.42*10^23 possible names.

I don't think we have to worry about running out of names any time soon - there are enough for each person alive to have [b]billions[/b] of usernames.

Also, I bet you just glossed over that huge number when you read it.

[hider=Maths]
How did I come up with that number?

Well, first, your username on the guild has to be from two to fifteen characters long.
Second, possible characters include a space, A-Z, and 0-9. It seems that capitalization does not matter, so Alex ALex alEX are all the same username (try logging in with strange capitalization). So 26 letters, ten numbers, and a space gives us 37 possible characters to work with.

The number of different usernames that can exist with two characters can be represented by
37*37
So, you can put any of our 37 characters in the first spot and any of the 37 characters in the second spot. (This isn't technically true - you can't have a name that has only spaces [i tried), and I bet you can't have a space preceding or following your username, usernames that are offensive and would be banned, and so on - but I think that these special cases aren't significant enough to really change that big number up at the top by a large amount.)

So then, we can see that the number of usernames that can exist for any length [i]k[/i] is 37^k.
So if we added up all the answers for k=2 and k=15...

Wait, this sounds like a [url=https://en.wikipedia.org/wiki/Geometric_series]geometric series[/url]!
In particular, this helps greatly:
[url]https://upload.wikimedia.org/math/0/2/9/029e68ec4c3a9c97b61c6d5b011efcfd.png[/url]
Replacing the pertinent numbers into the equation gives us:

(37^2-37^(15+1))/(1-37)
If you drop that into google, you'll get a number similar to what I have up top. I just calculated it out all the way.
[/hider]

Doing all that math reminded me of something I was working on a few weeks ago - someone on the IRC told me about [url=https://libraryofbabel.info/]The Library of Babel[/url] project, which apparently contains everything ever written (within the parameters they have currently). I was trying to figure out how large the library would have to be in meatspace (And now I realize I was going at it the wrong way!)

(p.s. if i'm wrong feel free to correct me i'd love to know what I messed up)
[/quote]

I personally get 324 Sextillion by using a network diagram approach of 36^2 + 36^2 *37^n for n = 1:13 (as you can't start or finish with a space and you can end at each value of n). Seems much to simple compared to your maths though. Although I'm not sure on the potential for more than one space in a row for a username. I have my doubts about the validity of yours as it  exceeds 37^15 = 333 Sextillion.