Like, you'd sign in with your shitty, number filled username, but you could set the actual name that appears to everyone else to be whatever you want, even if someone else already uses that name.

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 sextillion, 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 billions of usernames.

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

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 k is 37^k.
So if we added up all the answers for k=2 and k=15...

Wait, this sounds like a geometric series!
In particular, this helps greatly:
https://upload.wikimedia.org/math/0/2/9/029e68ec4c3a9c97b61c6d5b011efcfd.png
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.

Doing all that math reminded me of something I was working on a few weeks ago - someone on the IRC told me about The Library of Babel 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)

<Snipped quote by Alex>

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.

Another important thing is pronounceability. You may never have to enunciate your username, but people who make usernames generally make them capable of being enunciated nonetheless. This is achieved by way of constructing a username from consonant-vowel pairs, those 105 letter combinations that are phonetically viable. If we stray from pure mathematics by excluding phonetically infeasible names, we're left with a much smaller number than 324 sextillion, but still more than enough for us to never have to worry about running out of unique usernames.

Plus, the internet's so huge that having the same username as someone else doesn't really matter. Some people use names that are already associated with someone else completely intentionally, even. Say, for example, giving yourself the username of a Kingdom Hearts character.