This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
To solve Domino's puzzle, go into each library and read the names of the
files. Each library room has a certain theme (Materia Use, Shinra Weapon
Advancements, etc.). Look for the one file that has been misplaced (say,
a file about materia in a room that is about Industrial Developments).
Write down the first letter of that file's name. Find the misplaced
files in all four rooms, and write them down.
You need four letters to solve this puzzle. These four letters can be
obtained from the "incorrect" books. The letters that you use are an
equidistant number of letters from the left of each title, not counting
spaces or punctuation.
(In other words, if you have the sentences "Use Ultima to do damage to
all enemies." and "Teioh, the Black Chocobo, won the race." and the
number of letters from the left is eight, then you find the eighth
letter from the left of each sentence. In the first sentence, the letter
would be "m". In the other, the letter would be "e".)
So, how are you supposed to find the number of letters from the left to
use? You'll have to talk to Domino and get the possible answers in order
to find out.
Anyway, try to find and re-arrange the letters using the above criteria
into a word that would fit as one of Domino's answers. If you got it
right, Domino will give you the next Keycard and an item as a bonus.
You'll get a less-valuable item depending on how many times you mess up,
and eventually Domino will not give you anything aside from the Keycard
if you messed up badly.
(It should be noted that this puzzle changes every time you start a new
game. The answer is always randomly selected, and the books are always