What I'm going to say is too long for a comment so I guess I have to answer. This is funny because I'm the one who marked your question as favorite, just in case someone answers it, I wanted to know the answer, and here I am the one who's answering it!
Do you know what were the rules of these competitions ?
No but I'll keep searching, maybe I can provide a better answer later on. If you only want to read my answer to your question scroll down, it's the last part.
But I guess, that Alexander Alekhine did not look at anything when playing. I will tell you why I think he didn't look, I will tell you how I think he was able to do it. Lets look at other things that our mind can do, and we'll talk about chess in the end.
- Did you know that many Muslims are required to memorize each word of the Quran? They actually do competitions sometimes to see who can answer faster, many none Arabic Muslims do that even little kids, imagine how hard it would be for a small kid to memorize the entire Quran, in Arabic, a language that he doesn't even understand.
- How many songs can you memorize? at least 1000 but you don't know it yet. How many songs can singers memorize? How many music compositions can musicians play without looking at the notes?
- What Is the Memory Capacity of the Human Brain?
You can do this, untrained, normal person, what can a professional do?
To fully understand what the brain is capable of you should take a look at:
Lets talk about the second example in particular, that guy is autistic and blind, so the brain adapted to the situation and his memory becomes so good that he could play any music he ever heard any time, perfectly.
Those are more exceptional people but still untrained.
Lets talk about chess now, all the chess players, at least all the GMs are trained professionally from a very young age to play chess.
What would a trained chess kid be able to do? What would a 13 year old kid do nowadays? play xbox, listen to Justin Bieber, watch cartoon network...
What would a 13 years old chess player do? Well Fischer at the age of 13 did the most amazing checkmate ever against Donald Byrne, I think no supercomputer will be able to do what Fisher did, it was called the game of the century.
I talked about how good the brain could become at certain things like memorization, but I still haven't answer your initial question!
You should know that Alexander Alekhine isn't the current record holder.
I quote from the second article:
George Koltanowski set the world's blindfold record on 1937 by playing
34 chess games simultaneously while blindfolded, winning 24 games and
losing 10, over a period of 13 hours. The record was included in the
Guinness Book of Records and is generally accepted as the world record
to this day.
And most importantly
Miguel Najdorf increased this record to 45 opponents in 1947, with the
result of 39 wins, four draws and two losses.However, he had access to
the scoresheets, and there were multiple opponents per
board.Koltanowski claimed that he could have managed 100 games under
those conditions.
So you can see why i don't think Alekhine had access to the scoresheets, otherwise they wouldn't have accepted his record, which is exactly what happened with Najdorf. Besides the players who challenged Alekhine, would feel it's unfair to challenge a great GM, with access to scoresheets, he would beat them all so easily.
Now how do I personally think he did it?
- He challenged amateurs, 800 - 1400 Elo max
- He played one or two openings against all of them so he doesn't have to remember many openings
- He played aggressive opening that he knows so well so he can finish the games so fast and not having to worry about middle game calculations
And finally from Wikipedia I guess the answer you wanted to know
Occasionally, grandmasters have given blindfold simultaneous displays.
In such displays, the exhibitor does not look at any of the boards,
but retains all the moves of the games in his/her head. The opponents
utilize boards and pieces in the standard fashion, but their moves are
communicated verbally to the exhibitor by an arbiter or intermediary.
Note
I don't think that GMs rely on their visual memory so much, but rather they do something like meditation, where they create their own games inside their heads and visualize it.