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Like everybody else, you too can be unique. Just keep shuffling

The first take-home lesson of this note is that you too can be unique. You’ll have to keep shuffling to get there, but it is an attainable goal.

Several years ago it dawned on me that the number of possible ways to order or permute the cards in a standard deck of size $52$ was inconceivably large. Of course it was — and still is — $52!$. That’s easy enough to scribble down (or even surpass spectacularly) without understanding just how far we are from familiar territory.

Let’s start with something smaller: the number of possible ways to order or permute just the hearts is $13! = 6,\!227,\!020,\!800$. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those $13$ cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

Had a joker been thrown in too, it wouldn’t have worked out so well. Even today, with the population of the planet presumed to hover around 7 billion, there would have to be some sharing of the permutations on the list. In fact, since $14!$ is about 87 billion, it seems safe to predict that it will be a very long time indeed before the world’s population is large enough so that everyone gets just one such ordering.

Let’s put this in a musical context. It was also a decade ago, in the spring of 2002, that the Queens of the Stone Age recorded their Songs for the Deaf album. The standard release lists 13 tracks, but there is also a hidden 14th track at the end. They could have issued each person on earth their own personal copy, with their own personal track order, and also exhausted all of the possibilities in the process, assuming they still finished with that hidden track.

Adele ‘s recent 21 album, however, only has 11 tracks so, noting that $11! = 39,\!916,\!800$, she’d have had to settle for Poland or California if she wanted to achieve the same effect on both counts.

That’s something to think about it the next time you hit shuffle on your favorite music player.

The number of possible ways to order all the red cards in a deck is $26!$ which is about $4 \times 10^{26}$. How big is that? It’s certainly bigger than the number of grains of sand in Brighton, or Britain, or all of the beaches on earth for that matter.

You can be 100% sure that the compilers of the 26-track early Kinks set didn’t actually consider all possible track orders. To do so would have required making a list four times as long as a list of $10^{26}$ items. There simply isn’t enough paper, or computer memory. As anyone who has even compiled such a songlist knows, they probably decided on openers and closers and used something like chronological order for the rest.

Now consider taking out the four Aces from a deck, and putting the remaining hearts together in some order on the left followed by the other 26 cards in some order on the right. That can be done in over $10^{50}$ ways which exceeds the number of atoms on Earth.

As for playing with the full deck, note that $52!$ is about $8 \times 10^{67}$, which in the great scheme of things isn’t so far from $10^{80}$, the current estimate for the number of atoms in the universe.

Needless to say, nobody’s ever explicitly considered all of the options here either. Likewise, “the people in the office on the afternoon of Friday, September 3rd” over at Sub Pop, when coming up with the precise order for The Sub Pop List of the Top 52 Tracks Sub Pop Released in the ’90s.

What does this all mean? Well, if you were to hit “shuffle” (not allowing repeats) with a playlist consisting of those 52 tracks, it could be argued that you’d almost certainly hear a set of music that nobody has ever heard before. The same applies to the 52-track version of The World of Nat King Cole.

For a deck of cards, it means that there are far more shuffled states than have ever been written down. Likewise, the totality of all deck orders that have ever been achieved with actual decks in the history of the world is a very thin set within the set of all possible deck orders.

A well-shuffled deck, such as the one displayed here, which is far from being in any “obvious” or recognisable order, is probably unique in the sense that nobody else has ever come up with it before.1

As we’ve been saying all along, you too can be unique. Just keep shuffling. You’ll get there.

The other take-home lesson today is that you can shuffle till the cows come home, but you’ll still miss the vast majority of the possibilities. Or as Hamlet once said, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”

  1. I recall just using an order that resulted from numerous shuffles of an already seemingly well-jumbled deck []

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About the author

  • Dublin native Colm Mulcahy has been in the Department of Mathematics at Spelman College (Atlanta, Georgia) since 1988. His interests include algebra, number theory, geometry and mathematical card principles and effects. Follow him on Twitter at @CardColm and also check out @WWMGT.

5 Responses to “Like everybody else, you too can be unique. Just keep shuffling”

  1. Neil Calkin

    I like to point out to my discrete maths students that when I ask them to toss a coin a hundred times, and record the sequence of heads and tails, and then reduce it to just the number of heads and tails, they are taking it from an outcome which has most likely never been seen before, and never will be again, to a situation in which the class will almost certainly have at least two students with the same outcome.
    I find that it helps make the distinction between a discrete sample space and events.

    Reply
    • Rich

      I’m afraid I take issue with “As for playing with the full deck, note that 52! is about 8×10^67 , which in the great scheme of things isn’t so far from 10^80 , the current estimate for the number of atoms in the universe.”
      To my mind, 10^67 is massively massively massively far from 10^80.

      Reply

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