There are more ways to arrange a deck of cards than atoms on Earth


Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again.

At least until someone creates a self-replicating robot whose jobs are to create copies of itself and decks of cards.

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I’m told that if you want to say the precise number of potential shuffles out loud, it’s eighty unvigintillion, six hundred fifty-eight vigintillion, one hundred seventy-five novemdecillion, one hundred seventy octodecillion, nine hundred forty-three septendecillion, eight hundred seventy-eight sexdecillion, five hundred seventy-one quindecillion, six hundred sixty quattuordecillion, six hundred thirty-six tredecillion, eight hundred fifty-six duodecillion, four hundred three undecillion, seven hundred sixty-six decillion, nine hundred seventy-five nonillion, two hundred eighty-nine octillion, five hundred five septillion, four hundred forty sextillion, eight hundred eighty-three quintillion, two hundred seventy-seven quadrillion, eight hundred twenty-four trillion (or 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000). Now you can play Numberwang at home!


Isn’t that the plot to Grey Goo: New Vegas?

(on more further consideration than I can reasonably justify, I think I’ve come up with an…um…super convincing explanation for why there would be a ‘New Vegas’ after a grey goo incident, and what other plot elements would exist: Presumably if one were creating self-replicating shuffler robots, one would include a feature to ensure that the robots replicate as efficiently as possible by recognizing other robots of the same type and not wasting time breaking them down and re-replicating them, when the job was already done at least once.

If that is the case, locations on earth with extremely high concentrations of card-shuffling and similar machines already, notably the great casino-clusters of the world, would be overlooked during the unstoppable proliferation of self-replicating cardbots, known to survivors as “The Big All-In”. In Vegas, The Strip, and a few of the outer zones, would be spared for this reason, with a mixture of locals and stranded tourists in a world where, suddenly, what happens in Vegas stays in Vegas, because there isn’t anything outside, except rumors of ‘Macao’ still existing and a few offshore gambling boats that quickly descended into cannibalistic insanity, run aground on a slurry of self-replicating robots.

In what remains of Vegas, the Nevada Gaming Control board, the only entity with expertise in gambling machine programming and access to state violence steps in as the guardian, and government, of what remains of civilization. At the edges of the city, and in the wasteland, “Players”, rogue experts in machine manipulation indulge in the risky game of code-jacking ‘live’ nanite to insert their own directives. It’s a dangerous business. Most newbs are simply swallowed up and now shuffle cards with the rest of the earth’s crust. Sometimes an expert slips up and a more…twisted… fate befalls him (The player they call ‘The Joker’ is a living legend; but even he hasn’t been able to reverse what happened when the nanite he was 'jacking got out of control and reconstructed much of his body based on the templates for one of the deck’s face cards.) However, the risk is matched by the reward; because in a world whose industry, agriculture, and entire economy, save only hookers, gambling, and novelty restaurants was annihilated within hours, “tame” nanite is the most precious commodity there is, and a talented and lucky player can make a fortune supplying the city with this invaluable resource.

Within the city, of course, you have the former entertainers and casino operators, hustling to be the ones who get the job when a player makes good. You’ve got the Gaming Control board, which views the players as dangerous, liable to accidentally destabilize the grey goo and end things once and for all; but in no position to crack down too hard on people so vital to civilization. Adding to the general in-city atmosphere of mayhem and confusion, the destruction of almost everything that isn’t Vegas obviously rendered

Organized crime was there before the Big All-In, and some of it hangs on, trying to keep a finger in what pies remain. Some are merely crass and greedy. Some may be playing longer games…

On the edges of the city, where occasional nanite breakthroughs make things tenuous, you have the desperate and the idealistic. The bankrupts, the people who lost it all, the players who got burned (or worse), and the ‘reclaimers’ who use improvised camouflage barriers constructed of neutralized shufflers to ‘hide’ plots of land, and a combination of thermal destruction and whatever tame nanite they can beg(some players are mere mercenaries, some do have better natures, better even than one might expect, some are inscrutable; but are as likely, when they come to town, to give away an unimaginable fortune in tame nanite as they might be to deliberately breach the barriers without apparent reason and be taken down by Gaming Control, you just never know), borrow, buy, or steal to cleanse the ‘hidden’ plots and create arable land, industry, or other attempts to rebuild what was lost.

There are also groups you would be less advised to try to reason with: The various ‘stochastic Millenarian’ cults believe that the present hellish state is irredeemable until the last permutation has been dealt, and the planet redeemed, at which point the shufflers will cease their work and restore what was before. The (rare) agreement between the Gaming Control Board and the Players that the shufflers contain no such reconstruction code has failed to dampen their enthusiasm; but their passivity in waiting for the restoration has them viewed as largely harmless.

Unfortunately, not all the more shadowy cults that flourish in this environment of extreme dislocation are… against… the prospect of the final success of the shufflers and the hastening of their eventual success. The most open(but most controversial) of these is led by ‘The Player of Games’, a figure of undoubted brilliance, among the earliest and greatest of the players, who commands more nanite than almost anyone and plays a very ambiguous role: rumors of his association with the more extreme cults are too numerous and credible to deny; but (unlike some of the duller cultists) he is well aware that Earth doesn’t have enough time to complete the shuffling, whether or not a few extra cities are added, so his interventions have, so far, been overwhelmingly benevolent, if possibly in the direction of spreading even more virulent shuffler nanite beyond the planet and to the stars.

I don’t know exactly how a player-character would fit in; but that’s my general conception of Grey Goo: New Vegas.)


If you count whether the cards are right-side-up or not, the number gets even more ridiculous. And while it’s not very relevant for card games, it’s relevant if you’re just describing the number of ways to arrange a single deck.


$ python enc “Encoded via card order”
Message uses 136.2 of 225.6 bits total
8♥ A♦ 2♦ 5♥ 3♦ 2♥ 6♣ 4♦ A♣ 5♦ 6♦ J♥ 3♥ 7♦ 3♣ 9♣ 8♦ 9♦ 10♦ 8♣ 5♣ J♦ Q♦ 6♥ Q♣ J♣ K♦ 10♣ K♥ A♠ 2♠ 3♠ 4♠ 4♥ 4♣ A♥ 9♥ 5♠ 7♣ 10♥ 6♠ 7♠ 8♠ 9♠ 7♥ 10♠ J♠ Q♠ Q♥ 2♣ K♠ K♣


Atoms on earth? That’s almost 10^68 – probably within an order of magnitude of the number of atoms in the Galaxy. About a million million million times more than the number of atoms on earth (10^50 or so).


Also a nice demonstration that simply because something is very very unlikely we don’t need to invoke a supernatural explanation for its existence. I shuffle a deck and create something whose odds of existing are vanishingly small, but this is not a “miracle,” nor does this make me God.


Selection bias also plays a role: there are very few ways of shuffling a deck that will cause it to go “Wow, isn’t it amazing that we were shuffled just right so as to be able to appreciate having been shuffled just right? Miracle!”; but all the decks not so shuffled fail to remark on this, since they necessarily lack the capability.


QI has been on this. “here is how long it would take
to go through every possible permutation of cards. If every star in our
galaxy had a trillion planets, each with a trillion people living on
them, and each of these people has a trillion packs of cards and somehow
they manage to make unique shuffles 1,000 times per second, and they’d
been doing that since the Big Bang, they’d only just now be starting to
repeat shuffles.”




According to Ilya Prigogine, this is the real reason for the Second Law of Thermodynamics - and possibly the reason for the arrow of time. And you thought it was just cards.


There must be 50! ways to leave your lover.


The screen in front of you is capable of displaying only a finite number of images.


For comparison: when you want a new bitcoin address (which is effectively your bank account number), there’s no bank to ask for a new unique number. Instead you roll some dice to generate one of the 10^50 or so possible addresses. Chances are that nobody else is using it (or you’d both have a private key to the “account”, and be able to spend each other’s money!)

(Curiously, in the earlier post about the Tenzi dice game, there seem to be 64 dice in the set - just about right for letting you choose one of the 10^50 possibilities.)

Why, if I had a nickel for each way you can shuffle a deck of cards…


Post hoc ergo propter hoc

Seriously - doesn’t any animator in the world understand how the world actually turns? Nice job having sunrises over California. Makes the rest of the math seem, oh, so realistic and trustworthy.

(OK I"m not dissing the math but c’mon! The sun doesn’t rise in the east - the world just turns that way!)

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Other fun trivia: shuffling a deck of cards “perfectly” 12 times will return it to its original ordering.

“Perfect” meaning on each shuffle every single card is perfectly interleaved.

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Isn’t this principle the basis of a number of “pick a card” tricks? I have a card trick book here somewhere…

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