How many ways can you arrange a deck of cards?

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mrgoat
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How many ways can you arrange a deck of cards?

Postby mrgoat » April 6th, 2014, 5:07 am

80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 according to


http://boingboing.net/2014/04/05/there- ... range.html

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Re: How many ways can you arrange a deck of cards?

Postby Denis Behr » April 6th, 2014, 6:45 am

Of course it's 52! for 52 different cards - the exact number can be admired HERE.

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Q. Kumber
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Re: How many ways can you arrange a deck of cards?

Postby Q. Kumber » April 6th, 2014, 7:30 am

About the equivalent of books on card tricks.

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Andrew Pinard
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Re: How many ways can you arrange a deck of cards?

Postby Andrew Pinard » April 6th, 2014, 7:49 am

Slightly more ways than you can turnover a card:

How many combinations are possible in a simple shuffled deck of 52 playing cards?

'Tis but a factorial of 52 (52 x 51 x 50 x 49 x 48 x ...)

The number?

52!

Otherwise known as:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

Or it can be read:

80 unvigintillion 658 vigintillion 175 novemdecillion 170 octodecillion 943 septendicillion 878 sexdicillion 571 quindicillion 660 quattuordecillion 636 tredicillion 856 duodecillion 403 undecillion 766 decillion 975 nonillion 289 octillion 505 septillion 440 sextillion 883 quintillion 277 quadrillion 824 trillion

68 digits long.

How's them apples?

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Re: How many ways can you arrange a deck of cards?

Postby Brad Henderson » April 6th, 2014, 10:08 am

according to the confection licking owl I just spoke with the answer is: thhhererrreeee!

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Re: How many ways can you arrange a deck of cards?

Postby Ian Kendall » April 6th, 2014, 10:43 am

Aaronson
Mnemonica
Nikola
Bart Harding
Si Stebbins
NDO
Random

I make that seven.

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Re: How many ways can you arrange a deck of cards?

Postby R.E.Byrnes » April 6th, 2014, 10:51 am

I recalled reading in the most recent Persi Diaconis book a counterintuitive explanation as to why it's somewhat less than 52! but now can't find it; perhaps it was conditioned on beginning from a particular arrangement and spoke to the number of possible arrangements from there by way of a conventional riffle shuffle. Anyone know what I'm referring to?

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Re: How many ways can you arrange a deck of cards?

Postby AJM » April 6th, 2014, 11:15 am

Possibly 9 if you include: -

Osterlind
Joyal

or, indeed, not...
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Re: How many ways can you arrange a deck of cards?

Postby Bill Mullins » April 6th, 2014, 12:17 pm

To me, the amazing thing was that if you take a deck and give it a really good shuffle, the odds are that no deck in history has ever been in that particular order before.

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Re: How many ways can you arrange a deck of cards?

Postby Edwin Corrie » April 7th, 2014, 10:37 am

Bill Mullins wrote:To me, the amazing thing was that if you take a deck and give it a really good shuffle, the odds are that no deck in history has ever been in that particular order before.


Reported to me by a friend of my brother:

[quote]

There was a probability expert talking about this on Radio 4 about a year or so ago. He stated exactly that, that each well shuffled deck is likely to be a completely new combination. This is regardless of how many packs of cards there have ever been. For example, if every human being in existence (roughly 7,000,000,000) had 1000 packs of cards each. That would be 7,000,000,000,000 packs of cards. If each of those packs had been shuffled 1000 times each and the order of the cards noted down, that would be 7,000,000,000,000,000 combinations. That means that there would be about

11,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

combinations that had not yet been formed.

The universe is approximately 14,000,000,000 years old so, even if each of those people had existed since the beginning of the universe and shuffled each of their 1000 packs of cards 1000 times every year, it's unlikely that all combinations will have been "discovered" by now. That's some brain mashing figures!!

And to think it took me about 3 months to memorise just one particular order!

[...] I suppose the question is... How many cards do we actually need in a deck to make them good for gaming? Perhaps we only need 30 cards? That would still give us a gazillion and three combinations.

[unquote]

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Re: How many ways can you arrange a deck of cards?

Postby John Bowden » April 7th, 2014, 1:07 pm

Now remember we are magicians...................so sometimes we arrange decks with a card or cards face down in a face up deck.

This would play havoc with the possibilities as we could have any one of 52 cards face down in any one of 52 positions. This would be multiplied should there be two cards face down in different positions/not beside each other in the deck and the multiplication would get out of hand when we have 10 or 20 or even more cards reversed.

Now guys..........over to the mathematical geniuses who are posting on this subject.

I want answers.

John Bowden

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 7th, 2014, 1:15 pm

mrgoat wrote:80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 according to


http://boingboing.net/2014/04/05/there- ... range.html


The number is large. Countable yet somehow outside our notions of relevance. More than fingers, birthdays, family members, our yearly income ... number of visible stars at night...

It's like the grains of sand on the beach thing ... computable yet outside our perceived scope of relevance.
Mundus vult decipi -per Caleb Carr's story Killing Time

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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 7th, 2014, 1:29 pm

I'm probably wrong on this, but wouldn't allowing for face-up or face-down just introduce another factor of x 2 after every card in the original calculation? In other words, take 52! and allow for each card to be in one of two positions, i.e.:

Total combinations = 52! x 2^52 = 52 x 2 x 51 x 2 x 50 x 2 x 49 x 2 x 48 x 2 x … x 3 x 2 x 2 x 2 x 1 x 2

yielding a new 84-digit number:

363252127644247044041398160152368436824058904969853261166529262548680704000000000000

which is read as "three hundred sixty-three sexvigintillion, two hundred fifty-two quinvigintillion, one hundred twenty-seven quattuorvigintillion, six hundred forty-four trevigintillion, two hundred forty-seven duovigintillion, forty-four unvigintillion, forty-one vigintillion, three hundred ninety-eight novemdecillion, one hundred sixty octodecillion, one hundred fifty-two septendecillion, three hundred sixty-eight sexdecillion, four hundred thirty-six quindecillion, eight hundred twenty-four quattuordecillion, fifty-eight tredecillion, nine hundred four duodecillion, nine hundred sixty-nine undecillion, eight hundred fifty-three decillion, two hundred sixty-one nonillion, one hundred sixty-six octillion, five hundred twenty-nine septillion, two hundred sixty-two sextillion, five hundred forty-eight quintillion, six hundred eighty quadrillion, seven hundred four trillion."

We can ask Arthur Benjamin at Magic*Con this week just to be sure.

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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 7th, 2014, 1:34 pm

One more thing: that number I just gave is written with the American system. Folks elsewhere in the world would say/write the same number differently.

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 7th, 2014, 2:36 pm

And if the pack as a one way back design ... another factor of two raised to the fifty second power?
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Re: How many ways can you arrange a deck of cards?

Postby Carlo Morpurgo » April 7th, 2014, 4:37 pm

Jonathan Townsend wrote:And if the pack as a one way back design ... another factor of two raised to the fifty second power?


Correct...2^52 times 52!. For each of the 52! permutations there are 2^52 ways to switch each card up or down.

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Re: How many ways can you arrange a deck of cards?

Postby Kent Gunn » April 7th, 2014, 6:29 pm

I just counted,

That's a larger number than the number of atoms in the universe.

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Re: How many ways can you arrange a deck of cards?

Postby Brad Jeffers » April 7th, 2014, 6:58 pm

Kent Gunn wrote:That's a larger number than the number of atoms in the universe.


In the observable universe, yes.

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Re: How many ways can you arrange a deck of cards?

Postby Carlo Morpurgo » April 7th, 2014, 8:53 pm

As a side note, a 52-card deck needs 8 perfect faro shuffles in order to restore the deck to its original state. You would think that the more cards you have the worse it gets...wrong. For example you need 10 perfect shuffles for a 12-card deck, 100 shuffles for a 102-card deck, 10 shuffles for a 1024-card deck, and so on. I have not analyzed the problem rigorously, but it looks doable.

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Re: How many ways can you arrange a deck of cards?

Postby Carlo Morpurgo » April 7th, 2014, 9:07 pm

Ok someone has already thought about it... http://en.wikipedia.org/wiki/Out_shuffle
For a deck with n cards (n even) we need k shuffles, where k is the least number so that n-1 that divides 2^k-1.
Last edited by Carlo Morpurgo on April 8th, 2014, 6:49 am, edited 1 time in total.

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Re: How many ways can you arrange a deck of cards?

Postby Denis Behr » April 8th, 2014, 2:57 am

(Here's an old script of mine doing the calculation for a given number of cards (and an overview table for up to 1024 cards): http://www.denisbehr.de/faro/ )

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Re: How many ways can you arrange a deck of cards?

Postby Carlo Morpurgo » April 8th, 2014, 5:18 am

Denis Behr wrote:(Here's an old script of mine doing the calculation for a given number of cards (and an overview table for up to 1024 cards): http://www.denisbehr.de/faro/ )


Nice...do you actually run the shuffles until you get it right?

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Re: How many ways can you arrange a deck of cards?

Postby Denis Behr » April 8th, 2014, 7:26 am

For an even N, I'm just calculating the order of 2 mod(N-1) to get the number of out faros, and so on. In e.g. PHP this is a one-liner. I simply wrote it as a for-loop that way ($input being the even number of cards):

for ($n = 1, $t = 2; $t != 1; $n++, $t = bcmod(2*$t, $input-1)) {}
echo $n;

(If you're interested in the mathematics of the faro shuffle, get S. Brent Morris's book Magic Tricks, Card Shuffling and Dynamic Computer Memories, or the shorter version with less deep math Everything You Always Wanted To Know About The Faro Shuffle.)

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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 8th, 2014, 5:33 pm

Factoring in Jonathan's concern for one-way backs - meaning we are now accounting for the sequence of each card in terms of its face value, the "major" orientation of each card face-up vs. face down, and the "minor" orientation of each card as "right side up" or "upside down" with respect to one-way back design - we now get this lovely 100 digit number (split due to wraparound limitations):

16359421467001708365243762794498546079912166454178
26121290025525547977913533733414109184000000000000

which is (of course) read the American way as:

one duotrigintillion, six hundred thirty-five untrigintillion, nine hundred forty-two trigintillion, one hundred forty-six novemvigintillion, seven hundred octovigintillion, one hundred seventy septenvigintillion, eight hundred thirty-six sexvigintillion, five hundred twenty-four quinvigintillion, three hundred seventy-six quattuorvigintillion, two hundred seventy-nine trevigintillion, four hundred forty-nine duovigintillion, eight hundred fifty-four unvigintillion, six hundred seven vigintillion, nine hundred ninety-one novemdecillion, two hundred sixteen octodecillion, six hundred forty-five septendecillion, four hundred seventeen sexdecillion, eight hundred twenty-six quindecillion, one hundred twenty-one quattuordecillion, two hundred ninety tredecillion, twenty-five duodecillion, five hundred twenty-five undecillion, five hundred forty-seven decillion,
nine hundred seventy-seven nonillion, nine hundred thirteen octillion, five hundred thirty-three septillion, seven hundred thirty-three sextillion, four hundred fourteen quintillion, one hundred nine quadrillion, one hundred eighty-four trillion.

… and only a mere 8.361 duotrigintillion and change short of a googol.

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 8th, 2014, 6:41 pm

Is this a setup for a full deck "sympathetic cards" or prediction?
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Re: How many ways can you arrange a deck of cards?

Postby Bill Mullins » April 8th, 2014, 9:54 pm

I can see the arguments for considering if each card is face-up or -down, and if cards in a one-way deck are rotated end for end. However, consider a deck where the cards are randomly ordered and flipped face-up or -down. Now take the whole deck and flip it over. I'd consider it to still be the same arrangement. Likewise, take the deck, stand up, and turn around. Each card is now pointing north instead of south, but again I'd still consider it the same arrangement. So divide the big number by 4. So . . .

Total # of arrangments = 52! x 2^51 (each of the cards except the first can be face-up or -down relative to the first) x 2^51 (each of the cards except the first can be pointed either way relative to the first).

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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 8th, 2014, 10:54 pm

I respectfully disagree, Bill.

What you have is a reflection or complementary arrangement of the original, but it is not the same arrangement since the cards are always (for example) being numbered from 1 to 52 going top to bottom. Flipping the randomly ordered deck over and giving it an end-for-end turn yields new cards in every position with their own face-up/face-down and right-side up/upside down characteristics.

The new deck does, however, bear a strictly functional relationship with its predecessor in this case: new card #1 is the direct reflection/complement of old card #52, new card #2 is the direct reflection/complement of old card #51, new card #3 is the direct reflection/complement of old card #50, etc. - but they are still two totally unique arrangements.

Put another way, the strings: FeDcBa and AbCdEf contain the same letters but their case and sequence is reversed. But since English reads left to right, FeDcBa and AbCdEf are two distinct strings. Furthermore, AbCdEf would come before FeDcBa in sequence (alphabetically.)

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Re: How many ways can you arrange a deck of cards?

Postby Bill Mullins » April 9th, 2014, 8:26 am

When you shuffle a deck, the cards are arranged in a stack relative to each other.

When you start talking about the orientation of the individual cards, you have to be oriented relative to something else. Relative to what? The person holding the cards? The world?

If I'm standing with a deck, and I turn around, the whole deck turns with me. Is that a new arrangement? Suppose you and I are facing each other, and I hand you the deck. When I held it, all the one-ways were pointing at me. Now you hold it, and they are pointing away from you. Has the arrangement changed?

Suppose I'm holding a deck, and without letting go, I walk around the earth. Relative to the North Star, the deck has changed orientation, but not relative to me or to the earth. Is that a new arrangement, or not? Or maybe we are in the Space Station, floating. My head is at your feet, and vice versa. I hand you the deck -- every card that was face-up for me is now face-down, from your perspective. New arrangement, or not?

I contend that since we are talking about the arrangement of cards in a deck, the deck itself is the appropriate frame of reference, and all cards except the first are arranged relative to the first card. Thus 2^51 X 2^51, instead of 2^52 X 2^52, in which the first card itself can be oriented (but you have to define "oriented to what?").

I'm not saying that you are wrong. I am saying that if you contend that the orientation of the deck as a whole makes a difference, you have to be particular about your frame of reference.

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 9th, 2014, 8:35 am

It's simpler to imagine fifty two slots. Each takes one card. The card can be any of the available unassigned cards from the working pack, our universe of discourse. The pack is 'arranged' when all fifty two cards have been put in slots. I think we're good with the count of such arrangements.

As distinct matters - some of the arrangements may be related by operations of turning over, cutting or shuffling.
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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 9th, 2014, 6:12 pm

Bill,

I totally agree with you on orientation. When not explictly stated or implicitly inferred, it must be established.

I also agree that the deck itself makes a good (but only partial) reference. If I may be so bold, a deck held “face-down” in dealing position is the usual frame of reference for orientation. So if I open a brand new deck and hold the cards for a normal deal, all of the cards will be oriented with their faces down. Let’s also consider the top card (furthest away from my palm) in this orientation to be card #1 and the bottom card (next to my palm) to be card #52. Furthermore, if this fresh-from-the-box deck has one-way backs (for example, the lone word “MAGIC” emblazoned in non-ambiagram fashion across the center of an otherwise symmetrical pattern so the dealer perceives the word as either right-side up or upside down), it is quite likely that I will be holding a deck in which all of the MAGICs are right-side up. It is from such a confluence of agreements that we may proceed. Let’s call this “as viewed by the dealer.”

So if you’re holding the deck thusly and turn around, no – you do not have a different arrangement (although if you hold the deck the same way and turn a full 360 degrees it would be equivalent to giving the deck a single 360 turn on the same plane, but that’s another argument to be had.)

But if you hand me the deck face down without giving it a 180 turn (along the flat plane of the deck) then YES it IS a new arrangement – presuming we are both using “as viewed by the dealer” as our frame of reference. After handing it to me, while the face up/face down nature and the sequential order of values has not changed, every orientation was changed from when you were handling it. If you do give the deck that 180 turn, I would be “dealing” the exact same arrangement as you.

I again point out that I agree with you that the rules must be understood in advance. So to address your outer space scenarios, the answer is that it depends on what we/you agreed upon. Not sure that the North Star has any particular interest in this so I’ll give that one a “fine if you want that to be your reference, but I’d recommend we use others first.” If we were floating in the space station, again, it depends. But since you used the phrase “from my perspective” (which I am presuming you mean to be different from your perspective), then I can immediately conclude that YES it is another new orientation since you changed at least one of the cards in one of the changeable dimensions under consideration (or in this case again, all of them.)

I cannot agree, however, that the deck itself (and ONLY the deck itself) defines the orientation. By your own statement, you are contending that the first card is somehow different from the other cards and therefore should not participate in any of the computations. Sorry, but I can’t buy that. That would mean, for example, that the nine of diamonds figures into all the computations except when it’s the first card. Ditto for all the other cards. That’s an awful lot of combinations to leave out – and, quite simply, you can’t.

But the real important point (and again, I believe that we are in total agreement here) is that once you DO establish a (complete) frame of reference for orienting the deck, you stick with that frame of reference as you perform the calculations to arrive at the results we have previously. Within any complete frame of reference, the aforementioned math holds. But you have to compare apples to apples. If you start mixing or switching frames of reference midstream, you’ll just get a mess along with lots of errors.

Brian

ps: you were not very particular about your own frame of reference for the first card in the deck. How are we supposed to know which one is the first one when the deck is serving as its own frame of reference?

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 9th, 2014, 8:37 pm

Let's start with the pack defined as all fifty two stacked and in the case. Unless you want to go the way of Mertz :ugeek: and discuss all the ways one could arrange a pack of cards to make a work of art where at least one of the cards is visible to the viewer and the rest can be anywhere in the work, covered in paint, pasted to the canvas or the frame or ... somewhere in the gallery? :lol:
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Re: How many ways can you arrange a deck of cards?

Postby Bill Mullins » April 10th, 2014, 8:28 am

Jonathan Townsend wrote:L Unless you want to go the way of Mertz

Fred or Ethel?

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Re: How many ways can you arrange a deck of cards?

Postby Jonathan Townsend » April 10th, 2014, 8:55 am

Bill Mullins wrote:
Jonathan Townsend wrote:L Unless you want to go the way of Mertz

Fred or Ethel?


:) thanks Bill. typo - should be Merz

@Brian, but is the deck of cards as seen by one person necessarily the same as the deck as seen by another, or even the same person at a different time? Trying to sneak a countable infinity under the rug?
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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 10th, 2014, 7:54 pm

@Brian, but is the deck of cards as seen by one person necessarily the same as the deck as seen by another, or even the same person at a different time? Trying to sneak a countable infinity under the rug?


No I wasn't, but if we are going to go into vision (the stimulation of your photoreceptors / rods and cones versus the stimulation of mine) then we certainly do have a countable infinity to deal with. Of course, we could just agree that we're going to use the specific arrangement of the actual atoms that make up a particular deck of cards, but then again technically we don't really "see" them, do we?

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Re: How many ways can you arrange a deck of cards?

Postby Don Driver » April 10th, 2014, 10:49 pm

As many times as there are bad magicians in the world....no make that the U.S.A.

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Re: How many ways can you arrange a deck of cards?

Postby mrgoat » April 11th, 2014, 6:50 am

Don Driver wrote:As many times as there are bad magicians in the world....no make that the U.S.A.


I've got a bone to pick with you. My most recent obvious fake nick at the cafe was banned because they thought I was you!

I spent ages on that account!

:O

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Re: How many ways can you arrange a deck of cards?

Postby AJM » April 11th, 2014, 2:23 pm

brianarudolph wrote: (the stimulation of your photoreceptors / rods and cones versus the stimulation of mine)


We'll have enough of that behaviour here if you don't mind.
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Re: How many ways can you arrange a deck of cards?

Postby brianarudolph » April 11th, 2014, 6:23 pm

We'll have enough of that behaviour here if you don't mind.


Oh, admit it, AJM: you know that none of us can ever get enough. ;)

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Re: How many ways can you arrange a deck of cards?

Postby kaciecute » April 13th, 2014, 8:56 pm

I think there is a youtube video that explain all the possible permutations
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Re: How many ways can you arrange a deck of cards?

Postby Brad Jeffers » April 14th, 2014, 2:42 am

Here is a problem ...

What is the number of random deck arrangements needed to reach a point were there is a 50% chance that one of the arrangements will be duplicated.

Sort of a variation of The Birthday Paradox.


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