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The Magic Cafe Forum Index » » The workers » » Are there more ways to arrange a deck of cards than the number of atoms in the universe? (3 Likes) Printer Friendly Version

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ekgdoc
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I've told people that there are more ways to arrange a deck of 52 cards than there are number of atoms in the universe. Few believe me, but I believe it's true. The conditions I use for counting are (1) a face up card is different than a face down card (cards can be face up or face down), and (2) the order of face down cards matters (i.e., a different order of face down cards is a different arrangement). I calculate that the number of ways to arrange the cards under these conditions is (2^52) * (52!) = 3.63 x 10^83. According to a Google search, the upper limit for the number of atoms in the observable universe is 10^82. I find it remarkable that when you allow cards to be mixed face up and face down, the number of possibilities is so immense. I am hoping that someone can confirm my calculation.

David M.
MeetMagicMike
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You should post this in the puzzle section. The smart math guys hang out there. I have heard that when a deck is well shuffled it is likely that no other deck has ever been in that same order.
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Steven Keyl
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Here's an incredible way to visualize 52!

https://czep.net/weblog/52cards.html
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MeetMagicMike
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It is not really necessary to add face up and face down. 52 factorial is large enough to dazzle.
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Mike Powers
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If you google "the number of atoms in our galaxy" you'll find that it's about 1/3 of the way to 52! which is the number of ways to arrange the 52 cards in a deck. I used the reference Steven Keyl pointed to above in TESSSERACT to dazzle the reader with the incredible size of 52!.

SO 52! is not bigger than the number of atoms in the UNIVERSE. It's bigger than the number of atoms in our galaxy of two to four hundred billion stars. Our star, the sun, is about a million times the size of the earth. But check out Steven's reference. It's a mind blower.

Mike
dyoung
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This is usually true, unless you see a Nine of Diamonds on the bottom. Then chances are they've been in that order before Smile

//Dan
Mike Powers
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Good one Dan!
Greg Kiefer
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Dan’s quote usually goes over my head. I definitely think Juan Tamariz is smiling😀
Mike Powers
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Both Juan and Simon are smiling!

Mike
Scott Kahn
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No! More ways to arrange a deck than there are atoms on EARTH.
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Mike Powers
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Scott - do a google search on 52! and also on "number of atoms in the milky way galaxy." You'll see that there are more ways to arrange the cards in a deck (52!) than there are atoms in the entire galaxy of 200 to 400 million stars and all their planets etc.

Mike
ipe
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What would a real mindreader do?
Mike Powers
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The reference above just says that 52! is bigger than the number of atoms in our solar system. Just do a google search on "the number of atoms in the milky way galaxy." The estimate is less than 52! This is an estimate. So it's correct to say that 52! is "in the vicinity of" the number of atoms in our 200 to 400 billion star galaxy. It's not the number of atoms in our sun or the earth or the solar system. That's not even close. It's off by a factor of hundreds of billions.

Mike
Atom3339
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I would ask Allen Ackerman.
TH

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Chris K
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Is there is a qualitative difference between stating there are more ways to arrange a deck of cards than:
-grains of sand on the Earth
-atoms on Earth
-atoms in our solar system
-atoms in our galaxy
-atoms in our galactic cluster
-atoms in our universe
?

I think, qualitatively, they are the same to our spectators. I'll put it a different way: if somebody leaves your show and says "That was cool but it would have been more impressive if the odds of the effect were greater than the number of atoms in the galaxy, not just our solar system" then something is seriously wrong, and it isn't math.

There's enough literature about how humans can't conceive of astronomical numbers and I think that was born out in this discussion Smile. Heck, CARL SAGAN had to write a book, for viewers of Cosmos, explaining the difference (in actual scale, not just adding 3 zeroes) between a billion and a million. An actual interesting way to think about it is as follows:
- It takes a little over 11 and 1/2 days to count to a million (at 1 number/second)
- It takes over 31 and 1/2 years to count to a billion

All that being said, here are the actual numbers so there's no more argument. I'll even provide sources.

52!: 8.0658175e+67 from https://www.google.com/search?q=52!
The estimated number of atoms in the observable universe (10^80) from https://en.wikipedia.org/wiki/Large_numbers
Number of particles in the universe: 3.28 x 10^80 from https://www.popularmechanics.com/space/a......niverse/
Cells in the human body: ~4X10^13 from: https://pubmed.ncbi.nlm.nih.gov/23829164/

A generally fun NASA website (random entry selected): https://www.grc.nasa.gov/www/k-12/Number......rops.htm

I think the link Steven shares above illustrates giving people a reference to understand numbers versus just vomiting numbers at them (and almost nobody can conceptualize the number of atoms in the galaxy).

Here's another link that is interesting and illustrates a bit how people think about numbers: "Dealing with Big Numbers: Representation and Understanding of Magnitudes Outside of Human Experience" https://onlinelibrary.wiley.com/doi/full/10.1111/cogs.12388
Mike Powers
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Hi Chris K -

Yes. The number is incomprehensible. Nonetheless, you can tell people to google 52! and also to google "the number of atoms in our galaxy." You'll see that 10^67 power in both. Also, "the number of atoms in our galaxy of 200 to 400 billion stars" sound a lot bigger than "the number of atoms in our sun" or "the number of atoms in the earth." Also, it's just more accurate. Why not just give them the right information?

Mike
ASW
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And I believe this factoid comes from Darwin Ortiz’s original patter for an effect. Which is probably where the OP first heard it.
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ekgdoc
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Quote:
On Jun 16, 2020, Chris K wrote:
Is there is a qualitative difference between stating there are more ways to arrange a deck of cards than:
-atoms in our solar system...atoms in our galaxy...atoms in our universe?

There's enough literature about how humans can't conceive of astronomical numbers.


Humans can conceive of a deck that is dropped on the ground and then gathered up. Some cards will be face up and some will be face down. Easy concept. Only 52 cards. And yet the number of ways such a deck can be arranged is more than the number of atoms in the universe. I find this remarkable. Maybe that is just me. And the number I gave above for ways to arrange a dropped deck of cards (3.63*10^83) is correct. To my knowledge, this has never been published. The focus in the literature has been on 52 factorial, a big number for sure, but far less than the number of atoms in the universe.

David M.
Mike Powers
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Yes. Once you add in face up and face down cards, the number of arrangements is beyond the incomprehensible 52! by another level of incomprehensibility. If you have my book Tesseract, check out the little essay on 52!. It's another way to try to understand the mind blowing size of 52!.

Mike
ddyment
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Or, to be precise:

363252127644247044041398160152368436824058904969853261166529262548680704000000000000 ways!

:)
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