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Deck of cards vs. Rubik's Cube


Than_Bogan
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This really couldn't be any more off-topic, but the million seconds and trillion seconds references in another thread reminded me of some numbers-intuition things that I find kinda cool, and that eventually leads to a mildly-mind-blowing fact.

 

Just for fun, I'll start this with a poll. Please don't google or try too hard to figure it out, just vote with your intuition.

 

Which number is larger: The number of possible orderings of an ordinary deck of playing cards or the number of possible states of a Rubik's Cube?

 

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Seriously?

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Answers:

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The number of states that a Rubik's Cube can reach (without taking it apart) is 8! x 3^8 x 11! x 2^12 or about 43 quintillion or 4.3 x 10^19. So that's kind of a lot :smile:.

 

The number of possible orderings of a deck of cards is "simply" 52!, which turns out to be about 8.1 x 10^67.

 

So the correct answer to the opening question is that the deck is more than 1M times more. BUT I purposely misled you a bit with the choices. In fact, the number of states of the cube is approximately zero when compared to the number of possible deck orderings. Don't believe me? Try adding the smaller of those numbers to the larger one on your calculator and notice how it doesn't change. Your calculator doesn't display 48 significant figures, so you can't even observe the contribution of such a tiny number as 43 quintillion...

 

Put another way, the deck of cards has more than a million million million million million million million million million times as many possible orderings as the Rubik's Cube has possible states.

 

But that's not the mind-blowing part, which I'll dedicate a separate post to.

 

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So that was all build-up to emphasize just how obscenely large 52! is. A consequence of this ridiculous number is the following fact:

 

If you take an ordinary deck of cards and properly shuffle it so that the ordering is completely random, it is essentially certain that that exact ordering of a deck of cards has NEVER occurred before.

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Modestly entertaining permutation question, Permut(x,x) in Excel makes pretty quick work of it. Modestly interesting meaning more fun than work ;)

 

For the record, the North does Freeze, but it hasn't yet, still a few months to ski. Which is good since my scores have been a scatter gun this year with everything from missing my opener to 5@35...

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My son is a cubing nut, so I actually knew the 43 quintillion number... I picked that as my (wrong) answer, I mean what could possibly be higher than that!?

 

As an aside but still numbers related, I just introduced him to Chuck Norris jokes - and his favorite by far is that "Chuck Norris counted to infinity, twice."

 

Which leads me to a question I've never thought of - how would Chuck Norris solve a 3x3 cube?

 

Don't @ me bro - this whole thread is off-topic! B)

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Ok @Than_Bogan I'll play, I did get the poll right since I knew a bit about this.

 

a fun way to picture it that I've read...

 

Say you can shuffle a deck once a second. Pick a spot on the equator of the earth, and take one step every billion years (all while shuffling every second). When you make it all the way around the earth, remove one drop of water from the ocean. Do the same thing until the ocean is empty. Once you have the ocean emptied, law down a piece of paper flat on the earth, then fill the ocean and start again. Continue this process till your stack of paper reaches the sun. Guess what? Your number of shuffles still hasn't gotten to the left most digit of that number.

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