I understand the math behind it, I just can’t wrap my head around the fact that something so common and limited like a deck of cards can have more ways to be arranged than something so massive like the earth with all its oceans and mountains has atoms.
In my mind it would make more sense that even a little pond has more atoms than there are deck arrangements.
Could it be due to the fact that atoms have a lot of empty space in them?
In: 318
The trick and reason this seems counterintuitive is, because we tend to underestimate how quickly the number of ordering rises with each new card.
Two cards have 2 ways of being shuffled. Or 2! of arrangements.
1. (‘A’, ‘B’),
2. (‘B’, ‘A’)
Three cards have 6 ways of being shuffled, 3! arrangements
1. (‘A’, ‘B’, ‘C’), (‘A’, ‘C’, ‘B’)
2. (‘B’, ‘A’, ‘C’), (‘B’, ‘C’, ‘A’),
3. (‘C’, ‘A’, ‘B’), (‘C’, ‘B’, ‘A’)
Four cards have 24 ways of being shuffled 4! arrangements
1. (‘A’, ‘B’, ‘C’, ‘D’), (‘A’, ‘B’, ‘D’, ‘C’), (‘A’, ‘C’, ‘B’, ‘D’), (‘A’, ‘C’, ‘D’, ‘B’)
2. (‘A’, ‘D’, ‘B’, ‘C’), (‘A’, ‘D’, ‘C’, ‘B’), (‘B’, ‘A’, ‘C’, ‘D’), (‘B’, ‘A’, ‘D’, ‘C’)
3. (‘B’, ‘C’, ‘A’, ‘D’), (‘B’, ‘C’, ‘D’, ‘A’), (‘B’, ‘D’, ‘A’, ‘C’), (‘B’, ‘D’, ‘C’, ‘A’)
4. (‘C’, ‘A’, ‘B’, ‘D’), (‘C’, ‘A’, ‘D’, ‘B’), (‘C’, ‘B’, ‘A’, ‘D’), (‘C’, ‘B’, ‘D’, ‘A’)
5. (‘C’, ‘D’, ‘A’, ‘B’), (‘C’, ‘D’, ‘B’, ‘A’), (‘D’, ‘A’, ‘B’, ‘C’), (‘D’, ‘A’, ‘C’, ‘B’)
6. (‘D’, ‘B’, ‘A’, ‘C’), (‘D’, ‘B’, ‘C’, ‘A’), (‘D’, ‘C’, ‘A’, ‘B’), (‘D’, ‘C’, ‘B’, ‘A’)
I will stop making the graphical representations, but at this point I think you can see, that adding a increases the number of combinations by a factor equal to the order of the card (second card doubles the preceding number, third car triples preceding number, fourth quadruples ….)
Fifth card gives us more then hundred combinations
Sextuple that and you get almost a thousand combinations
septuple that and you are half way to ten thousands.
multiplication by 8 gets you almost half way to hundred thousand
9 gets you third of a million
10 gets you 3 millions every successive number therefore increases the preceding number by more than an order of magnitude (add a zero to a number)
18th card, will get you a number of approximately 6 moons worth of atoms
after 20th card (we are now at 2.43×10^18, number of transistors we produced per year in 2008), each new card will increase the order of magnitude and double it. and yet we are not even half way done.
42nd card will get you about number that represent number of atoms on 10 Earths
46th card number of atoms in 5 Suns
47th card would allow you (according to Archimedes) fill a quarter of two light years wide cosmos with grains of sand.
And finally 52 will get you about number that represent number of atoms in 3 milky ways
According to random page I found, a number of atoms in observable universe is somewhere between 10^78 – 10^82 meaning you would need between 58 and 61 cards to have a comparable number of shuffelings
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