52 Factorial
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52 Factorial
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Re: 52 Factorial
#2So if I want a collection of card decks in every possible combination I should start collecting now, got it.
Re: 52 Factorial
#3So if I want a collection of card decks in every possible combination I should start collecting now, got it.
With this, you could invent a very unique magic trick where your spectator well-shuffles a deck of cards, then you can secretly swap your deck with one of the decks from your collection to reveal that yours was shuffled in the exact same way.
Re: 52 Factorial
#4Is it safe to say that it's almost certain that no two people have ever shuffled a deck of cards in the exact same order?
Re: 52 Factorial
#5Is it safe to say that it's almost certain that no two people have ever shuffled a deck of cards in the exact same order?
No, due to suboptimal shuffling strategies. If you shuffle exactly the wrong way, you can even end up with the cards in their original order.
Re: 52 Factorial
#6In terms of actually playing games with cards, the effective number of permutations can be much smaller (though still large enough to be going on with). In many card games, the suits are distinct but functionally identical; you could swap spades rank-for-rank with hearts and get a functionally equivalent deck. In Klondike solitaire, the tableau is concerned with red cards and black cards, not all four suits.
I imagine somebody has done research on this: for a given card game, how many functionally distinct shuffles there are.
Re: 52 Factorial
#7In terms of actually playing games with cards, the effective number of permutations can be much smaller (though still large enough to be going on with). In many card games, the suits are distinct but functionally identical; you could swap spades rank-for-rank with hearts and get a functionally equivalent deck. In Klondike solitaire, the tableau is concerned with red cards and black cards, not all four suits. I imagin…
I remember looking at some early Draw Poker machines in Las Vegas back in the late 80's/early 90's, and thinking about pseudo-random number generation (as it existed at that time). If it was using a standard linear-congruential RNG with a 16 bit value, there would be only 65536 possible seeds, and hence only that many distinct sequences of random numbers, and hence only that many possible shuffles. Even a 64-bit RNG would have only 1.844674407370955e+19 possible shuffles, far lower than 52!.
Re: 52 Factorial
#8So if I want a collection of card decks in every possible combination I should start collecting now, got it.
mass of Earth is 6×10^27 kg
mass of the Milky Way is 6x10^42 kg
mass of the visible Universe is 1.5×10^53 kg
Quality deck of cards weighs 0.3 kg. All deck combinations would weigh 2.4×10^67 kg.
= 10^14 times the mass of the visible Universe.
You need many multiverses to store all those cards.
Re: 52 Factorial
#9In terms of actually playing games with cards, the effective number of permutations can be much smaller (though still large enough to be going on with). In many card games, the suits are distinct but functionally identical; you could swap spades rank-for-rank with hearts and get a functionally equivalent deck. In Klondike solitaire, the tableau is concerned with red cards and black cards, not all four suits. I imagin…
You're right. There are also game rules that can act as multipliers (e.g. who has the blind affects the game state in poker - so that means that there are actually multiple "games" for each given shuffle).
Something you may find interesting is formal methods - what we're talking about here are "equivalence classes". In formal methods you want to analyze the totality of a system and understand every outcome based on every possible state within the rules of the system. Clearly this is a combinatoric nightmare in anything interesting (even a simple system like a card game and 52 cards, as we can see). Equivalence classes are a way of knowing distinct, but not actually different, states, and grouping them together so that they can all be considered just once.
Re: 52 Factorial
#10This thought experiment gets even wilder if you like to live dangerously and leave two jokers and the rules card in the deck. 55 factorial is even larger