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After thoroughly shuffling, the exact order of the deck is one of 52! (52 factorial, or 52 * 51 * 50 * ... * 2 * 1) possible combinations. That is such a large number that it's possible, even likely, that the exact ordering of your deck has never existed before and will never exist again.
In the future you can denote your factorial with 51! through single inline backticks for clairity!
True, but due to the Birthday Paradox, the chance of any two people shuffling a deck the same way at some point is a lot higher than you might think.
I never learned how to properly shuffle cards that way. My hands just fail at the basic mechanics. Perhaps coincidentally, I would be mortified if something like that were done to the vast majority of the games in my collection. That ain't no $2 Bicycle deck, mi amigo.
What if I just tossed the cards into the air and then picked them up? Would one toss be sufficient? 🤔
No. Surprisingly much of the order is preserved.
Only if you pick them up with your eyes closed
It’s also not sufficient to randomize a deck of cards using a 32-bit seed as was once common in software.
Indeed, even with a 64 bit seed, it is not sufficient.
Some quick math tells me you need 256 bits. Big numbers are wild
Numberphile has a bunch of videos on it, and yes 7 the accepted number because more shuffling doesn't increase the randomness in an effective manner.
I always thought riffle shuffles were super ineffective. Most of the cards remain in each others vicinity. What is a better way?
You're essentially splitting the deck and recombining the two halves imperfectly multiple times in a row. Like if a riffle was perfect, you would get the cards from both halves equally distributed, but nobody can do it perfectly, so they actually end up properly randomized. After 7 imperfect riffles, the entire deck is unpredictable.
After 4 perfect ABAB riffle shuffles, you would end up with the same order as you started with. If your shuffles are imperfect, your deck becomes more random every time.
I read ABAB as “Assigned Bitch at Birth” and immediately was like “hey that’s me”
I don't know much about card tricks, just that many appear to use non-random cuts and those ABAB shuffles to get cards where they need to go. This one 'The Hotel' might even be easy enough for me to learn:
It's the best way, period. Even as inefficient as it may seem, inexpertly done, seven shuffles and you're good.
With or without cutting the deck in-between each shuffle?
That’s called a “box.” I believe the box/riffle action counts as one shuffle.
And how many boxes make a riffle? I always do three or four between each riffle, and wonder how much it changes.
This is oddly satisfying to know. Thank you.