[HN Gopher] The Math of Card Shuffling
___________________________________________________________________
The Math of Card Shuffling
Author : Anon84
Score : 95 points
Date : 2024-07-03 16:32 UTC (4 days ago)
(HTM) web link (fredhohman.com)
(TXT) w3m dump (fredhohman.com)
| matsemann wrote:
| Once saw a talk by Roger Antonsen where he did various
| visualizations of shuffles (but using 64 cards I think for the
| even splits). Quitw cool images
| https://rantonse.org/art/2018-07-25
| veunes wrote:
| The deep connection between mathematics, art and magic
| slaymaker1907 wrote:
| I wish the 7 shuffle thing would stop getting repeated
| everywhere. It's based off a faulty model, particularly for
| sleeved cards, since if we consider the deck as two piles left
| (L) and right (R), Pr[L|R] (the probability of taking from the
| left pile given the last draw was from the right), should be
| about 0.5 but is nearly 1 in many cases.
|
| The riffle shuffle also has to be one of the only things you
| actually get worse at the more you practice it.
| waynecochran wrote:
| You get worse w more practice? Explain. I have been practicing
| the riffle shuffle w/o using the table and keeping the cards
| contained -- am I going to get worse at it?
| slaymaker1907 wrote:
| Yes, you'll get the alternating pile phenomenon more and more
| often. In reality, big "clumps" of cards from one pile or the
| other actually improve the randomization. In the extreme,
| this becomes the Faro shuffle
| https://en.m.wikipedia.org/wiki/Faro_shuffle
|
| From a probabilistic standpoint, even if it isn't a perfect
| Faro shuffle, almost-Faro shuffling clearly introduces far
| fewer bits of entropy which is less than ideal.
| cpeterso wrote:
| To try to account for that, I purposely alternate thin and
| chunky interleaving when I riffle shuffle (the customary
| seven times :).
| Orlan wrote:
| I wish! Yes, the better you get at riffle shuffling, the
| more it will approximate a faro shuffle (meaning less
| "clumps" and more singles alternating) in some parts of the
| deck, but it's the repetition that makes it random. If you
| have accidentally mastered a perfect tabled faro without
| deliberate practice, I'd like to shake your hand and read
| your book (or video) on the technique!
|
| Perhaps we can caveat that the person is not cutting
| perfectly at 26 every time and performing perfect out
| faros, then we can assume they'll be mixed enough. :-)
|
| It's been a while since I've read Persi Diaconis' paper
| (who coined the 7 shuffles thing), but I believe all your
| main concerns were addressed. Not sure about the sleeved
| deck thing, that wasn't probably a consideration.
| slaymaker1907 wrote:
| Yes, sleeves weren't much of a thing until the 2000s and
| they do seem to make a big difference in my experience.
| trombonechamp wrote:
| Here is a bit more realistic model of the riffle shuffle. These
| results show that you need between 3 to 8 riffle shuffles,
| depending on your skill. Notably, the number of needed shuffles
| actually does get lower as you get better, i.e., less clumping
| is better:
| http://blog.maxshinnpotential.com/2017/11/05/optimality-in-c...
| askvictor wrote:
| Definitely haven't been getting worse at the riffle shuffle.
| But one of my more nerdy gaming buddies doesn't allow any of
| his cards to be riffled as it degrades the cards.
| ballenf wrote:
| The other thing is that the chaotic pile on the table, well-
| mixed, is about as fast as 7 riffles mixed with cuts and
| probably (my personal experience) equally mixed.
|
| Just makes you feel like a child.
| slaymaker1907 wrote:
| The other thing I don't like about it is that many cards will
| be flipped upside down which is very annoying for games like
| MTG.
| waynecochran wrote:
| I have played Pinochle w friends who always wanted the dealer to
| shuffle the deck less since they believed that this would more
| likely result in good/interesting hands (four of kind, straight,
| pinochle...). This seems to stem from the notion that randomness
| looks different or is less interesting. If all sequence of cards
| are equally likely, the ratio of interesting hands to all hands
| seems worse for a less shuffled deck since the cards are
| interleaved throughout all the players.
| remram wrote:
| This is true if you deal one card at a time. If you deal 2 by 2
| or 3 by 3 on the other hand...
| less_less wrote:
| There's a Dutch card game called Rikken that intentionally
| incorporates bad shuffling. It's a trick-taking game, so the
| cards get clumped by suit during the hand, and you shuffle very
| lightly between hands. After shuffling, the cards are dealt
| several at a time to each player, so players tend to get _much_
| more suited hands than with a well-shuffled deck.
| nyc_pizzadev wrote:
| Check out Jason Ladanye. He's a magician who uses shuffle math to
| place cards exactly where he wants them to be in the deck. Both
| impressive and scary.
| veunes wrote:
| Jason Ladanye is a remarkable magician
| khazhoux wrote:
| Ha, came here to post about Ladayne. I've been watching all his
| 2-minute shorts and I _still_ can't wrap my brain around the
| mental /mathematical skill, or the astounding precision of his
| fingers.
|
| Another takeaway is the jaw-dropping amount of practice he's
| put into it, and his total dedication to perfection (zero
| mistakes allowed, ever). He speaks at length about this in
| interviews, and in fact part of his act is to explain that it
| is _not_ magic at all, but his full commitment to mastering the
| craft. Something we can all reflect on.
| veunes wrote:
| Do not know why but still it's a problem for me to shuffle cards
| indoorcomic wrote:
| My mom put a deck of cards in my hand before I could walk, so i
| ended up being the one to teach all my friends in school. It
| takes a while to get it down
| jwood27 wrote:
| I tried to make a model of my own shuffling. Blog post in work,
| but some interesting bits:
| https://jacobw.xyz/posts/005_shuffle/#data_exploration
| Syzygies wrote:
| Pretty funny that this cites a 2015 Numberphile video,
| considering that my paper with Persi Diaconis was published in
| 1992.
|
| (You know you're on to something when so many people misquote and
| misinterpret your result, and say it sucks!)
|
| Of the many formulations of Gilbert-Shannon-Reeds, an easy one to
| generalize is running the shuffle in reverse. One could flip a
| fair coin for each card, to decide if it flies up to the right or
| left hand. Equivalently, let heads fly up to the same hand as
| before, and let tails fly up to the other hand. As an unfair coin
| approaches always tails, the inverse shuffle becomes smoother,
| approaching a perfect shuffle, which Persi can easily do.
|
| One way to measure how smoothly one shuffles is to flip one
| hand's packet over before shuffling, then count face up / face
| down runs. For a perfect shuffle, there will be 52 singleton
| runs: up, down, up, down... For GSR shuffles there will be 26 1/2
| runs on average. Experienced human shuffles tend to exhibit 30 to
| 40 runs. While only GSR can be solved in closed form, one can run
| simulations to see what happens with smoother shuffles. As a
| somewhat surprising coincidence, one approaches randomness most
| quickly with shuffles corresponding to 30 to 40 runs.
| rahimnathwani wrote:
| Is this the paper?
|
| https://doi.org/10.1214/aoap/1177005705
|
| And this? https://www.nytimes.com/1990/01/09/science/in-
| shuffling-card...
| cal85 wrote:
| > ...an ordering that has likely never existed before. In other
| words, it's unlikely you'll ever shuffle two decks the same.
|
| This is an understatement. In fact it is virtually certain that
| no two shuffled decks of 52 cards have _ever_ been in the same
| order, across all history. That's how big 52! is. In fact, to get
| a 50% chance of _any_ two shuffles matching, you'd need to
| shuffle a billion decks of cards every second for 4x10e17 years.
___________________________________________________________________
(page generated 2024-07-07 23:02 UTC)