#permutation

[knotty-et-al]

Visualization of the Rubik's cube

[what-about-zaladane]

Entropy, Intuition, and The Art & Science of Shuffling A Tarot Deck (Or: How Many Times Should You Shuffle It, Mathematically?)

This post is specifically about riffle shuffling -- not overhand or pile shuffles.

tl;dr: If you care about randomness and making sure all the cards have a fair chance of coming up, 9-10 times.

But how much should you really care about randomness?

Let's back up.

Mathematicians Gilbert, Shannon and Reed developed a model of randomness for riffle shuffling (i.e. the snappy type of shuffling you do with a playing card deck). They found that the first few shuffles of a deck of playing cards only rearrange the order of the cards a little…but by the 7th shuffle, the deck's order is almost indistinguishable from random.

You might ask -- but a tarot deck has 26 more cards! Does that matter?

Yes, it does.

Each added card increases the permutation space of the deck. You go from 52! (a number with 68 digits) to 78! (a number with ~115 digits). The number of possible orderings explodes -- as does the time and effort it takes to "mix" those orderings.

The Math

If you do a perfect riffle shuffle, the deck becomes increasingly mixed. The GSR Theorem uses total variation distance from uniform randomness to measure just how mixed it is. Total Variation Distance is a number between 0 and 1 that tells you how different two probability distributions are. The two distributions we are comparing are:

The probability distribution of the deck after n shuffles

A perfectly random deck, where every single card has an equal chance of being in every position.

What do different values of TVD mean?

If TVD = 1 -> the two distributions are completely different

If TVD = 0 -> the two distributions are essentially the same

Imagine you’re drawing the top card of a tarot deck after shuffling.

If the deck isn’t well shuffled yet, certain cards are more likely to be on top.

If the deck is perfectly shuffled, every card has exactly a 1 in 78 chance to show up.

Total Variation Distance measures:

"How far are we from that perfect 1-in-78-for-every-card situation?"

TVD is basically asking: how unfair is this shuffle still?

If you shuffle enough times, you'll get close to 0.

TVD Convergence Chart

Here’s what the convergence of TVD looks like for 52- and 78-card decks:

You can see that the first few shuffles don't bring us much closer to a totally "fair" deck -- but once we hit 4 or 5 shuffles, we start to converge to 0 rather quickly!

Back to the Shuffling

GSR modeled the distribution of permutations using the random riffle shuffle model, which assumes:

You cut the deck into two halves with a binomial split

You then interleave the cards in all possible ways

(Btw: This gives you a Markov chain on the symmetric group S_n, where n is the number of cards. If you just flinched at that sentence -- don't worry about this! Forget you read it! It doesn't come up again in the post.)

What about a 78-card Deck?

According to Bayer & Diaconis (1992) in “Trailing the Dovetail Shuffle to its Lair”:

The number of riffle shuffles needed to mix a deck of n cards is approximately:

So 9 to 10 riffle shuffles are needed to randomize a 78-card tarot deck.

Okay But…How Important is Randomness in Tarot, Anyways?

Honestly, it depends on you.

If you believe in pure chance, then randomness is the point. You want the cards to speak without being nudged by muscle memory or old orderings.

But if you believe in "divine" order, then even imperfect shuffles are sacred. Every shuffle is a divination. Every card is a mirror.

So shuffle 3 times. Shuffle 9. Shuffle until the deck feels right.

But just keep in mind: mathematically, 10 riffle shuffles are optimal if you want to approach full chaos. And chaos is very good at telling the truth.

What Do I Do Personally?

You'll notice that the graph of TVD converges pretty fast after about 6-7 shuffles. Personally, when I'm feeling the need to completely reset the deck, I shuffle 10 times. Any other time, I shuffle 8. I'd love to tell you that there's some fantastic reason for this, but it's mostly because I'm Chinese-American and I'd probably do everything eight times if I could. :D

[mondfamilie]

permutierter fünfzeiler

Eine frühmorgendliche Betrachtung der Möglichkeiten einer permutativen Gedichtgenerierung (aus gegebenem Anlass)

raptorenvögel jagen den schlaf liegt zerhackt unwiederbringlich

Wörter für Permutationen: Zeile 1: raptorenvögel, erinnerungen, die lichter der stadt, ufos aus rosswell, schlechte pilzragouts, maschinentexte Zeile 2: jagen, preisen, trinken, treiben, lesen, füllen Zeile 3: schlaf, herz, mond, tier, lust, sinn Zeile 4: zerhackt, gelöst, gehemmt, ergötzt, verteilt, begrünt.

Der Artikel in Zeile 2 wird Zeile 3 zwanglos angegendert. Sechs mal sechs Tabellen zu 6x6 Feldern, 36x36‎ = 1.296 Kombinationen. Sechs Hefte zu sechs Tafeln à 36 Gedichten. Die Stimmung durch die Auswahl steuern? Bin ich für alle möglichen Kombinationen verantwortlich? Disclaimer!? Eine vierdimensionale Tabelle ist schwer vorstellbar/sich vorzustellen: "Nutzen sie einen Würfel." 1-1-1-1 bis 6-6-6-6

maschinentexte füllen den sinn liegt begrünt unwiederbringlich

[knightofleo]

Amon Tobin | Nova

[frederikschubert]

96 Band IV Kreis

[drondskaath]

Permutation | Transcience | 2023

German Melodic Death/Black Metal

[louis-hypo]

The Largest Conceivable Number (Pi Day 2025)

Happy Pi Day. I couldn’t think of anything Pi-related to say that hasn’t already been said, but in fairness the number of words written about it over the centuries is about Graham’s Number. Ok, that’s an exaggeration, after all Graham’s Number is so big that the space needed to write it out is more than the observable universe. However, I was watching the new Numberphile video today and it got me…

[fjordstan]

imagine fourth wheeling a tricycle

[febos]

ARTEM: THE tool for RNA 3D motif identification, with backbone permutations!

ARTEM identifies RNA tertiary motifs based purely on their isostericity. We applied it to search for kink-turn-like motifs and discovered two new kink-turn topologies, multiple no-kink variants of the motif, and showed that a ribosomal junction in bacteria forms either a kink or a no-kink variant depending on the species (see the Figure). We also identified kink-turns in the catalytic core of group II introns, whose structures have not previously been characterized as containing kink-turns.

ARTEM opens a fundamentally new way to study RNA 3D folds and motifs and analyze their correlations and variations.

Want to learn more? - Check out our preprint on bioRxiv doi.org/10.1101/2024.05.31.596898 - ARTEM is available for you on GitHub https://github.com/david-bogdan-r/ARTEM https://github.com/febos/ARTEM-KT

[visenyaism]

asha at the kingsmoot

[vranias]

this is the only version that makes sense to me no i will not be accepting constructive criticism at this time

[inkskinned]

there's this video you've probably seen already where a woman is shaking in front of a microphone and delicately tries to ask - how can i make my husband listen to me, i've tried everything, i don't want to seem ungrateful and the other man laughs - the problem is that you married a man, we're only listening 25% of the time and we only understand 5% of that! and the audience laughs and the woman laughs and you just sat there, phone in your hand, letting the sound of it echo

and the thing is that people make think-pieces about it (isn't this one of them) and satire versions and "flipping the script" which is good and fun but at the end of the day, there's some truth in that man's response about men-not-listening. and you have tried to language that feeling for years, this sense that you can only take up 33% of a conversation before others view it as being "dominating".

it's not that they aren't listening, it's that the action they're taking is purposefully silencing. it's different. you accidentally-don't-listen a lot; just because the world is loud and you're distracted. you don't mean anything by it. and the truth is that the man who spoke is relying on that to be true of you; the way it's true of everyone. but there is a different undertone to his kind of not-listening. what he means is they don't respect you and you shouldn't expect them to. there is a difference between oh shit i forgot to take the trash out and why didn't you remind me to do it, just like there is a difference between i didn't realize you wanted to go out this weekend and why do you expect me to plan things why can't you just tell me where we're going.

and the thing is that it isn't just him, and it's actually not just because of your gender - your skin, your class status, your weight, their ableism - it happens often. so often it feels like a tightness around your throat and a weight in your stomach. you're not even "really" allowed to be upset about it, because to them it's a joke. and they laugh. and you know exactly the amount of work that goes into every conversation. how you have to work to condense down your thoughts into intelligent, crisp soundbites; worried someone will try to swoop in and cut you off. and there's this sense from everyone else - oh stop being so sensitive, are you really upset just because they weren't listening and you don't know how to say the way that feels when it happens constantly.

there's that video of the science summit where a woman in the audience finally says let her speak please! and the whole crowd bursts into applause and the man leading the summit holds up his hands and bows his head and says oops, sorry! like what he did was awkward and embarrassing, a little social gaffe that happens easily. later in your meetings, you're asked to take notes, and you don't say anything, you just hear let her speak please! ringing in your head and know that you'll never be brave enough for that kind of thing. and besides. think of all the people who agree this was a one-off, he just got excited and all of the people who say one man is not indicative of all of society

at the dinner table you're talking about someone you don't like and how he's not good to his girlfriend and how she always has to remind him to put the effort in and before him, she was glowing with curiosity and passion but now she just seems... tired, unhappy. that he likes the way she burns out; she stays home and takes care of him and their 2 kids. and your father sniffs and says that men take a while to learn those kinds of things. and you just stare at him and think about your childhood and are like - no wonder i turned out like this

and you want to say - there's no fucking secret school or mystic form of communication. i was not sent to Rearing a Child University. i did not graduate from Getting Chores Done College. i ask questions and i listen and i pay attention, because that's basic fucking human decency. it stems from respect, and how i respect others and their agency. i clean the house because someone should clean. not because it comes "naturally".

hell, you had to google "how to boil an egg" the other day, just because you usually make them scrambled. you can never remember which of the 2 bathroom cleaners make chlorine gas, only that two of them definitely do. you've accidentally bleached your clothes. it took you like 3 years of self-teaching before you figured out how to actually cook things correctly - for that whole time, you burnt or undercooked everything. but you did teach yourself; just like you taught yourself how to listen with empathy. just like how you taught yourself to think before you speak. to be kind first, to be better at communicating. it seemed like a good thing, an adult thing.

the joke the man in the video makes is that women say i'm fine! when they are not fine. and you think about the 150 conversations that happened around that; about how she probably has had so many arguments with her husband. how she said i'm upset you don't take me anywhere and he got mad at her because of course i do, you made me go to that stupid restaurant like last week and she probably said that's not what i'm saying and he said now i'm supposed to be psychic or something and she said no of course not and he said how am i supposed to know what to do when you don't even like everything and she said i do like things and he said well how am i supposed to win? and her pastor probably told her to be more grateful because they do things at all, even if she has to plan them and her mom probably told her that's just how men are honey and she probably cried over her journal, trying to figure out why the fuck she "has everything" and is still so bitterly, horribly unhappy

and how, in your life, for so many reasons, you looked down the barrel of another argument; of explaining yourself and being vulnerable and begging for help again. how many times you just said i'm fine because it was better than doing that again; it was better than wringing yourself out when it's literally easier to just pretend. because he wasn't going to listen. your father wasn't going to be better and your boyfriend wasn't going to be better and your boss wasn't going to be more respectful.

and you sit in front of a video of a woman shaking, looking horrible and guilt-wrought that she's even asking this question. and you know; deep in your heart - that's you. in a different life, you are her. you've stood in her spot. and you had to listen while someone else cackled - why would we bother to notice when you talk?

[somekindofhannibalcannibal]

hannibal: the fly is a creature named, in english, for the most notable thing, to us, that it does. what does it mean to be known only for what you do that is visible to others? what is your defining trait or ability, that you may not even consider a particularly special skill, but simply something you are naturally equipped with, as essential to your function as breathing? and what would you be without this ability, still living but having lost your defining trait? tell me, will, what would one call a fly without wings?

will: hannibal, don't...

hannibal: a walk?

(all of the dogs barrel into the room at once)

[fuckyeahisawthat]

It must have been Jayce who found Viktor passed out at the lab, right? Like let's be real that stubborn mf is not dragging himself to a hospital if he can help it at all. So the most likely people to find him and make him get medical care would be either Jayce or Sky, and it's established that Sky's left for the night.

Whenever Jayce leaves Mel's bed, it doesn't wake Mel up, so I don't think it's the case that anyone came to her door looking for him if anyone would have even known to look for him there. Nah I think Jayce woke up and quietly left in the middle of the night and in the moment Mel thinks it's because he's a fuckboi who gets his dick wet and then bounces but I think actual possible scenarios include: (1) he got a brainwave in the middle of the night and both he and Viktor are used to popping by the lab at stupid o'clock whenever inspiration strikes; (2) woke up with a weird bad feeling of suddenly needing to check on Viktor, in that uncanny way you sometimes get a call from a relative and just Know it's bad news before you pick up the phone, and lately Viktor's been sleeping at the lab more than at home so that's where he goes first; or (3) especially if you think this is Jayce's first or close-to-first sexual experience, he's having some big confusing feelings and needs to take a walk about it, and he just ends up at the lab on autopilot and since he's there he's just gonna pop in and check on this one thing...

Whatever the exact sequence of events that is one hell of an emotional rollercoaster for one night.

[frederikschubert]

96 Band 3 Dreieck