#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

[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

[astral-herald]

viktor season 2 lookbook

do we think he served in every universe, in every timeline?