How many times do you ACTUALLY need to shuffle?

[CLARIFICATIONS AND CORRECTIONS]

Clarification at 2:50 – "probability mass" just means the probability of a particular outcome.

Clarification at 6:41 – This is also because the events which bear witness to the distance being high after only a few shuffles are complicated, like the one described in the skit, and so they can’t really be seen in the gameplay of most games (I don’t have a good source for this unfortunately). Also, the point of this message is not to say that you shouldn’t shuffle more if you want to, but that you should not be too upset if someone else only shuffles a few times in a casual context.

[CREDITS]

Vilas Winstein: black nails, script, animation, voiceover, editing.
Anisah Awad: blue and green nails.

Special thanks to Alistair Sinclair for teaching me about this subject.

Card Decks used in the video:

[SOURCES]

Bayer, Dave, and Persi Diaconis. "Trailing the dovetail shuffle to its lair." The Annals of Applied Probability (1992): 294-313.

Martin, Aigner, and M. Ziegler Günter. "Chapter 31: Shuffling Cards." Proofs from the book–Sixth Edition (2018): 219-228.

[SECTIONS]

0:00 Introduction
1:31 What Is Shuffling?
3:19 Total Variation Distance – Introduction
4:22 Total Variation Distance – Definition 1
5:00 Total Variation Distance – Definition 2
6:13 Skit (Don't Think Too Hard)
7:07 Total Variation Distance – Definition 3
8:20 Coin Flip Example – Introduction
8:50 Coin Flip Example – Definition 1
9:03 Coin Flip Example – Definition 2
9:31 Coin Flip Example – Definition 3 (Independent Coupling)
10:25 Coin Flip Example – Definition 3 (Optimal Coupling)
11:48 Total Variation Distance – Review
12:27 Plan for the Rest of the Video
13:23 Reversing the Shuffle
14:45 Top-To-Random Coupling – Introduction
15:09 Top-To-Random Coupling – Description
16:05 Top-To-Random Coupling – Analysis
18:23 Skit (Don't Shuffle This Way)
18:41 Riffle Shuffle – Model Description
20:03 Riffle Shuffle – Coupling Description
21:42 Riffle Shuffle – Coupling Analysis
23:37 Paper by Bayer & Diaconis
24:14 Conclusion

This video is an entry in the πth Summer of Math Exposition, #SoMEpi #SoMEπ

[lonestarr]

Me, about 25 minutes ago:
"Curious. How did this video end up in my feed? I don't remember this channel... Oh, it's the one with the awesome percolation video from a year ago that made me subscribe instantly! So I guess I'm learning something about shuffling cards now."
And learn something I did. That were 25 minutes well spend.

[objectobject]

I remember when Persi Diaconis mentioned the 7 shuffles of the Riffle Shuffle in Numberphile. In one of the videos he says that someone asked him how many shuffles were needed for a 250 card deck, which is the usual size of a Battle of Wits Magic the Gathering deck.

[lexinwonderland]

This is an amazing video, obviously, but I wanted to mention that I love both of y'all's nail paint! It's actually quite helpful for keeping track watching your hands while you shuffle, and the second hands' paint colors are my <3 i'm so excited to see you post again, keep up the great work!!

[codescent]

Speaking non-exaggeratedly, this is my favourite maths channel on youtube. You're really good at this and should keep going.

[cartatowegs]

I appreciate you going in depth on the probability distance section. Had to watch ot a couple times but it was very helpful.

[diablmaster]

That table at the starts should have been caveated with "under some reasonable assumptions about shuffling"

[drdca]

Very nice! I had not expected a proof of such a thing to be as understandable as this!

[calvinkonchar]

This video was a fantastic find, seriously great job on everything. subscribed

[Krunschy]

Boy, do you know how to pick thrilling subjects. I was always curious about this, so much so that I tried understanding a paper on this subject years back, but eventually gave up after an hour.
Didn't even finish the video yet, but I've already gotten so much closer to getting it than I ever have.
Edit: Turned out it was the exact paper you sourced.

[SugarGlider]

8 - 52 perfect shuffles and I'll be back to square one.

[fibbooo]

Awesome video! I learned a lot, thank you!

[daan]

Great video! How do you decide which topics to cover? I think they are all super interesting.

[ArgumentumAdHominem]

You mentioned that in real life, the number of shuffles only really matters if players know the initial state. This, actually, more often than not, is the case. And this is a completely normal scenario, not targeted cheating or card counting. In many games, after the game, the cards follow some kind of order. In solitaire, the order is perfect or close to perfect. In trick-based games like bridge or whist, the cards of the same suit cluster together, because of how people order their hands, and because you typically answer suit with the same suit. In resource-based games like magic the gathering, at the end the resources (e.g. lands) are fully separate from non-resources (e.g. spells). While shuffling as means of preventing exploitation by opponent is a valid reason, I would argue that the more common reason to shuffle is to avoid pathological games. For example, in magic the gathering, it is pathological to have no resource cards in your starting hand, and in a perfectly random game should not occur more than in a few percent of games, which is considered tolerable. Instead, if a deck is poorly shuffled inbetween the games, the number of such pathological hands dramatically increases, which completely invalidates the game design.

[librarianmage]

16:22 the decks also certainly align once n-1 cards are chosen at least once, right? the last unpicked card in both decks are at the bottom of each deck. (I imagine this was not mentioned because it does not shift the upper bound too far)

[jakethewolfie]

During the Coin Flip Example, why was it okay to couple the coins in this way, especially as this method of coupling eliminated the possibility that coin p is tails while coin q is heads?