The Math of Card Shuffling

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.

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.

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.

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.

> …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.

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

Do not know why but still it's a problem for me to shuffle cards

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