Comment by Syzygies

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.