4 min read

How to shuffle cards well when you never shuffle cards

I play board and card games rarely—maybe four days a year in a very heavy year. This is about the most I've ever played board games. I played a little more in college, but not by a huge margin, and before that I played even less. I play whenever friends are doing, say, a board game night. I don't seek out board or card games.

I sometimes think it would be nice if I could shuffle cards for games that use them, Shuffling is a fiddly bit that often needs doing, and the easy way (overhand shuffling) seems to have terrible randomization properties (video).

That video claims it takes 10,000 complete overhand shuffles to fully randomize a deck of 52 cards. That's absurd and obviously not practical. Riffle shuffles do better (same video), taking only ~8 shuffles to reach proper randomization, but bend the cards, which isn't ideal for game-specific cards—they're more expensive to replace if damaged. Also, riffle shuffles are tricky to do properly, the more so given I would practice at most four times a year. I can do them, but there's a risk of damaging the cards and the cards often fall in clumps because I haven't practiced much.

Can we do better?

Three things matter here:

  1. Skill. Whatever method I learn, it should take as little skill as possible, given I won't have many natural chances to practice it.
  2. Time. It has to randomize in little enough time that I can finish shuffling within a minute or two. Faster is better.
  3. Wear. It should cause minimal wear on the cards.

In investigating this, I looked at two earlier write-ups about shuffling: Max Shinn's 2017 post about optimal card shuffling and Lucas M's 2015 Board Game Geeks forum post about how you should be shuffling. These posts analyze shuffling using computer simulations. I also looked at a second video on table riffle shuffling without bending the cards, which explains a technique that may be useful.

Unlike the video, Max's post focuses on finding the best shuffling technique for games between friends—so we don't need quite as strong a randomization guarantee. the compares riffle shuffling, overhand/Hindu shuffling (effectively identical in their mixing properties), and pile shuffling. For terrible riffle shuffling, Max found that it takes 10 shuffles to properly randomize—only slightly more than with good technique. Meanwhile, it takes around 20 overhand/Hindu shuffles to properly randomize the deck assuming you make 5 cuts. This suggests that overhand shuffles may be more practical for friendly gaming than one might guess based on the "10,000 shuffles" number. Max found pile shuffling inefficient.

Lucas M's post focuses on a similar setting. It's impossible to read the exact results because the images have been deleted from TinyPic, but the conclusions suggest results broadly consistent with Max's but with a slightly more optimistic interpretation of the pile shuffling results. Besides the methods they both tried, Lucas M also describes an experiment with top-bottom shuffling and finds it works quite well: "achieves almost the same randomness [as] riffle shuffle with 7 iterations." Lucas M also experimented with combinations of techniques and found, contrary to Max's post, that a mix of overhand and riffle shuffling "was not really fast but satisfactory."

The second video describes a table riffle shuffle that doesn't bend the cards (what I'll call a bevel shuffle). Instead you bevel the card stacks over your thumb and release them with minimal bending. This is a nice alternative to the riffle shuffle if you can perform it properly. Unfortunately, it is tricky enough to do well that it's not the obvious winner for my situation.

To check time to shuffle, I timed myself performing various shuffles an appropriate number of time. I performed 11 riffle shuffles, 25 overhand shuffles, 25 Hindu shuffles, and 10 bevel shuffles—in other words, always at least the minimum for full randomization. Each round of shuffling I fumbled a few times. It took me 2 minutes and 3 seconds to perform 11 riffle shuffles (11 seconds/shuffle), 2 minutes and 42 seconds for the 25 overhand shuffles (6.5 seconds/shuffle), and 2 minutes 12 seconds for the 25 Hindu shuffles (5.3 seconds/shuffle). It took me 2 minutes 28 seconds to do 10 bevel shuffles (14.8 seconds/shuffle), which often degenerated into short-side weave shuffles.

So where does that leave us? Let's make a table of the techniques:

Technique Skill needed Time to randomize Wear on cards
Riffle High Low (123 seconds) High
Bevel High High (148 seconds) Low
Overhand Low High (162 seconds) Low
Hindu Low Medium (132 seconds) Low

Overall, the Hindu shuffle looks like the winner for me personally. I suspect top-bottom may be even better, but without good simulation results I can't be sure. I doubt it is quite as good as riffle- or bevel-shuffling. But I don't need "the best technique"—I need the one I can actually do given how much work I'm willing to put in. For now, that seems to be the Hindu shuffle.

(I want to come back for the bevel shuffle even though it's not worth it. I'll learn it properly someday.)

[An idea for future work: Re-implement and replicate Max's simulation results in Python or R, then extend by adding a top-bottom shuffle with 5 cuts. Time myself doing the top-bottom shuffle. Add such information to the table above. The top-bottom shuffle is described in the Board Game Geeks forum post, but the mathematical modeling seems to assume we draw at most 2 cards at a time, which sounds way too time-consuming to actually do. It doesn't seem to match top-bottom shuffling as I've attempted it.]