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According to mathematicians, the most effective card shuffle involves using the “Riffle Shuffle” seven times. But can math really help us shuffle cards more effectively? I put it to the test…

Can Math Really Help Us Shuffle Cards More Effectively?

But first… what’s this about math and shuffling cards?

I first heard about the seven shuffle theorem on the Numberphile YouTube channel. In this video, Persi Diaconis from Stanford University discusses the best (and worst) ways to shuffle cards.

Drawing on his knowledge of complex mathematical equations and algorithms, he confirmed the number of times you need to shuffle a deck of cards to get a random result differs based on what shuffle you do:

His reasoning was based on the number of configurations of a full deck of cards being a number so big, it had over 60 digits in it. Apparently, that’s more than the number of particles in the universe.

Okay, mind officially blown. But what does all this mean in real life? Let’s test the theory to find out.

Testing The Seven Shuffle Theorem

To conduct my highly UN-scientific test, I started with a deck of cards that was ordered by number and suit. i.e. Ace of Spades, 2 of Spades, 3 of Spades, etc.

The Riffle Shuffle

First, I performed the “Riffle Shuffle” seven times as per the seven shuffle theorem and recorded my results:

Testing the seven shuffle theorem - results 1

As you can see, my results were okay. Now, I’m no card shark but I also think I’m okay at the “Riffle Shuffle” as I’ve been doing this shuffle several times a month for many years. I’m a big fan of card games!

But I had issues with both the Clubs and Spades. The order of the Clubs hardly changed at all, while several Spades stayed together as well. Not great.

Now, it’s highly unlikely that you’ll be starting with a deck of cards that are all in order (unless you’re using a brand-new deck, of course). So, I also performed the “Riffle Shuffle” seven times on the newly ordered deck. Here’s what I got:

Testing the seven shuffle theorem - results 2

This was much better! While a few cards here and there stayed together, it’s not nearly as bad as the first time I shuffled the cards. I’d argue this result is much more random.

Most people say the “Riffle Shuffle” is the best way to shuffle cards. But the “Overhand Shuffle” is one of the most common ways to shuffle cards. So I wanted to compare the two.

The Overhand Shuffle

Once again, I ordered the cards by number and suit and performed the “Overhand Shuffle” seven times. Here’s what happened:

Testing the seven shuffle theorem - results 3

Looking at these results, I definitely had more issues than with the “Riffle Shuffle”. This time, most of the Diamonds stayed together, as well as some Spades, Clubs and Hearts. This hardly looks like a randomised deck, does it?

I then performed another round of “Overhand Shuffles” (again seven times) using these newly ordered cards. Here’s how they came out this time:

Testing the seven shuffle theorem - results 4

Much better. Although some of the Diamonds stayed together, as well as a few other suits here and there, this looks much more random than before.

But according to the Numberphile video I mentioned earlier, I should’ve had to use the “Overhand Shuffle” more than 10,000 times to get a truly random result. And yet, I achieved fairly random results after just seven shuffles. Not too shabby then.

Note: I didn’t test “Smooshing” as this is a less common way to shuffle cards. Also, this shuffle requires a smooth surface to be effective and then you struggle for a bit picking up the cards afterwards. The other shuffle techniques mentioned above are usually far quicker than this one.

So, can math really help us shuffle cards more effectively?

In conclusion, I’m a blogger with far too much time on my hands to even think this experiment was worth doing. But in all seriousness, I found this test interesting.

The video mentioned I’d have to do an “Overhand Shuffle” over 10,000 times versus just seven “Riffle Shuffles”, so I genuinely thought there would be more of a difference between the two results.

Yes, the “Riffle Shuffle” was better, but it wasn’t perfect. Okay, okay, that could be on me being a novice. But still. We’re not all going to be card sharks playing card games with our friends and family, are we?

Most argue math doesn’t lie and I would agree with that. But at the end of the day, probability is still a thing.

We can use math to find the best probability of achieving the best standard deviation of a deck of cards. And we can use math to work out the probability of things.

But we can’t use math to determine an outcome.

There’s always going to be an element of unpredictability and change. Especially when it comes to who’s actually shuffling the cards and whether they’re an expert or not.

Maybe using the “Riffle Shuffle” seven times works if you’re an expert card shuffler. But beginners probably need more shuffles than that to get the best outcome. I reckon I needed to shuffle my cards an extra one or two times for it to be most effective.

One comment underneath the YouTube video particularly stuck out to me, which was from a user called @Mkaali. They said: “The problem with riffle shuffling is that the order on the bigger scale stays relatively the same. The bottom cards stay near the bottom and the top cards stay near the top. They can traverse the deck only very slowly, while the cards towards the middle get more randomness. 7 passes might be enough, but what I like to do is to throw a few of those overhand shuffles in there or even just cut the deck at some points to alter the macro scale order more.”

With 65 likes at the time of writing, it’s not just me who thinks this is a good way to fix the issues I encountered in my tests. I also usually do a “Riffle Shuffle” several times plus a few “Overhand Shuffles”. I imagine this is probably quite common (outside of casinos, of course).

Maybe a combination of the two shuffles is the best way to do it. I’d argue the “Riffle Shuffle” is probably better than the “Overhand Shuffle”. But only just.

If you’re struggling with the “Riffle Shuffle”, it doesn’t really matter. Just do a combination of the two or the “Overhand Shuffle” seven times and you’ll probably get a fairly decent shuffle out of your cards. Or just use a card shuffling machine.

Over to you now – do you fancy putting the seven shuffle theorem to the test? Let me know how you get on…

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