What’s the best way to shuffle?

There was a bit of chatter recently on Facebook about shuffling and the right and wrong ways to shuffle decks.  There seemed to be a bit of disagreement about what you should do, so I thought I’d do a bit of maths to answer the question: how should I shuffle my deck?

I’m going to assume that you want your deck to be random.  Another way to think about this is: how much should I shuffle my opponent’s deck to make sure their deck is random.

There’s three standard options in card games for shuffling.  The overhand, the pile and the riffle. I won’t explain how these work here, so if you’re not sure, check out the Wikipedia page on shuffling.

The Overhand

The overhand shuffle is much more common and one almost everyone uses.  But how good is it at randomising a deck? Some very clever mathematicians have been looking at shuffling, and have an equation for working out how many times you have to do an overhand shuffle to truly randomise a deck.  Plugging the numbers in for a deck of 30 cards, I calculate you need to perform an overhand shuffle 1000 to 2000 times to completely randomise the deck.

Noone’s going to sit around while you shuffle their deck 1000 times, so let’s say we just want the deck to be shuffled ‘well enough’.  But what does that mean?

Let’s say your deck contains 15 pairs of cards.  Most players don’t want their cards to come up in pairs during a game, but to deliberately split the pairs would be cheating.  Therefore, I’m going to make my goal to split up the pairs as close to what you’d get with a random shuffle as possible. I think this is a good proxy for randomness.

If you completely randomise the deck, 90% of the time 0, 1 or 2 pairs will be next to each other, with an average of almost exactly 1 pair together.  So the goal is to get close to this range when starting off with a deck that’s entirely in order (which it normally is at the start of a tournament after verifying your deck).  

I wrote a little simulation for overhand shuffling.  I assumed that each overhand shuffle consists of 5 packets of cards being transferred from one hand to the other.  Below is a graph showing the range of number of pairs sitting next to each other in your deck after a certain number of shuffles (grey) with the average number as a green line. I’ve added a red line to show the target of 1 pair together:

Overhand shuffling

This means that if you overhand shuffle 10 times with a sorted deck, you’ll have somewhere between 2 and 6 pairs next to each other (most likely 4), which isn’t great.  It looks like you’ll need to shuffle about 20 times to make the deck close to good enough (though still averaging closer to 2 pairs than 1), and 32 times to meet my target of between 0 and 2 pairs together with an average of 1.  That’s a lot of shuffling.

The Pile Shuffle

There’s two ways to pile shuffle.  Put your cards into piles in order, or try to put your cards into random piles.  It’s pretty obvious that no matter how bad you are at trying to randomly put cards into piles rather than in order, it’s going to be more random than if you put them in order.

I like pile shuffling as it also allows me to count the cards in my deck.  However, since consecutive cards are always put in separate piles, does it split pairs so much that I’m cheating?

If each pile shuffle involves dealing your cards into 6 piles of 5 cards (in a different order each time) here’s what you get:

Pile shuffling

The plot is the reverse of the overhand, as after one pile shuffle it’s impossible for there to be any pairs.  After this, pairs quickly creep back in over time through chance. It looks like you only need to pile shuffle twice to hit the target of between 0 and 2 pairs together with an average of 1 pair.  This is really fast. In fact, random pile shuffling is a slightly better method of randomisation than the riffle.

While 2 pile shuffles alone won’t be enough to make the deck truly random, it will be as close to random as can be expected.

While this might look better from a stats point of view, doing multiple pile shuffles is time consuming.  Also, it’s tricky to make your piles truly random, and sticking to just pile shuffling might make it look like you were cheating (as it’s very easy to keep track of a couple of key cards when pile shuffling).

It looks like we might need a pragmatic compromise.

Pile then Overhand

If you were to do a single pile shuffle followed by a few overhand shuffles, this is the result:

One pile shuffle then overhand shuffling

It looks like if I’m only going to do a single pile shuffle, I’m going to need to overhand shuffle my deck at least 5 times afterwards to get the deck closer to 1 pair together than 0 on average, and around 10 times afterwards to make the deck random enough to avoid inadvertently cheating.  This will be as effective as 30 overhand shuffles, and by starting with a pile shuffle I get to confirm that my deck contains 30 cards.

The Riffle

This is what you’ll see people do at most poker tables as it is the best way to quickly randomise the cards, but is almost never used in CCGs as you’ll bend and possibly damage cards.  Please don’t riffle your opponent’s deck at a tournament as they probably won’t be happy.

However, there is a version of the riffle you can do with sleeved cards.  If you split the deck in two and push the two halves together, this is effectively akin to the riffle shuffle.  Again, I wouldn’t recommend doing this with your opponent’s deck, as they might worry about cards getting bent. However, if you were to use this yourself, how good is it?

Riffle shuffling

It looks like it’s very good.  After 3 riffles, you will have an acceptably randomised deck, and after 5 it’s pretty much completely random.  This looks like a good bet for quick and highly effective shuffling, though it is a bit more fiddly than the overhand shuffle.

How I’ll be shuffling in tournaments

I think on the basis of these results, I’m going to be pile shuffling followed by at least 10 overhand shuffles with my deck.  It lets me check I’ve got 30 cards, it randomises my cards nicely, and it’ll probably take less time than 30 overhand shuffles.

When my opponent presents their deck to me to shuffle, if they look like they’ve shuffled properly I’ll not pile shuffle (as it takes a while and may be frowned upon) but will overhand shuffle about 10 times to mix it up sufficiently.

However, if I had any suspicion my opponent was cheating (possibly keeping cards aside or deliberately tracking cards) or thought they had just shuffled poorly, I would either: pile shuffle (to check they have 30 cards) then overhand shuffle 10 times; or overhand shuffle at least 20 times.

On casual nights, I’ll just cut the deck.

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