The Shuffler Really is Broken: Finding the Best Method for Shuffling Cubes

I’ve been playing Cube for quite a few years, and among new Cube owners, I hear two questions over and over. The first is “how many basics do I need?”, and the second is “how do I even begin shuffling this thing?”

Shuffling a single deck of cards by riffle or mash shuffling is easy, and mathematicians have proven that seven riffles will fully randomize a deck of poker cards. But to shuffle a cube of hundreds of cards, we have to shuffle smaller, deck-sized portions and then mix them together. If the cube isn’t randomized enough at the end of the process, then you run the risk of unshuffled chunks showing up in your cube packs, causing your innocent drafters unneeded psychic pain.

So which shuffling method offers the best mix of speed and randomness? If you want to skip ahead to our endorsed method, we have a separate primer at the ready. Otherwise, buckle up for stochasticity and simulation!

We Cube owners almost universally shuffle the same way, by recruiting our friends before the draft. Everyone grabs a stack of cards, shuffles them, and occasionally swaps some of their cards stack with someone else.

Alternatively, a motivated cube owner might shuffle their cube independently before a draft, to be a great host and be sure it’s shuffled — spending hours shuffling and randomly recombining stacks of cards in much the same way. But this leaves the question of how much of this shuffling is enough. I was always curious whether this “casual chaos” was as timely as it could be, or even effective at randomizing.

The goal of shuffling is to get a stack of cards from an ordered state (either last week’s decks, or a freshly sorted cube) to a random state. At first, we can visualize an ordered cube like this, with some color and ordered white dots to give each card a unique label:

A final, randomized state looks more like this. Now, there’s no way to tell where a card ended up based on its starting location:

As the cube is shuffled, we’ll be able to see how randomized it is by looking for clumps of the same or a few colors, or bands of white dots marking the cards. Cards are still near to their initial neighbors. This is the danger of insufficient shuffling!

To optimize our shuffling method, we can simulate the process of randomization using these abstracted cubes. For example, take that “casual chaos” method that struck me as being inefficient. Let’s visualize the entire cube alongside the piles that our players are riffling and occasionally passing around:

The cube is divided into 8 piles. All the piles are simultaneously shuffled by players. At random, two players will exchange half of their stacks of cards.

This model is simpler than reality. Our diligent simulated shufflers work like ​clockwork, never missing a beat. No cards are left idle, and our shufflers are equally likely to pass to any other player, not favoring their nearest neighbors. In a real cube night, some cards might be passed back and forth between just two shufflers, or even forgotten altogether. If our optimistic simulation doesn’t randomize well, then our real shufflers are in trouble.

So how long does our model suggest it takes to fully shuffle a cube? 30 minutes!? An hour!? Even with 8 people helping!? If this is the murkiest reflection of reality, we can clearly do better.

Because of the randomness of exchanges, each time it runs, the simulation will be a bit different, but in most iterations, the cube isn’t randomized even after tens of minutes. Cards of the same color appear in chunks, or someone’s sorted deck is still mostly intact.

In theory it’ll get there eventually, but not quickly. It’s clear from the simulation that players shuffling the same stack without making frequent exchanges is an issue. Returns quickly diminish on one player shuffling the same set of cards. If a cube is starting from an already somewhat mixed up state this might not be obvious in the final packs, but imagine one chunk of cards is the mono-red deck from the last draft. It’s easy for someone passively helping shuffle to keep them clumped together.

On top of being inefficient, this method is also unpredictable. No one enjoys wading through the unclear minutes of “is it shuffled enough yet?” before starting to deal out packs. We can’t give that answer reliably, unless we’re comfortable with saying “No, it’s not” and spending even more time to be safe.

The biggest issues with the former method are that individual players may sit shuffling the same stack of cards for a long time, and large portions of the cube are often not in contact with other portions.

The intuitive fix, then, is to add a little structure to the same basic process. Rather than exchanging cards haphazardly at each player’s discretion, everyone could periodically pass cards in the same direction around the table.¹

A little bit of structure does make things much better! We don’t have the same issues of completely isolated sections and the shuffling proceeds in a predictable way so we can answer how much shuffling is necessary.

But, it still takes a long time. Cards slowly migrate around the table, smeared across the piles being shuffled only fairly gradually diffusing into noise.

By simulating different numbers of piles and shuffles performed between passes, there two critical things we can observe to improve this strategy. We can also apply them generally to others.

1. Less shuffling by each player before passing is much more effective. Anything above 3 shuffles has diminishing returns, wasting time.

2. Fewer people shuffling larger stacks of cards is much more effective. More stacks means a much longer train of exchanges for all parts of the cube to contact the rest.

Both of these can be counterintuitive. Ensuring a particular stack of cards is thoroughly shuffled feels more effective. In reality, it keeps one set of cards isolated and preserves any bias it has, such as being disproportionately made of cards from a specific deck in a previous draft.

Players want to be helpful. It feels better to have more people contributing and not have idle hands. But if fewer players can each handle larger stacks of cards, everyone is much better off. There’s probably something else you can do. Sort tokens! Get snacks!

While this is an improvement, and works quite well if four players can manage the whole cube, the “Pass Left” method is still fairly slow. With 8 players shuffling, it takes about 10 passes before any cards starting from one particular player even begin to make it around the full circle. It takes about 25 iterations before the cube starts looking uniformly shuffled. If we add up the estimated time it takes for each shuffle and pass thats upwards of 25 minutes. This, again, is all assuming our shufflers work like clockwork. My one experience with this method, players failed to keep to, or were confused by and didn’t follow, the ‘only 2 riffles’ guideline.

A huge gain this structure provides: it’s at least consistent, so we can answer the question of when the cube is shuffled enough. With 8 helpers, make like a burn player and count to (at least) 20. The fact that Pass Left is easy to explain, and coordinate is a huge boon. In some contexts those might make it the best compromise, keeping the two key lessons in mind, even if we can technically squeeze out some more efficiency.

The big limitation here is it takes time for cards to slowly process around the table. An alternative method designed by Heather Zeis adjusts how cards are being passed. The first iteration is the same as Pass Left. Next, players pass half two seats to the left, then four seats, shuffling after each exchange. (Your mnemonic: a hop, a skip, and a jump.) By passing packs further, the cube gets more uniformly mixed much more quickly. She’s concluded that just these four iterations are sufficient.

For more details check out Heather’s thorough analysis. In addition to this variation, she also independently, came up with the same solution below!²

Pass Left is an improvement over unstructured shuffling, but still takes more time to reach randomization than most groups want to spend. Let’s adjust the structure and remix cards in a way that most quickly mixes every part of the starting state with every other part. Rather than shufflers distributing cards to only one other, each shuffler distributes their stack evenly among all shufflers (including themselves).

I’m calling this Broadcast Shuffling, since each pile gets “broadcast” evenly into the next iteration. I’ve made a few other changes to demonstrate what I’ve found to be the most effective method, but added some controls to explore variations. The cube is divided into only four stacks, each is shuffled 3 times in sequence as if shuffled by a single person, before being divided into a new row.

Broadcast is markedly quicker than Pass Left. Even after one iteration, the cube is looking pretty well shuffled; I recommend 3 or 4 iterations. The downside of the Broadcast Shuffle is it’s more complex to manage, and more difficult to explain to potential helpers.³ But Broadcast Shuffling is so much more efficient that it can actually be faster for one person to shuffle alone than for 8 players Passing Left! It can take as little as ten minutes, so it may be preferable as a cube host to shuffle as folks arrive. Or, recruit a couple instruction-following helpers to get it done in just a few minutes.

For direct explanation of this method, I’ve separated the more practical details of this shuffle strategy onto a separate, dedicated page for easy reference.

Looking at the three methods side-by-side shows a progression of improvement.

This shows a version of the three methods discussed, each shuffled by 8 players simultaneously, keeping to a strict 3 shuffles per iteration. The contrast between the unstructured shuffling and the other two methods is stark. Broadcast beats Pass Left by a solid margin, taking 3 iterations rather than 25. In practice, 8 shuffler Broadcast is difficult to coordinate, but with fewer shufflers, Broadcast is a clear winner.

As Cube grows in popularity, shuffling is a small hurdle for new players entering the format. Cube events have popped up like weeds and fallen into a regular cadence, presenting their own challenges: cubes need to be shuffled and ready to go, even sometimes starting fully sorted, and to be reset quickly between back-to-back drafts.

With a little systematic care, fortunately, shuffling doesn’t have to be difficult or time consuming. Broadcast Shuffling effectively randomizes your cube, while making the most of your time. Spend less time shuffling and more time drafting!

Each simulation shows 540 cards by default. The individual riffle shuffles of stacks are modelled the same way, by dividing the stack in two, and combining them by randomly pulling a card from one of the two halves. Only how, and how often, cards are remixed between piles varies between methods.

The simulations are not in real time, and the animation pace is tailored to make what’s happening clear. The real time it takes players to shuffle varies wildly. I’ve picked rough numbers to approximate the time it takes to riffle shuffle a pile or redistribute cards in order to provide rough numbers for comparison, but these will vary between players and groups. The number of actions it takes to shuffle each is a more concrete indicator.

Our goal is to get the whole cube uniformly shuffled. It’s the baseline you can dealing out packs for a draft or that you can pull cards from for other formats. Some Cube designers collate or seed packs to mimic sealed packs’ rarity or ensure evenly mixed colors, but this is outside the scope of our investigation.

At a clear point in each of these simulations, more shuffling stops making the cube look more shuffled, but you might still notice some quirks: a few clumps of similar cards that happen to fall together, shuffle after shuffle.

“Random”, as it turns out, is particularly unintuitive. If you have a person create a ‘random’ set of coin flips, the human-made random sequence will be visibly different from true random.⁴ People will try to maintain an even, random-feeling mix. After a few “heads” their more likely to pick a “tails”. With real coins, anything is possible. Theres a 50% chance to flip “heads”. 25% chance to flip two in a row. 10 in a row has a 0.0977% chance. A very small number that’s extremely different from 0. When looking at a long sequence of flips, it’s even more likely to find some runs because there are many opportunities for them to happen.⁵

Magic players will swear to no end that “the shuffler is broken,” but a shuffled set of cards is like a set of random coin flips. It has many chances to have some quirks. Shuffled decks still get mana screwed. Random packs will still be heavy on a color every once in a while. When you open the occasional weirdo pack, take comfort knowing it’s a fun result of expansive combinatorics, and not half of someone’s deck from the last draft.