Can You Count All the Ways to Shuffle a Deck of Cards We Bet You Cannot

So, how long do you think it would take to shuffle a deck of cards into every possible order? We will make things easy for you by assuming you can shuffle cards faster than everyone else can. Every second, there is a brand fresh shuffle. Do you think you will be able to handle it?

What if we told you; you could shuffle 1,000 times each second? What if we used the world’s fastest computer – which is now combating a coronavirus in Japan, but we are confident this is more essential – to mimic shuffles at a pace of 415,530,000,000,000,000 per second? What do you believe the period would be? Here is the thing: even if you (or the Fugaku supercomputer) had begun the task soon after the big bang, you would not have completed it. That is not even close. It all boils down to a single exclamation point – and the immense power of numbers it may unleash.

Let us break down the situation to see what is going on. How many different ways can you shuffle a single card, like the ace of spades? It may appear to be a trick question, but it is not; it is truly that simple. One approach to “shuffle” a single card is to place the spadille on the table and proclaim the work done. Two cards are a bit challenging but not significantly so: you have two alternatives, which are determined which card you place first.

When three cards used, though, things get a bit more interesting. Any of the three can take the lead. Then there are two alternatives for the second card for each of those. After those two places filled, there is only one option for who comes out last. There are six distinct ways to shuffle this three-card mini-deck when we add up all the potential permutations (that is a valid math phrase, by the way, so feel free to use it at parties to impress your friends).

Let us break down what we are doing here: each card we lay down reduces the shuffle’s degree of freedom by one. The six alternatives discovered by multiplying three, two, and one, as shown in the picture. Mathematicians call it a factorial when we multiply numbers like this, taking a value and multiplying it up by every positive whole number below it. It expressed like this.

So far, we have taken one, 2, and 6 seconds at a pace of one shuffle per second — hardly a long time, right? However, factorials can quickly grow in size. We will need roughly two millennia to lay out all potential shuffles by the time we finish one suit. We need to calculate out 52! – Remember, that 52 factorial, not 52 in an enthusiastic voice – to figure out how many individual orders you can get using all 52 cards. This equates to about 8 1067 or 80 thousand vigintillion distinct shuffles. That is right: it is a figure so large that you have never heard of the term used to describe it.