From 'Blackjack' To The Universe (Pt. 1)

Ever get that feeling, half-an-hour later, when you think "I wish I'd said this"? It takes me a bit longer (about two days).

I think what I have learned is that I shouldn't get drunk and start talking about maths. We were having a chilled-out night, having a few drinks and playing some board games. A pack of cards came out. And I drop the bomb:

Did you know? "It is very unlikely that any two randomly shuffled decks [of 52 cards] will ever have had the same order in the history of the world – even if the world's population had started playing cards at the Big Bang" – Alex Bellos. Alex's Adventures in Numberland. pg339.

I don't think the significance of this properly hit. Or maybe it did and I just couldn't answer the questions. "No, really? No way!" "Since the beginning of the universe?". I mumbled something about a number being followed by 67 zeroes, but I don't think the true awesomeness of this really hit. Top tip for me: don't get pissed before spouting off about numbers.

But, now I want to analyse what I said, and try to make up for my mistakes in communicating. Let's start at the beginning. How many different ways are there to order a deck of 52 cards? This is quite a big number to start with, let's begin with a smaller number. If I gave you two coloured pieces of card (red and blue) and asked you to show me every unique combination for those two colours, I'm sure you would easily show me something that looks like Figure 1A. You can have  red—blue or blue—red. What if I give you three colours, and asked the same question? Well, it might take longer to get there, but you would eventually give me the six combinations shown in Figure 1B.

Figure 1. When there are two ways of arranging two colours (red–blue; shown in A). If we add another item (green), as shown in B, there are now six unique ways of arranging the colours.

So, we can see that there are two unique ways of arranging two objects. When another object is added, we get three times as many unique ways of arranging the set of three. This jump in the number of unique ways of arranging the objects can be explained by thinking about what is happening when we pick a particular arrangement. When drawing these arrangements from our set of three colours, we start by picking one of the three colours to be the first. There are now two possible colours to pick for the second colour (as we have already used one of the original three colours). The last colour has already been picked for us, we have used two of the three colours and can only use the remaining third colour. The total number of arrangements can be calculated by multiplying the number of options at each stage. So, we have six ways of arranging three objects (mathematicians call the "number of ways of arranging objects" the number of "permutations"):

3 x 2 x 1 = 6

This can be generalised to say that for n objects (where n can be any whole, positive number), we have the following number of permutations:

n x (n–1) x (n–2) x ... x 3 x 2 x 1 = n!

The exclamation mark at the end is a mathematical notation and is just a short way of showing the expression above. The number n! is pronounced "n factorial". Factorials get large, fast. Very fast.

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5,040

8! = 40,302

9! = 362,880

10! = 3,628,800

Adding three objects to one object results in an increase in 23 permutations. But if we add three objects to a group containing seven objects, we see an increase in the number of permutations of over three-and-a-half million!

And now we can go back to the deck of cards. A standard deck contains 52 cards. To find the number of permutations for a deck of 52 cards, we need to find 'fifty-two factorial' (52!).

52! = 52 x 51 x 50 x ... x 3 x 2 x 1 = 8.07 x 10⁶⁷

8.07 x 10⁶⁷ is a mind-blowingly huge number and is roughly equal to 8 followed by 67 zeroes. This is about as many as the number of atoms in the entire Milky Way (I'll write another post soon showing my working out). Edit: I made some BIG assumptions in this calculation! Edit2: I've looked at this again and have realised that I calculated it wrong (the number, 52! is about a tenth of the number of atoms in the Milky Way). Sorry for this mistake, I will write another blog post showing how I calculated this.

52! is roughly equal to the number of atoms in the whole of the Milky Way. Image reproduced under a Creative Commons License from http://www.digitalskyllc.com

We now know how many permutations of 52 cards there are. And my next blog post will attempt to show you how we can figure out the probability that any two randomly shuffled decks will be the same. I will then try to show why it is highly unlikely that any two randomly shuffled decks of cards in the whole of history have ever been the same.