The Mathematics of Card Shuffling

Shuffling a pack of cards isn’t as easy as you think, not if you want to truly randomise the cards.

Most people will give a pack a few shuffles with the overhand or riffle methods (where the pack is split and the two halves are interweaved).

But research has shown this isn’t enough to produce a sufficiently random order to make sure the card game being played is completely fair and to prevent people cheating.

As I wrote in a recent article about card counting, not having an effective shuffling mechanism can be a serious problem for casinos: Players have used shuffle tracking, where blocks of cards are tracked so that you have some idea when they will appear.

If you are given the option to cut the pack, you try and cut the pack near where you think the block of cards you are tracking is so that you can bet accordingly.

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A variant on this is to track aces as, if you know when one is likely to appear, you have a distinct advantage over the casino.

So how can you make sure your cards are well and truly shuffled?

To work out how many ways there are of arranging a standard 52-card deck, we multiply 52 by all the numbers that come before it (52 x 51 x 50 … 3 x 2 x 1).

This is referred to as ’52 factorial’ and is usually written as ’52!’ by mathematicians.

The answer is so big it’s easier to write it using scientific notation as 8.0658175e+67, which means it’s a number beginning with 8, followed by 67 more digits.

To put this into some sort of context, if you dealt one million hands of cards every second, it would take you 20 sexdecillion, or 20,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, years to deal the same number of hands as there are ways to arrange a deck of cards.

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