Shuffling a Deck of Cards, to the End of Space and Time

If you sat down to shuffle this unordered deck of cards, and you wanted to deal them out in the exact same order twice, you would be sitting shuffling and laying out cards past the death of the Universe. As a matter of fact, I’m not totally sure of the math, but I think if we had started at the Big Bang, if indeed that’s where the Universe had started, we would never have shuffled the cards in the same order.

You have never shuffled and dealt a deck of cards in the same order twice. You just haven’t. Or rather, it’s so near to being impossible that, well, it’s impossible.

Start with just 3 cards. Shuffle them. Lay them out. What are the chances you could do it the same way twice?

Here is one possible order of these three cards. In the first position, you will have dealt one of three cards. Two cards are left. After you have dealt the second one, only one is left to fill the third spot.

Imagine three empty boxes: the first one can be filled three ways, the second one two ways, the last one with the card that’s left. 3x2x1=6 possible orders.

Every six deals, you should get the same order.

The math rapidly gets out of control here. Shuffle four cards, already there are four times as many combinations.

52! or 52 factorial is the number of unique orders of a full deck. You can’t draw that as a tree diagram. There is no number that big needed for any purpose here on Earth. There are not even that many stars in the Universe.

Will a little patience though, you can do it. Patiently record all your 52 card deals, and watch, from your chair, as the lights go down. When the Big Bang happens again (if indeed it will), different laws of probability may apply.

K-2-A-3–4…