It fascinates me because 52! is such a large number. Here it is in full 8.0658* 10^67 or 80,658,175,170,943,878,571,660, 636,856,403,766,975,289,505,440,883, 277,824,000,000,000,000. That is the possible number of ways of shuffling a pack of cards.
It means that when you shuffle a deck of cards, it’s possible that you are the first person on Earth to ever get that particular arrangement. It’s the kind of fact that amazes me. Another one is that it takes a very long time for particles emitted from the centre of the sun to reach the surface and blast into space. On the order of many many years. (Thousands of years!)
Playing cards have only been around maybe 500 years as we know it (52 card deck) though date back to 9th century China for their invention. If there had been a billion shuffles each day during that 500 years, that’s only 1.8 x 10^14 shuffles. That is a minuscule fraction of the possible number of arrangements so the chances are that any shuffled arrangement is new is pretty high.
It’s accepted that seven is the number of riffles needed to perfectly shuffle a pack of cards. A riffle is where you split the deck in two and then merge the two halves back into one deck as in the photo I took.
I proved this once by writing a program to simulate riffles and looking how far cards have moved after seven. In fact a card at the top of the deck moved to the bottom after only six riffles. I’ll try and write that in C and will publish it here in a day or two.
Other shuffling techniques like smooshing (spreading out all the cards on the table with their backs face up and then pushing them together) are nowhere near as efficient. It’s estimated it can take thousands of smooshes to properly shuffle a pack. It’s not easy to simulate, though one of these days I’ll have a go and see if I can come up with a more accurate estimation.