## How to measure how shuffled a deck of cards is

I first thought about this when I wrote the program to shuffle  deck of cards using a riffle shuffle. If you are given a deck of cards (or pack of cards as us Brits say), how do you discern just how shuffled the pack is? Can you calculate a numeric value for it say a % ranging from 0 to 100?

I believe it’s possible.  Here’s how.

1. Start with a default pack of cards in perfect sorted order. Out of curiosity I found an unopened deck oif Waddington’s cards and opened it as the photos show. The cards in the pack were arranged in order King, Queen, Jack down to Ace in each of the four suits Heart, Clubs, Diamonds and Spades in that order. Let’s reverse the card rank ordering so a full deck starts with Ace Hearts through to King Hearts, Ace of Clubs to King of Clubs and so on with the last card being the King of Spades.
2. Instead of referring to cards by their rank and suite lets just number them 0-51. 0= Ace of Hearts, 51 = King of Spades.
3. When a deck is shuffled, each card can move to any other position. So a measure of shuffledness is calculating how far the cards moved in aggregate.
4. However the card movements have to be “normalized”. Cards 0 and 51 can move to any of 51 positions while cards 26 (King of Clubs) and 27 (Ace of Diamonds) can only move a maximum of 26 places.
5. I’m looking at the absolute value of a movement so if card 3 moves to position 47, it has moved 44 places and likewise card 47 moving to position 3 moves 44 (not -44) places.
6. So to normalize a card’s move, divide its move by its maximum possible distance it can move. So wherever card 0 moves divide it by 51, card 1 by 50, card 26 ‘ move by 26.
7. Sum up all 52 normalized move’s and multiply by 100. That is the measure of shuffledness.

I’ll write a C program to measure how shuffled a deck is and publish it in a day or two. Also here is a conversation on Reddit about shuffling cards.

## How many riffles are needed to shuffle a deck?

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.