## 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.

- 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.
- Instead of referring to cards by their rank and suite lets just number them 0-51. 0= Ace of Hearts, 51 = King of Spades.
- 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.
- 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.
- 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.
- 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.
- 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.