## More thoughts on shuffledness

Thinking about what I said, it can’t be just about how far a card moves when shuffled but how much disordering occurs as well. If you shuffled the deck and discovered 7 cards in a sequence like 2,3,4,5,6 of hearts deep into the deck, they would have moved from near the top of the deck (an “ordered deck” is A-K in Hearts, Clubs, Diamonds and Spades) then it probably wasn’t a very good shuffle.

One way to measure “disorder” is to subtract each card’s ordinal value from it’s successor and sum up the absolute value of them . Ordinal value is the value 0-51 where 0 = Ace Hearts, 1 = 2 Hearts and so on. In an ordered deck the sum of all these subtractions will be 51. In a well shuffled deck I would guess it might be something like 1352 (52 x 26). Absolute value is the abs(x) function of x which is always positive if x is negative. abs(-8) = 8. abs(8) = 8.

So now for shuffledness we have two values. Displacement and disorder. If both are calculated as a value between 0 and 1 then multiplying them together might work as a measure of shuffledness.

Here’s a very simple example of calculating the value for disorder. So far the calculations have never got a total much above 900, and not near 1352. I’ll run a few experiments and see the range of values.

```
// shuffledness.c : This measures how shuffled a deck of cards is
// cards are held as values 0-51 in a 52 int array.
#include <time.h>
#include <stdio.h>
#include <stdlib.h>
int deck[52];
time_t t;
// Order the cards 0-51. 0 = Acew Hards,13 = Ace Clubs,26 = Ace of Diamonds and 51 = King of Spades
void InitDeck() {
for (int i = 0; i < 52; i++) {
deck[i] = i;
}
}
// Works on glovbal array deck and calculates a value for how displayed each card is
float CalcDisorder() {
int total=0;
for (int i = 1; i < 52; i++) {
total += abs(deck[i] - deck[i - 1]);
}
printf("Total = %d\n", total);
return total / (52.0f * 26.0f);
}
// Shuffle a deck by swapping two random cards a specified number of times
void ShuffleDeck(int numtries) {
int firstIndex, secondIndex;
for (int i = 0; i < numtries; i++) {
do {
firstIndex = rand() % 52;
secondIndex = rand() % 52;
}
while (firstIndex == secondIndex);
int value = deck[firstIndex];
deck[firstIndex] = deck[secondIndex];
deck[secondIndex] = value;
}
}
int main() {
/* Intializes random number generator */
srand((unsigned)time(&t));
InitDeck();
ShuffleDeck(1000);
printf("CalcDisorder() == %f\n", CalcDisorder());
return 0;
}
```

That’s about 50 lines long. I’ll work on it and add the displacement calculation and try and get a maximum figure the the disorder calculation.