What is the minimum number of times the deck needs to be shuffled?
Ordering the deck
Each time you play through the ZAR deck, you create an ordered Playstack. The stack has been sorted such that every card is next to a card with a matching color or symbol (with the occasional Power card thrown in). If you were to simply deal out the Playstack to start a new hand without first shuffling it, then the game play would be substantially degraded. The object of shuffling is to remove a significant amount of this order before dealing.
The most obvious symptom of an inadequately shuffled deck is that an inordinately large number of matches are occuring — especially being called by a player sitting directly next to the player being matched. Another common symptom is being dealt a hand consisting of cards which are all the same color. (I consider that to be the worst possible ZAR hand.) In order to avoid this, extra care must be taken when shuffling the deck. In practice, this means that a few more shuffles than some people are accustomed to doing must be performed.
Creating disorder
There are several mathematical approaches to calculating a measure of disorder. A good one for card shuffling is to calculate the distance from uniform distribution. If the probability of any card being next to any other card is equal for each card in the deck, we can say that the deck is uniformly distributed. We'll assign a range of numbers from 1.000 (the deck is ordered) to 0.000 (the deck is uniformly distributed). The question now becomes: how many shuffles do we need to get this number reasonably close to zero?
The normal (and very effective way) to shuffle cards is by using a riffle shuffle. In a riffle shuffle, the deck is cut into two (possibly unequal packets), and the two packets are then raggedly interleaved. Using Markov Chains we can model the distance from uniform distribution after any number of sequential riffle shuffles. The following table is the result of that calculation for decks of various sizes.
| n | k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 52 | 5.927 | – | – | – | – | 0.924 | 0.614 | 0.334 | 0.167 | 0.085 | 0.043 |
| 62 | 6.191 | – | – | – | – | 0.982 | 0.733 | 0.423 | 0.218 | 0.110 | 0.056 |
| 93 | 6.799 | – | – | – | – | – | 0.964 | 0.694 | 0.391 | 0.201 | 0.102 |
| 124 | 7.230 | – | – | – | – | – | 0.999 | 0.888 | 0.564 | 0.305 | 0.155 |
| 155 | 7.565 | – | – | – | – | – | – | 0.973 | 0.726 | 0.414 | 0.214 |
| 186 | 7.839 | – | – | – | – | – | – | 0.997 | 0.851 | 0.528 | 0.280 |
Notes:
- A – indicates a number ≥ 0.9995.
- There is a cutoff point at k = 3⁄2(log n) after which the distance to uniformity decreases by ≈ log₂ for each additional shuffle.
Disordered enough
In this type of analysis, a fascinating phenomenon becomes evident — in some systems the switch to randomness is abrupt. Observe in the above table that the first 5 shuffles do virtually nothing! In the case of an ordinary 52 card deck, after 6 riffle shuffles, the deck is still visibly ordered. But after the 7th, it suddenly becomes random.
So, how disordered is enough? At absolute minimum, we would like at least 50% of the order removed, i.e. a distance ≤ 0.500. This means that nothing less than 7 shuffles will do. Also, because this is after the cutoff point, any additional shuffles get us logarithmically closer to uniformity — a rapidly decreasing return on our effort. Moreover, this means that we want the lowest number of shuffles that result in numbers ≤ 0.500 and optionally at most one more shuffle past that.
Recall that the riffle shuffle works by cutting the deck into two packets which are then shuffled together. Notice that the order of cards in each packet does not change relative to itself during the shuffling part. Therefore, the only time that order changes is during cuts. Adding additional cuts allows cards to move further faster relative to each other. Therefore, because this is both quite useful and easy, I specify that a few extra cuts be performed while shuffling.
Finally, it should be noted that the shuffling model uses an average of all possible cuts and shuffles. A very careful shuffle, which comes close to getting alternating single cards from each packet, could possibly achieve the same level of randomness in fewer shuffles. Conversely, someone who is not very good at shuffling may need to add more shuffles to get the same effect. You may be interested in looking at my simulation of a riffle shuffle.
Recommended shuffle
| n | Deck type | Shuffles |
|---|---|---|
| 62 | Single | 7 |
| 93 | Deck and a Half | 8 |
| 124 | Double Deck | 9 |
Shuffle the deck Shuffles times, (or optionally Shuffles + 1 times,) cutting it after each 2 shuffles.
If you didn't just cut the deck, cut it once more just before dealing.