Of fact, our method has only provided an upper bound on the distance between Rk and U. Indeed, J. Doyne Farmer and Alexander Sesonske have shown that randomized card decks do not always converge to the same distribution after repeated shuffling.

A deck of cards will be shuffled at random by 236 single-card riffles on average. A standard deck contains 52 cards, so there is one card per riffle position on average.

There are two ways to shuffle a deck of cards: randomly and thoroughly. Random shuffling means selecting cards at random from **the whole deck** and placing them in **new order**. This can be done either by hand or using **an automated device**. The purpose of thorough shuffling is to mix up all the cards so that none are sorted into groups.

Card sorting is the process of grouping cards with **similar values** or patterns into sets. These sets may be based on color, rank, suit, or any other factor. Card sorting can be useful for organizing cards into teams or racks. It can also help to identify problems with the distribution of cards in a deck. For example, if cards are often found in pairs then this might indicate that the deck has been tampered with.

When shuffling a deck of cards, it is important to not only mix up the cards but also find out whether they have been grouped together already. This can be done by cutting the deck into several smaller batches and sorting each batch separately.

Jim Reeds of Bell Laboratories demonstrated that shuffled between 5 and 20 times, a deck is perfectly mixed. Dr. Diaconis then collaborated with Dr. Aldous to demonstrate that it takes 5 to 12 shuffles to correctly mix a deck. He also showed that for certain popular card games like Blackjack, the number of shuffles is related to the probability of getting a particular hand - the more times you shuffle, the better your chance of obtaining any given card value.

In general, the more times you shuffle a pack or bag of cards, the more thoroughly they will be mixed. However, too many shuffles are time-consuming and some people may find them tiring. Shuffling is usually not necessary for randomized hands of cards; merely cutting the deck into stacks or groups without further sorting it out is sufficient. But for some decks, such as bridge decks, only a single shuffle will do - the cards must stay together in their original order for them to be valid for play.

The best way to cut down on shuffling time is by using electric or automated shufflers. These devices use sensors or lasers to detect the presence of cards in **each subsequent pocket** of **the shaker table**, thus eliminating the need for **manual placement** of each card. Some players prefer not to use automatons because they believe human judgment is essential for good game play.

For most games, four to seven riffle shuffles are needed; for **unsuited games**, such as blackjack, four riffle shuffles are sufficient, while seven riffle shuffles are required for suited games. However, there are some games where even seven riffle shuffles are inadequate. For example, in pinochle seven riffle shuffles can result in **two cards** ending up in the same hand.

In general, the more times you shuffle a deck of cards, the more likely it is that two cards will end up in the same place. Therefore, it is important to shuffle enough times to avoid duplicates, but not so many times that you slow down the game too much.

There are several ways to shuffle cards. The simplest way is called a "shuffle" and it requires no special equipment. You simply spread out the cards and randomly pick them up one by one. This is enough to get rid of any ordering and ensure that any two cards picked at random are going to be put into different hands.

However, this method takes **a long time** because there are 52 cards and they have to be shuffled one by one. It is easy to see how this would take **a very long time** if you were trying to shuffle a whole deck of cards. Even though it may seem like a long time, it's really not when you consider that card games involving shuffled decks are often over before they start!