## Shuffling Cards

Cutting cards:

What exactly does cutting a deck of cards do? In some games (bridge in particular) you cut the cards by taking the top cards (more than four cards and less than 48 cards) in your hand, and place them on the table next to the bottom cards, on the side nearest the dealer. The dealer then finishes the cut by placing the cards that were on the bottom on top of the other cards. The cut does not change any of the bridge hands, it just determines who will get them. If every fourth card is a spade, producing one hand of all spades, then cutting will determine who gets 13 spades. People who attempt to shuffle by repeated cutting do not actually mix the cards. Repeated cutting is the same as cutting once in a different place. In games where not all of the cards are dealt out, cutting does significantly change the nature of every player's hand.

Riffle shuffle:

Shuffling is the actual mixing of cards. This can be done in a number of ways. The Laws of Contract Bridge do not specify how shuffling is to be done, just that the cards must be thoroughly mixed. Several shuffles are required in order to mix the cards.

The riffle shuffle (sometimes called a faro shuffle) is the traditional method of shuffling. The deck is divided approximately in half, one part going into the left hand and the other part going into the right hand. Then by riffling the thumbs over the edges of the cards, the two halves of the deck are interleaved together. With practice a person can sometimes achieve a perfect shuffle, with every other card coming from alternating hands. This is not recommended, as will be shown below. Sometimes, computer programs that shuffle cards do several riffle shuffles, rather than just determine each card at random.

One riffle shuffle reorders the cards significantly. But the reordering is not random, by any means. The top card is still near the top. The bottom card is still near the bottom. Most pairs of cards are either still adjacent to each other, or not very far apart. And so, several shuffles are necessary. How many shuffles? Mathematics shows that at least seven shuffles are needed to randomize the cards. More shuffles than that do not affect the randomness. Bridge players often complain about the poor hands that result from computer dealt cards. This would seem to indicate that these people are used to inadequate shuffling.

A perfect shuffle occurs when the deck is divided exactly in half, and the cards are perfectly interlaced, with one card coming from one hand, then one card coming from the other hand, then one card coming from the first hand, etc. There are two types of perfect shuffles, the "in-shuffle" and the "out-shuffle." Let's assume that the deck is divided in two with the top cards going into the left hand and the bottom cards into the right hand. Then an in-shuffle begins with the first card coming from the left, the second from the right, the third from the left, etc. An out-shuffle begins with the first card coming from the right. If the right hand originally took the top cards, then the definitions are reversed (the in-shuffle begins with the first card coming from the right...). It has been shown that eight perfect out-shuffles returns the 52-card deck to its original order. Apparently, it takes more in-shuffles to do that. So, perfect shuffles do not randomize a deck, far from it.